Catalog of Models

page-cat-models-catWhether you know it or not, you deal with models every day. Your weather forecast comes from a meteorological model, usually several. Mannequins are used to display how fashions may look on you. Blueprints are drawn models of objects or structures to be built. Maps are models of the earth’s terrain. Examples are everywhere.

Models are representations of things, usually an ideal, a standard, or something desired. They can be true representations, approximate (or at least as good as practicable), or simplified, even cartoonish compared to what they represent. They can be about the same size, bigger, or most typically, smaller, whatever makes them easiest to manipulate. They can represent:

  • Physical objects that can be seen and touched
  • Processes that can be watched
  • Behaviors that can be observed
  • Conditions that can be monitored
  • Opinions that can be surveyed.

The models themselves do not have to be physical objects. They can be written, drawn, or consist of mathematical equations or computer programming. In fact, using equations and computer code can be much more flexible and less expensive than building a physical model.

Stats with Cats Models 10-23-2017

Classification of Models

There are many ways that models are classified, so this catalog isn’t unique. The models may be described with different terms or broken out to greater levels of detail. Furthermore, you can also create hybrid models. Examples include mash-ups of analytical and stochastic components used to analyze phenomena such as climate change and subatomic particle physics. Nevertheless, the catalog should give you some ideas for where you might start to develop your own model.

Physical Models

Your first exposure to a model was probably a physical model like a baby pacifier or a plush animal, and later, a doll or a toy car. From then, you’ve seen many more – from ant farms to anatomical models in school. You probably even built your own models with Legos, plastic model kits, or even a Halloween costume. They are all representations of something else.


Physical models aren’t used often for advanced applications because they are difficult and expensive to build and calibrate to a realistic experience. Flight simulators, hydrographic models of river systems, and reef aquariums are well known examples.

Conceptual Models

Strat modelModels can also be expressed in words and pictures. These are used in virtually all fields to convey mental images of some mechanism, process, or other phenomenon that was or will be created. Blueprints, flow diagrams, geologic fence diagrams, anatomical diagrams are all conceptual models. So are the textual descriptions that go with them. In fact, you should always start with a simple text model before you embark on building a complex physical or mathematical model.

Mathematical and Computer Models

ROCKWARE strat_fence_cage_01Theoretical Models

Theoretical models are based on scientific laws and mathematical derivations. Both theoretical models and deterministic empirical models provide solutions that presume that there is no uncertainty. These solutions are termed exact (which does not necessarily imply correct). There is a single solution for given inputs.

Analytical Models

Analytical models are mathematical equations derived from scientific laws that produce exact solutions that apply everywhere. For example, F (force) = M (mass) times A (acceleration) and E(energy) = m (mass) times c2 (speed of light squared) are analytical models. Probably, most concepts in classical physics can be modeled analytically.

Numerical Models

Numerical models are mathematical equations that have a time parameter. Numerical models are solved repeatedly, usually on a grid, to obtain solutions over time. This is sometimes called a Dynamic Model (as opposed to a Static Model) because it describes time-varying relationships.

Empirical Models

Empirical models can be deterministic, probabilistic, stochastic, or sometimes, a hybrid of the three. They are developed for specific situations from measured data. Empirical models differ from theoretical models in that the model is not necessarily fixed for all instances of its use. There may be multiple reasonable empirical models that can apply to a given situation.

Deterministic Models

Deterministic empirical models presume that a mathematical relationship exists between two or more measurable phenomena (as do theoretical models) that will allow the phenomena to be modeled without uncertainty (or at least, not much uncertainty, so that it can be ignored) under a given set of conditions. The difference is that the relationship isn’t unique or proven. There are usually assumptions. Biological growth and groundwater flow models are examples of deterministic empirical models

12-sistwins-cats.w710.h473Probability Models

Probability models are based on a set of events or conditions all occurring at once. In probability, it is called an intersection of events. Probability models are multiplicative because that is how intersection probabilities are combined. The most famous example of a probability model is the Drake equation, a summary of the factors affecting the likelihood that we might detect radio-communication from intelligent extraterrestrial life

Stochastic Models

Stochastic empirical models presume that changes in a phenomenon have a random component. The random component allows stochastic empirical models to provide solutions that incorporate uncertainty into the analysis. Stochastic models include lottery picks, weather, and many problems in the behavioral, economic, and business disciplines that are analyzed with statistical models.

Comparison Models

Bombay-cat-3In statistical comparison models, the dependent variable is a grouping-scale variable (one measured on a nominal scale). The independent variable can be either grouping, continuous, or both. Simple hypothesis tests include:

  • c2 tests that analyze cell frequencies on one or more grouping variables, and
  • t-tests and z-tests that analyze independent variable means in two or fewer groups of a grouping variable.

Analysis of Variance (ANOVA) models compare independent variable means for two or more groups of a dependent grouping variable. Analysis of Covariance (ANCOVA) models compare independent variable means for two or more groups of a dependent grouping variable while controlling for one or more continuous variables. Multivariate ANOVA and ANCOVA compare two or more dependent variables using multiple independent variables. There are many more types of ANOVA model designs.

Classification Models

Classification and identification models also analyze groups.

Clustering models identify groups of similar cases based on continuous-scale variables. There need be no prior knowledge or expectation about the nature of the groups. There are several types of cluster analysis, including hierarchical clustering, K-Means clustering, two-step clustering, and block clustering. Often, the clusters or segments that are used as inputs to subsequent analyses. Clustering models are also known as segmentation models.

cute-dog-and-cat-hd-wallpaperClustering models do not have a nominal-scale dependent variable, but most classification models do. Discriminant analysis models have a nominal-scale dependent variable and one or more continuous-scale independent variables. They are usually used to explain why the groups are different, based on the independent variables, so they often follow a cluster analysis. Logistic regression is analogous to linear regression but is based on a non-linear model and a binary or ordinal dependent variable instead of a continuous-scale variable. Often, models for calculating probabilities use a binary (0 or 1) dependent variable with logistic regression.

There are many analyses that produce decision trees, which look a bit like organization charts. C&R (Classification and Regression Trees) split categorical dependent variables into its groups based in continuous or categorical-scale independent variables. All splits are binary. CHAID (Chi-square Automatic Interaction Detector) generates decision trees that can have more than two branches at a split. A Random Forest consists of a collection of simple tree predictors.

Explanation Models

Explanation models aim to explain associations within or between sets of variables. With explanation models, you select enough variables to address all the theoretical aspects of the phenomenon, even to the point of having some redundancy. As you build the model, you discover which variables are extraneous and can be eliminated.

page-cat-models-kittenFactor Analysis (FA) and Principal Components Analysis (PCA) are used to explore associations in a set of variables where there is no distinction between dependent and independent variables. The two types of statistical analysis:

  • Create new metrics, called factors or components, which explain almost the same amount of variation as the original variables.
  • Create fewer factors/components than the original variables so further analysis is simplified.
  • Require that the new factors/components be interpreted in terms of the original variables, but they often make more conceptual sense so subsequent analyses are more intuitive.
  • Produce factors/components that are statistically independent (uncorrelated) so they can be used in regression models to determine how important each is in explaining a dependent variable.

Canonical Correlation Analysis (CCA) is like PCA only there are two sets of variables. Pairs of components, one from each group, are created that explain independent aspects of the dataset.

Regression analysis is also used to build explanation models. In particular, regression using principle components as independent variables is popular because the components are uncorrelated and not subject to multicollinearity.

Prediction Models

catSome models are created to predict new values of a dependent variable or forecast future values of a time-dependent variable. To be useful, a prediction model must use prediction variables that cost less to generate than the prediction is worth. So the predictor variables and their scales must be relatively inexpensive and easy to create or obtain. In prediction models, accuracy tends to come easy while precision is elusive. Prediction models usually keep only the variables that work best in making a prediction, and they may not necessarily make a lot of conceptual sense.

Regression is the most commonly used technique for creating prediction models. Transformations are used frequently. If a model includes one or more lagged values of the dependent variable among its predictors, it is called an autoregressive model.

Neural Networks is a predictive modeling technique inspired by the way biological nervous systems process information. The technique involves interconnected nodes or layers that apply predictor variables in different ways, linear and nonlinear, to all or some of the dependent variable values. Unlike most modeling techniques, neural networks can’t be articulated so they are not useful for explanation purposes.

Picking the Right Model

There are many ways to model a phenomenon. Experience helps you to judge which model might be most appropriate for the situation. If you need some guidance, follow these steps.

  • maxresdefaultStep 1 – Start at top of the Catalog of Models figure. Decide whether you want to create a physical, mathematical, or conceptual model. Whichever you choose, start by creating a brief conceptual model so you have a mental picture of what your ultimate goal is and can plan for how to get there.

If your goal is a physical or full blown conceptual model, do the research you’ll need to identify appropriate materials and formats. But this blog is about mathematical models, so let’s start there

  • Step 2 – If you want to select a type of mathematical model, start on the second line of the Catalog of Models figure and decide whether your phenomenon fits best with a theoretical or an empirical approach.

If there are scientific or mathematical laws that apply to your phenomenon, you’ll probably want to start with some type of theoretical model. If there is a component of time, particularly changes over time periods, you’ll probably want to try developing a numerical model. Otherwise, if a single solution is appropriate, try an analytical model.

  • Step 3 – If your phenomenon is more likely to require data collection and analysis to model, you’ll need an empirical model. An empirical model can be probabilistic, deterministic, or stochastic. Probability models are great tools for thought experiments. There are no wrong answers, only incomplete ones. Deterministic models are more of a challenge. There needs to be some foundation of science (natural, physical, environmental, behavioral, or other discipline), engineering, business rules, or other guidelines for what should go into the model. More often than not, deterministic models are overly complicated because there is no way to distinguish between components that are major factors versus those that are relatively inconsequential to the overall results. Both Probability and Deterministic models are often developed through panels of experts using some form of Delphi process.
  • Step 4 – If you need to develop a stochastic (statistical) model, go here to pick the right tool for the job.
  • Step 5 – Consider adding hybrid elements. Don’t feel constrained to only one type of component in building your model. For instance, maybe your statistical model would benefit from having deterministic, probability, or other types of terms in it. Calibrate your deterministic model using regression or another statistical method. Be creative.



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How to Describe Numbers

Data catSay you wanted to describe someone you see on the street. You might characterize their sex, age, height, weight, build, complexion, face shape, hair, mouth and lips, eyes, nose, tattoos, scars, moles, and birthmarks. Then there’s clothing, behavior, and if you’re close enough, speech, odors, and personality. Your description might be different if you’re talking to a friend or a stranger, of the same or different sex and age. Those are a lot of characteristics and they’re sometimes hard to assess. Individual characteristics aren’t always relevant and can change over time. And yet, without even thinking about it, we describe people we see every day using these characteristics. We do it mentally to remember someone or overtly to describe a person to someone else. It becomes second nature because we do it all the time.

Most people don’t describe sets of numbers very often, though, so they don’t know how easy it actually is. You have to consider only a few characteristics, all of which are fairly easy to assess and will never change for the dataset. Once you learn how, it’s hardly a challenge to get it right, unlike describing the hot young guy who just robbed a bank wearing a clown costume.

What’s involved in describing a dataset? First, before considering any descriptive statistics, you have to assess two qualities.

  • Phenomenon and population or sample
  • Measurement scale

From this information, you’ll be able to determine what descriptive statistics to calculate.

Phenomenon and Population or Sample

This is a thinking exercise; there are no calculations.

First, determine what the numbers represent. What is the phenomenon they are related to? If there’s no context for the numbers, like it’s just a dataset for a homework problem, that’s fine too. But if you know something about the data, you might be able to judge whether your answer makes sense later when the calculations are done.


Next, think about the population from which the data were obtained. How is the population defined? Do you have all the possible measurements or entities? If not, you have a sample of the population, hopefully a sample that is a good representation of the population. This knowledge will help you judge whether your answer makes sense and will be consistent with other samples taken from the same population. Again, if there’s no context for the numbers, that’s fine. Now, all you have to decide is whether you want to describe the population or just the sample of the population for which you have measurements. If you’re not sure, assume you want to describe the population. All the fun stuff in statistics involves populations.

Measurement Scale


Scales of measurement express the phenomenon represented by the population. Simply put, scales are the ways that a set of numbers are related to each other. For example, the increments between scale values may all be identical, such as with heights and weights, or vary in size, such as with earthquake magnitudes and hurricane categories. The actual values of scales are called levels.

You have to understand the scale of measurement to describe data. There are a variety of types of measurement scales, but for describing a dataset you only need to pick from three categories:

  • Grouping Scales – Scales that define collections having no mathematical relationship to each other. The groups can represent categories, names, and other sets of associated attributes. These scales are also called nominal scales. They are described by counts and statistics based on counts, like percentages.
  • Ordered Scales – Scales that define measurement levels having some mathematical progression or order, commonly called ordinal scales. Data measured on an ordinal scale are represented by integers, usually positive. Counts and statistics based on medians and percentiles can be calculated for ordinal scales.
  • Continuous Scales – Scales that define a mathematical progression involving fractional levels, represented by numbers having decimal points after the integer. These scales may be called interval scales or ratio scales depending on their other properties. Any statistic can be calculated for data measured on continuous scales.

There are other scales of measurement but that’s all you’ll need at this point.

Descriptive Statistics

Now you can get on to describing a set of numbers. You’ll only need to consider four attributes – frequency, central tendency, dispersion, and shape.

 Frequency refers to the number of times the level of a scale appears in a set of numbers. It is used mostly for nominal (grouping) scales and sometimes with ordinal scales. The level with the highest frequency is called the mode. Frequency is used most effectively to show how scale levels compare to each other, such as with percentages or in a histogram.


Central Tendency refers to where the middle of a set of numbers is. It is used mostly for continuous (interval or ratio) scales and often with ordinal scales. There are many statistics that may be used to describe where the center of a dataset is, the most popular of which are the median and the mean. The median is the exact center of a progression-scale dataset. There are exactly the same number of data values less than and greater than the median. You determine the median by sorting the values in the dataset and counting the values from the extremes until you find the center. The mean, or average, is the center of a progression-scale dataset that is determined by a calculation. There may not be an equal number of data values less than and greater than the mean. You determine the mean by adding all the values in the dataset and dividing that sum by the number of values. The mean or the median is used in most statistical testing to find differences in data populations.

 Dispersion refers to how spread out the data values are. It is used for continuous (interval or ratio) scales but only rarely with ordinal scales. There are many ways to describe data dispersion but the most popular is the standard deviation. You calculate the standard deviation by:

  1. Subtracting the mean of a dataset from each value in the dataset
  2. Squaring each subtracted value
  3. Adding all the squared values
  4. Dividing the sum of the squared values by the number of values in the dataset (if you’re describing a sample) or by the number of values in the dataset minus 1 (if you’re describing a population).

The standard deviation is used in statistical testing to find differences in data populations.

5518606-pics-of-kittensShape refers to the frequency of the values in a dataset at selected levels of the scale, most often depicted as a graph. For ordinal scales, the graph is usually a histogram. For continuous scales, the graph is usually a probability plot, although sometimes histograms are used. Shapes of continuous scale data can be compared to mathematical models (equations) of frequency distributions. It’s like comparing a person to some well-known celebrity; they’re not identical but are similar enough to provide a good comparison. There are dozens of such distribution models, but the most commonly used is the normal distribution. The normal distribution model has two parameters – the mean and the standard deviation.

There are many other statistics that can be used to describe datasets, but most of the time, this is all you need:


For example, a nominal-scale dataset would be described by providing counts or percentages of observations in each group. An ordinal-scale dataset would be described by providing counts or percentages for each level, the median and percentiles, and ideally, a histogram. A continuous-scale dataset would be described by providing the closest distribution model and estimates of its parameters, such as “normally distributed with a mean of 10 and a standard deviation of 2.” Continuous-scale datasets can be described so succinctly because the distribution-shape specification contains so much of the telling information.

Now isn’t that a lot easier than describing that hot bank robber wearing a clown costume?


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Visualizations versus Infographics

Visualizations and infographics are both visual representations of data that are often confused. In fact, there is not a clear line of demarcation between the two. Both are informative. Both can be static or animated. Both require a knowledgeable person to create them.


Visualizations Explore

Data visualizations are created to make sense of data visually and to explore data interactively. Visualization is mostly automatic, generated through the use of data analysis software, to create graphs, plots, and charts. The visualizations can use the default settings of the software or involve Data Artistry and labeling (i.e., these Enhanced Visualizations fall in the intersection of the two circles in the figure). The processes used to create visualizations can be applied efficiently to almost any dataset. Visualizations tend to be more objective than infographics and better for allowing audiences to draw their own conclusions, although the audience needs to have some skills in data analysis. Data visualizations do not contain infographics.

Infographics Explain

Infographics are artistic displays intended to make a point using information. They are specific, elaborate, explanatory, and self-contained. Every infographic is unique and must be designed from scratch for visual appeal and overall reader comprehension. There is no software for automatically producing infographics the way there is for visualizations. Infographics are combinations of illustrations, images, text, and even visualizations designed for general audiences. Infographics are better than visualizations for guiding the conclusions of an audience but can be more subjective than visualizations.

Visualization Infographic
Objective Analyze Communicate
Audience Some data analysis skills General audience
Components Points, lines, bars, and other data representations Graphic design elements, text, visualizations
Source of Information Raw data Analyzed data and findings
Creation Tool Data analysis software Desktop publishing software
Replication Easily reproducible with new data Unique
Interactive or Static Either Static
Aesthetic Treatment Not necessary Essential
Interpretation Left to the audience Provided to the audience


img_8475c (1)

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How to Analyze Text

Statisticians love to analyze numbers, but what do they do when what they want to explore is unformatted text? It happens all the time. The text may come from opencat-diary-ended responses on surveys, social networking sites, email, online reviews, public comments, notations (e.g., medical, customer relations), documents and text files, or even recorded and transcribed interactions. But before anything can happen, you have to accomplish three tasks:

  • Get the text into a spreadsheet or other software that you can use to manipulate it.
  • Break the text into analyzable fragments – letters, words, phrases, sentences, paragraphs, or whatever.
  • Assign properties to the text fragments

How you might complete these tasks depends on what you want to do and the software you have. Nonetheless, you’ll be surprised by how much you can do with just a spreadsheet and an internet connection if you have the time and focus. This article will show you how.


Ther0402a6_fd87fbc829ec41faaf10aa7aa1cbed88-mv2_d_2000_1333_s_2e are several ways that you can analyze text. You can:

  • Count the occurrence of specific letters, words, or phrases, often summarized as Word Clouds. There are quite a few free web sites that will help you construct word clouds.
  • Categorize text by key themes, topics, or commonalities, called Text Mining.
  • Classify attitudes, emotions, and opinions of a source toward some topic, called Sentiment Analysis or opinion mining. There are many applications of sentiment analysis in business, marketing, customer management, political science, law, sociology, psychology, and communications.
  • Explore relationships between words using a Word Net. The relationships can reflect definitions or other commonalities.

Some of these analyses can be performed using free web apps, others, require special software.

Specialized Software

Some text analytics can be performed manually, but it is a time consuming process so having software can be crucial. Unfortunately, the biggest and best software is proprietary, like SAS and SPSS, and costs a lot. There are also free and low-cost alternatives, as well as free web sites that preform less sophisticated analyses. There are a lot of software options so there are probably a lot of people analyzing text. Let Google be your guide.

Manual Analyses

Even if you don’t have access to specialized software for text analyses, you can also still perform two types of analyses with nothing more than a spreadsheet program and an internet connection. You can count the number of times that a letter, word, or phrase appears in a text passage. Word frequency turns out to be relatively easy to produce but once you have the counts, the analysis and interpretation may be a bit more challenging. You can also do simple topic analyses or sentiment analyses. Parsing the sentences or sentence fragments and analyzing them is straightforward but time consuming, though the interpretation is usually easier.

Word Counts

If you are just looking for keywords or counting words for some diagnostic purpose, you’ll find that it’s not that difficult. Here’s how to do word counts.

Step 1 – Find the text you want to analyze.

This is usually easy except for there being so many choices. You have to start with an electronic file. If you have hard copy, you’ll have to sc
an it and correct the errors. If you have text from separate sources, you’1399360333213ll want to aggregate them to make things easier. If you have text on a website, you can usually highlight it and copy it using <ctrl-C>. If the passage is long, you can use <ctrl-A> to select everything before copying it, but you’ll have to edit out the extraneous material. You can do these operations in most word processors.

Step 2 – Scrub the data

You should scrub the text to be sure you’ll be counting the correct things. Take out entries that aren’t part of the flow of the text, like footnotes and section numbers. Correct misspellings. Take out punctuation that might become associated with words, like em dashes.

Step 3 – Count the words.

The quickest way to count words is to go to an Internet site for that purpose. Just copy your scrubbed text, paste it into the box on the site, and press submit. You’ll get a column of words and their frequencies. Parse the numbers from the text and you’re ready to analyze the data. It’s a good idea to review the results of the counting to be sure no errors have crept into the process.

Another way to do this solely in a spreadsheet is to replace all the punctuation with blanks and then replace the blanks with paragraph marks. This will give you a column of words. Copy it and remove the duplicates then you can use a formula to count each word.

Once you have the counts, the analysis is up to you. You can compare word statistics from different sources or analyze word frequencies within a single source. The possibilities are endless. Interpretation is another matter. Here are some examples.


One thing you can do with word counts is to produce a word cloud. There are many web sites that will generate these graphics. My favorite is Wordle, but be advised, you have to use Internet Explorer for it to work. Here’s an example of a word cloud produced with Wordle.


Text Mining

Topic or Sentiment Analyses are straightforward but more time consuming than word counts. Unless you are analyzing text for work or school, relax and turn on Netflix. This isn’t very sophisticated, but it’ll take a while and you’ll need frequent breaks to maintain your focus.

There are six steps.

Step 1 – Get the Data into a Spreadsheet

As with word counts, you have to get the text file into a text manager, preferably a spreadsheet. Highlight your text or use <ctrl A> and then <ctrl C> and <ctrl V>. You’ll need to parse any block text into sentences or whatever length fragment you want to analyze. You can usually do this by replacing periods with paragraph marks. Start with a small dataset, perhaps fewer than fifty fragments, until you get used to the process.

Step 2 – Scrub the Responses

Format the fragments into a single column with one fragment per row. Delete extraneous fragments. Don’t worry about misspellings and punctuation. If you make a mistake, <ctrl Z> will undo it.

Step 3 – Assign Descriptors

In a column next to the column with the fragments, enter your first descriptor. It can be a keyword, theme, sentiment, length, or whatever you want to analyze. Unless you have predetermined descriptors you are looking for, don’t worry too much about the descriptors you use. You’ll review and edit them in the next step.

cat-writingStep 4 – Count the Fragments Assigned to Each Descriptor

When you count the fragments assigned to each descriptor, you’ll probably find a few descriptors with only a few fragments. Consider combining them with other descriptors. When you’re satisfied with the assignments, you might want to subdivide the descriptor groups with another set of descriptors.

Step 5 – Repeat Steps 3 and 4

You can repeat the last two steps as many times as you feel is necessary. You can use these hierarchical descriptor groups to characterize subsets of the text so don’t have too many or too few fragments in each descriptor group. When you’re done, your data set would look something like this.


If you have a predetermined set of descriptors, you can assign each one to a column of the spreadsheet and code them as 0 or 1 for presence or absence.

Step 6 – Analyze

Once you have built your data set, you can analyze it statistically by counts and percentages, or graphically using word clouds. Consider this example. On December 29, 2016, Tanya Lynn Dee asked the question on her Facebook page, “Without revealing your actual age, what [is] something you remember that if you told a younger person they wouldn’t understand?” There were over 1,000 responses (at the time I saw the post), which I copied and classified into common themes. The results are here.

To learn more about analyzing text for its sentiment, read Sentiment Analysis
nearly everything you need to know by MonkeyLearn.

So, try analyzing some text (and other things) at home. You won’t need parental supervision.


Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data analysis at, or other online booksellers.

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Top 50 Statistics Blogs And Websites on the Web

Number 28

Reading Stats with  Cats

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Hellbent on Measurement

tape-measure-on-a-catAny variable that you record in a dataset will have some scale of measurement. Scales of measurement are, simply put, the ways that associated numbers relate to each other. Scales are properties of numbers, not the objects being measured. You could measure the same attribute of an object using more than one scale. For example, say you were doing a study involving cats and wanted to have a measure of each cat’s age. If you knew their actual birth dates, you could calculate their real ages in years, months, and days. If you didn’t know their birth dates, you could have a veterinarian or other knowledgeable individual estimate their ages in years. If you didn’t need even that level of precision, you could simply classify the cats as kittens, adult cats, or mature cats.

Understanding scales of measurement is important for a couple of reasons. Use a scale that has too many divisions and you’ll be fooled by the illusion of precision. Use a scale that has too few divisions and you’ll be dumbing down the data. Most importantly, though, scales of measurement determine, in part, what statistical methods might be applied to a set of measurements. If you want to do a certain type of statistical analysis on a variable, you have to use an appropriate scale for the variable. There are a few intricacies involved with measurement scales, so for now, just know that you have to understand a variable’s scale of measurement in order to analyze those data and interpret what it all means.

sound-27302bac00000578-3021300-image-a-35_1427885938930In Statistics 101, you’ll learn that there are four types of measurement scales – nominal, ordinal, interval, and ratio. This isn’t entirely true. The four-scale classification, described by Stevens (1946)[1], is just one way that scales are categorized, though it’s mentioned in almost every college-level introduction to statistics. There are actually a variety of other measurement scales, some differing in only obscure details.

The most basic classification of measurement scales involves whether or not the scale defines (1) groups having no mathematical relationship to each other, called grouping scales, or (2) a progression of measurement levels within a group, called progression scales.

Grouping (Nominal) Scales

Grouping scales define groups, which are finite, usually independent, and non-overlapping (discrete). Nominal scales are grouping scales. They represent categories, names, and other sets of associated attributes. None of the levels within a grouping scale have any sequential relationship to any of the other levels. One level isn’t greater than or less than another level.

Examples of properties that would be measured on a qualitative scale include:

  • Names—Kyle, Stan, Eric, Kennycup-beaker-vv9areh
  • Sex—female, male
  • Identification—PINs, product serial numbers
  • Locations—Wolf Creek, Area 51, undisclosed secure location
  • Car styles—sedan, pickup, SUV, limo, station wagon
  • Organization—company, office, department, team

Grouping scales are sometimes subdivided by the number of measurement levels. Discrete scales have a finite number of levels. For example, sex has two levels, male and female. Discrete scales with two levels are also called binary or dichotomous scales. Discrete scales with more than two levels are called categorical scales.

Variables measured on grouping scales can be used for counts and statistics based on counts, like percentages. They are also used to subdivide variables measured on progression scales.


Progression Scales

Progression or continuous scales define some mathematical progression. The number of possible levels may be finite or infinite. They can be limited to integers or use an integer and any number of decimal points after the integer. Ordinal, interval, and ratio scales are all progression scales.

Ordinal Scales

Ordinal scales have levels that are ordered. The levels denote a ranking or some sequence. One measurement may be greater than or less than another. However, the intervals between the measurements might not be constant.

Examples of properties that would be measured on an ordinal scale include:

  • Time—business quarter, geologic period, football quarters
  • Rankings—first place, second place, third place …
  • Thickness—geologic strata, atmospheric layers
  • Survey responses—very good, somewhat good, average, somewhat bad, very bad

Sometimes the intervals between levels of an ordinal scale are so different they can be treated as if they were grouping scales. Consider geologic time. It’s divided into eon, eras, periods, epochs, and ages, but the divisions aren’t the same lengths. Some periods are four times longer than others and the lengths can change as more is learned about the history of Earth. The units of the scale are also different in different parts of the world. Then there’s Moh’s scale of mineral hardness. It consists of ten levels. However, the interval between levels 1 and 8 is about the same as the interval between levels 8 and 9. The interval between levels 9 and 10 is four times greater than the interval between levels 8 and 9. Geologists must be a bunch of really creative people who aren’t bound by convention.

More frequently, the intervals between levels of an ordinal scale are the same, in theory or reality. Rankings, game segments like innings and periods, business quarters and fiscal years, are all examples.


Counts and statistics based on medians and percentiles can be calculated for ordinal scales. This includes most types of nonparametric statistics. However, there are situations in which averages and standard deviations are used. Surveys present one of those situations because the responses can be considered to be either grouping or progression scales depending on how the levels are defined. Say you have a survey question that has five possible responses:

  • Very good
  • Good
  • No opinion
  • Poor
  • Very poor

This is a grouping scale because the No Opinion response is not part of a progression. But, if the responses were:scale-twitch_scale

  • Very good
  • Good
  • Fair
  • Poor
  • Very poor

The scale could be recoded as Very Good=5, Good=4, Fair=3, Poor=2, and Very Poor=1 allowing statistical analyses to be conducted. If it were believed that the intervals between levels were not constant, analyses should be limited to counts and statistics based on medians and percentiles. If the intervals between levels were believed to be fairly constant, calculating averages and standard deviations might be legitimate. This is one of the points of contention with Stevens’s categories of scales. A given measurement’s scale might be perceived differently by different users.

Ratio Scales

Ratio scales are the top end of progression scales. Their levels consist of integers followed by any number of decimal points. Ratios and arithmetic operations are meaningful. Zero is a constant and a reference to an absence of the attribute the scale measures.

Measurements made by most kinds of meters or other types of measuring device are probably ratio scales. Examples of variables measured on ratio scales include:five

  • Concentrations, densities, masses, and weights
  • Durations in seconds, minutes, hours, or days
  • Lengths, areas, and volumes

Any type of statistic can be calculated for variables measured on a ratio scale.

Other Scales of Measurement

Understanding different types of measurement scales can help you select appropriate techniques for an analysis, especially if you’re a statistical novice. Stevens’s classification of scales works for many applications but it should be viewed as guidance rather than gospel. Interval scales in particular are an exception to the progression of scales form ordinal to ratio scales, and there are other exception scales as well. The following sections describe interval scales and a few scales that don’t quite fit into Stevens’s taxonomy.

Interval Scales

Interval measurements are ordered like ordinal measurements and the intervals between the measurements are equal. However, there is no natural zero point and ratios have no physical meaning. The classical example of an interval scale is temperature in degrees Fahrenheit or Centigrade. The intervals between each Fahrenheit degree are equal, but the zero point (-32 degrees) is arbitrary. Elevation is sometimes considered to be an interval scale temperature-should-hospital_e2d565717fa09970because the choice of sea level as the zero elevation is arbitrary. Time can also be thought of as an interval scale.

Some statisticians consider log-interval scales of measurement, in which the intervals between levels are constant in terms of logarithms, to be a subset of interval scales. Earthquake intensity (Richter and Mercali scales) and pH are examples of log-interval scales.

Statistics for ordinal scales and statistics based on means, variances, and correlations can be calculated for interval scales.



Counts are like ratio scales in that they have a zero point, constant intervals and ratios are meaningful, but there are no fractional units. Any statistic that produces a fractional count is meaningless. The classic example of a meaningless count statistic is that the average family includes 2.3 children. Counts are usually treated as ratio scales, but the result of any calculation is rounded off to the nearest whole unit.

Restricted-Range Scales

A constrained or restricted-range scale is a type of scale that is continuous only within a finite range. Probabilities are examples of constrained scales because any number is valid between the fixed endpoints of 0 and 1. Numbers outside this range are not possible. Percentages can be considered constrained or unconstrained depending on how the ratio is defined. For example, percentages for opinion polls are restricted to the range 0 to 100 percent. Percentages that describe corporate profits can be negative (i.e., losses) or virtually infinite (as in windfall profits). Restricted-range scales must be handled with special statistical techniques, such as logistical regression, that account for fixed scale

Cyclic Scales

Cyclic scales are scales in which sets of units repeat.

Repeating Units

Some cyclic scales consist of repeating levels for measuring open-ended quantities. Day of the week, month of the year, and season are examples. Time isn’t the only dimension with repeating scales, either. Musical scales, for instance, repeat yet have very different properties compared to time scales.

Repeating scales can be analyzed either by (1) treating them as an ordinal scale or (2) ignoring the repeating nature of the measure and transforming them into non-repeating linear units, such as day 1, day 2, and so on, or using a specialized statistical technique. The objective of the statistical analysis dictates which approach should be used. The first approach might be used to identify seasonality or determine if some measurement is different on one day or month rather than another. For example, this approach would be used to determine if work done on Fridays had higher numbers of defects than work done on other days. The second approach might be used to examine temcompass-20130531-182857poral trends. The third approach is used by statisticians who want to show off.

Orientation Scales

Orientation scales are a special type of cyclic scale. Degrees on a compass, for example, are a cyclic scale in which 0 degrees and 360 degrees are the same. Special formulas are required to calculate measures of central tendency and dispersion on circles and spheres.

Concatenated Numbers and Text

Concatenated numbers and text are not scales in the true sense of variable measurement, but they are part of every data analysis in one way or another. Concatenated numbers contain multiple pieces of information, which must be treated as a nominal scale unless the information can be extracted into separate variables. Examples of concatenated numbers include social security numbers, telephone numbers, sample IDs, date ranges, latitude/longitude, and depth or elevation intervals. Likewise, labels can sometimes be parsed into useful data elements. Names and addresses are good examples.

Time Scales

Time scales have some very quirky properties. You might think that time is measured on a ratio scale given its ever finer divisions (i.e., hours, minutes, seconds), yet it doesn’t make sense to refer to a ratio of two times any more than the ratio of two location coordinates. The starting point is also arbitrary. This sounds like an interval scale.


Time is like a one-dimensional location coordinate but it can also be linear or cyclic. Year is linear, so it’s at least an ordinal scale. For example, 1953 happened once and will never recur. Some time scales, though, repeat. Day 8 is the same as day 1. Month 13 is the same as month 1. So, time can also be treated as being measured on a nominal scale.

Time units are also used for durations, which are measured on a ratio scale. Durations can be used in ratios, they have a starting point of zero, and they don’t repeat (eight days aren’t the same as one day).

Time formats can be difficult to deal with. Most data analysis software offer a dozen or more different formats for what you see. Behind the spreadsheet format, though, the database has a number, which is the distance the time value is from an arbitrary starting point. Convert a date-time format to a number format, and you’ll see the number that corresponds with the date. The software formatting allows you to recognize values as times while the numbers allow the software to calculate statistics. This quirk of time formatting also presents a potential for disaster if you use more than one piece of software because different programs use different starting dates for their time calculations. Always check that the formatted dates are the same between applications.location-d71_2271

Location Scales

Just as there is time and duration, there is location and distance (or length), but there are a few twists. Time is one-dimensional; at least as we now know it. Distance can be one-, two-, or three-dimensional. Distance can be in a straight line (“as the crow flies”) or along a path (such as driving distance). Distances are usually measured in English inches, feet, yards, and miles or metric centimeters, meters, and kilometers. Locations, though, are another matter. Defining the location of a unique point on a two-dimensional surface (i.e., a plane) requires at least two variables. The variables can represent coordinates (northing/easting, latitude/longitude) or distance and direction from a fixed starting point. Of the coordinate systems, only the northing/easting scheme is a simple, non-concatenated scale that can be used for classical statistical analysis. However, this type of scale is usually not used for published maps, which can be a problem because virtually all environmental data are inherently location-dependent and multidimefly-c2bac65889b946dec4996a0a248e2ba0nsional. Thus, coordinate systems usually have to be converted for one to the other. Geostatistical applications, for example, are based on distance and direction measurements but these measurements are calculated from spatial coordinates.

At least three variables are needed to define a unique point location in a three-dimensional volume, so a variable for depth (or height) must be added to the location coordinates. Often, however, a property of an object occurs over a range of depths (or heights or elevations) rather than a finite point. Unfortunately, depth range is a concatenated number (e.g., 2-4 feet). It’s always better to use two variables to represent starting depth and ending depth. Thus, it may take four variables to define an environmental space, such as the sampled interval of a well or soil boring.

Selecting Scales

In the simplest taxonomy, almost all scales act either to group data othe_cat_stairsr represent the progression in a variable’s attribute, whether simple, ordinal-scale levels or more expansive ratio-scale levels. One way to view these differences is this: nominal (grouping) scales are like stone outcrops, randomly scattered around a garden area. Ordinal scales are like garden steps. You can only be on a step not between steps, and the steps lead progressively upward or downward. There may be many steps or just a few. Ratio scales are like a garden path or ramp. You can be anywhere along the path, at high levels or low. You can move forward or back, in small or large intervals.

Somewhere between those simple, discrete ordinal scales and the finely-divided ratio scales, however, are quite a few types of scales that don’t meet either definition. Just ask yourself these questions to understand the scale you will be dealing with:

  • Does the scale represent a progression of values? If not, the scale is a grouping scale.
  • Are the scale intervals approximately equal? If not, the scale is may be treated as a grouping scale.
  • Is there a constant zero (or other reference point) representing the absence of the attribute being measured? If not, the scale is may be treated like an interval scale.
  • Are the limits of the scale limited in any way? Is there a scale minimum or maximum? Are negative numbers prohibited? If so, you may have to use special statistical approaches to analyze data measured on the scale.
  • Are the scale values cyclic or repeating? If so, you may have to use special statistical approaches to analyze data measured on the scale.
  • Are ratios and other mathematical operations that produce fractional scale levels permissible? If so, you have a ratio scale.

Some people think that an attribute can be measured in only one way. This is untrue more often than it is not. Consider the example of color. To an auto manufacturer, color is measured ontape-itskhnz a nominal scale. You can buy one of their cars painted red or blue or silver or black. To a gemologist, the color of a diamond is graded on an ordinal scale from D (colorless) to Z (light yellow). To an artist, color is measured on an interval scale because their color wheel contains the sequence: red, red-orange, orange, orange-yellow, yellow, yellow-green, green, green-blue, blue, blue-violet, violet, and violet-red. To a physicist, colors are measured by a continuous spectrum of light frequencies, which employ a ratio scale.

Using a different scale than what might be the convention can provide advantages. Consider this example. Soil texture is usually measured on a nominal scale that defines groups such as loam, sandy loam, clay loam, and silty clay. The information can be made quantitative by recording the percentages of sand, silt, and clay (which define the texture) instead of just the classification. The nominal-scale measure is much easier to collect in the field and is one variable to manage rather than three. On the other hand, the progression-scale measures can be analyzed in more ways. Correlating the clay content of a soil to crop growth, soil moisture, or a pollutant concentration can be done only if soil texture is measured on a progression scale.


If a choice can be made on which type of scale to use, use a ratio scale. Ratio scales are usually best because they provide the most information and can be rescaled easily as ordinal scales. For example, many sports organize contestants using weight class, measured on an ordinal scale, instead of weight, which is measured on a ratio scale. Weight is still measured at weigh-in using a ratio scale but is converted to the ordinal-scale weight classes for simplicity. In contrast, it’s usually not possible to upgrade an ordinal scale to a ratio scale unless the ordinal scale has equal intervals and calculation of percentages or z-scores makes sense. You couldn’t just estimate a contestant’s weight of 178.2 pounds from a weight class of 170-185 pounds.

If you can’t measure an attribute on a ratio or interval scale, think about hobook-92e6f20565c2a9472dda6410939a44a6w an ordinal scale could be applied. You can almost always devise an ordinal scale to characterize an attribute; you just have to be creative. Think of opinion surveys. If you can measure opinions, you can measure anything.

[1] Stevens, S. S. 1946. On the theory of scales of measurement. Science v. 103, No. 2684, p. 677–680.

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data analysis at, or other online booksellers.


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Searching for Answers

scaleNeed to find something out, just Google it. Now that Google is a verb as well as a noun, it’s easy. But …

It Hasn’t Always Been Easy

Adults under 30, Millennials, grew up with smartphones, laptop and tablet computers, and the Internet. As a group, they’ve never known a time when technology wasn’t integral to their existence. For those of us who finished school before the 1980s, personal computers were a rarity and the Internet was only then being developed for the military-industrial complex. Browsers didn’t appear until the early 1990s. You couldn’t buy a book from Amazon until 1995.

student-using-the-card-catalog-1971So, it hasn’t always been easy to find information. For most students, searching for information before 1980 usually involved a trip to the library. There, you would thumb through the 3×5” cards in the drawers of the card catalog looking for information by keywords. You would write down the title of the book referenced on a card along with its location classification (Dewey, Library of Congress). Then you would go to the location in the book stacks and retrieve the book, unless it was already in use, checked out, misplaced, or stolen. Finding enough information to fulfill a need might take hours or days or longer. Then you had to lug the books to a place where you could read them, extract the information you needed, and write it all down on paper. Needless to say, things have changed for the better. Now you can enter your keywords into an Internet search engine, and in a fraction of a second have references to hundreds, if not hundreds of thousands, websites, articles, blogs, books, images, and presentations. You can bookmark sites to read later or just save the relevant information to the cloud. That process might take minutes and will return more relevant information than you could ever access a generation earlier.


What People Looked For

Not only can people search more information sources faster than ever before but now Big Business and Big Government collects data on all those searches. For example, keeps track of the number of visitors to the Stats with Cats bcat-using-iphonelog site, what country they accessed the blog from, the search terms they used to find the site, and the blogs they visited. This is useful because it reveals what people are looking for, at least those people who ended up at the Stats with Cats blog.

Here are the frequencies for pertinent search terms from May 2010 through June 2016 and the associated word cloud (produced at; works best in IE).

keywords2Perhaps not surprisingly, the most common terms are associated with topics students would search if they were confronted with taking their first statistics class – statistics or stats, school or class, graph or chart, data, variable, and correlation. This may reflect the overpowering anticipation of learning about the some of the fascinating aspects of statistical thinking or, more likely, the fear of number crunching.

People searching for “report” are probably trying to figure out how to convert their statistical results into some meaningful story. How to Write Data Analysis Reports is probably much more than they might have expected.

People searching for the number 30 are looking for the reason they were told that their statistical analysis must have at least 30 samples. They might not like the answer at 30 Samples. Standard, Suggestion, or Superstition? but at least they’ll understand where it started, why they keep hearing it, and why the real answer is so unsatisfying.

What They Found

There were over 76,000 referrals from 255 sites, of which 97% came from Google. Bing and Facebook each contributed about 1%. Five Things You Should Know Before Taking Statistics 101 was viewed over 100,000 times in five and a half years. Secrets of Good Correlations had nearly 70,000 views in six years.


The following table summarizes the views and the views per year for 56 Stats with Cats blogs.



Total Views

Years Available

Views per Year

Five Things You Should Know Before Taking Statistics 101 109,329 5.5 19,878
Secrets of Good Correlations 69,212 6.1 11,377
How to Write Data Analysis Reports 32,253 3.5 9,774
How to Tell if Correlation Implies Causation 10,552 1.5 7,035
30 Samples. Standard, Suggestion, or Superstition? 18,151 6.1 2,984
Why Do I Have To Take Statistics? 13,645 6.1 2,243
Ten Fatal Flaws in Data Analysis 13,618 6.1 2,239
Fifty Ways to Fix your Data 11,067 6.1 1,819
Six Misconceptions about Statistics You May Get From Stats 101 8,011 5.5 1,457
Regression Fantasies 7,117 5.5 1,294
The Right Tool for the Job 5,586 6.1 918
The Best Super Power of All 3,511 4.5 780
Why You Don’t Always Get the Correlation You Expect 1,450 2.5 580
Looking for Insight through a Window 224 0.5 448
A Picture Worth 140,000 Words 2,292 5.5 417
The Heart and Soul of Variance Control 2,248 6.1 370
O.U..T…L….I……E……..R………………..S 907 2.5 363
The Five Pursuits You Meet in Statistics 2,005 6.1 330
Ten Ways Statistical Models Can Break Your Heart 144 0.5 288
The Zen of Modeling 1,731 6.1 285
The Foundation of Professional Graphs 1,226 4.5 272
Assuming the Worst 1,550 6.1 255
It’s All Relative 1,303 5.5 237
There’s Something About Variance 1,424 6.1 234
The Measure of a Measure 1,180 6.1 194
Purrfect Resolution 1,167 6.1 192
The Data Scrub 1,145 6.1 188
Limits of Confusion 1,030 5.5 187
Try This At Home 1,133 6.1 186
Grasping at Flaws 1,009 5.5 183
Consumer Guide to Statistics 101 984 5.5 179
It’s All Greek 1,058 6.1 174
It was Professor Plot in the Diagram with a Graph 1,028 6.1 169
Weapons of Math Production 934 6.1 154
Polls Apart 819 5.5 149
You’re Off to Be a Wizard 881 6.1 145
Samples and Potato Chips 866 6.1 142
Time Is On My Side 865 6.1 142
You Can Lead a Boss to Data but You Can’t Make Him Think 833 6.1 137
Types and Patterns of Data Relationships 323 2.5 129
The Santa Claus Strategy 741 6.1 122
It’s All in the Technique 693 6.1 114
The Data Dozen 603 5.5 110
Becoming Part of the Group 589 5.5 107
Reality Statistics 618 6.1 102
Aphorisms for Data Analysts 524 5.5 95
Ten Tactics used in the War on Error 520 5.5 95
The Seeds of a Model 478 6.1 79
Ockham’s Spatula 389 5.5 71
Statistics: a Remedy for Football Withdrawal 384 5.5 70
Many Paths Lead to Models 370 6.1 61
Dealing with Dilemmas 283 5.5 51
Perspectives on Objectives 251 6.1 41
Tales of the Unprojected 241 6.1 40
Getting the Right Answer 197 5.5 36
Resurrecting the Unplanned 202 6.1 33

The message these statistics are sending appears to be that the Stats with Cats blog attracts introductory students who don’t know what to expect from their statistics class or need help in understanding challenging statistical concepts. In contrast, experienced students are acquainted with more statistics professors and students. They own more statistics textbooks and have visited more educational web sites. And as a consequence, they search for more specific statistical terms, like tolerance limits and autocorrelation, that beginners wouldn’t know. It’s ironic, then, that Stats with Cats was written for students who had completed Statistics 101 and were looking for some help in applying what they had learned. Interesting … sometimes statistical analyses reveal things you don’t expect.


Read more about using statistics at the Stats with Cats blog. Read them to your cats. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at,, or other online booksellers.

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Predict the Next President of the United States

Cat for presidentThe American Statistical Association is sponsoring a new statistics contest for high school and college students. The contest, known as Prediction 2016, challenges students to use statistics to predict the next president of the U.S. The purpose of the contest is to get more students interested in statistics by showing them how it can apply to the real world. It’s part of the larger student education campaign This is Statistics. Here’s more information:


ASA Announces Prediction 2016, a National Student Contest to Predict the Next President of the United States


Sponsored by the American Statistical Association, Prediction 2016 is a contest for high school and undergraduate college students to predict the winner of the U.S. presidential election using statistical methods. Winners will receive a variety of prizes and perks, including exposure to the nation’s leading statisticians and data scientists.


One winner will be chosen among high school contestants and one among college contestants. Those with the most accurate predictions developed with sound statistical methods will win the contest.


October 24, 2016 at 5pm — Deadline for submitting predictions.

October 27, 2016 — ASA announces which candidate wins in the student predictions.

November 9, 2016 — ASA announces contest winners.

Learn more at ASA spokespersons are available for interviews about the contest, as well as trends in statistics education and careers that are shaping the economy and workforce.

Media Contact:

  • Sarah Litton
  • (202) 851-2479
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Regression Fantasies

Common Reasons for Doubting a Regression Model

Finding a model that fits a set of data is one of the most common goals in data analysis. Least squares regression is the most commonly used tool for achieving this goal. It’s a relatively simple concept, it’s easy to do, and there’s a lot of readily available software to do the calculations. It’s even taught in many Statistics 101 courses. Everybody uses it … and therein lies the problem. Even if there is no intention to mislead anyone, it does happen.

Here are eleven of the most common reasons to doubt a regression model.

Not Enough Samples

Accuracy is a critical component for evaluating a model. The coefficient of determination, also known as R-squared or R2, is the most often cited measure of accuracy. Now obviously, the more accurate a model is the better, so data analysts look large values for R-squared.

R-squared is designed to estimate the maximum relationship between the dependent and independent variables based on a set of samples (cases, observations, records, or whatever). If there aren’t enough samples compared to the number of independent variables in the model, the estimate of R-squared will be especially unstable. The effect is greatest when the R-squared value is small, the number of samples is small, and the number of independent variables is large, as shown in this figure.

The inflation in the value of R-squared can be assesses by calculating the shrunken R-square. The figure shows that for an R-squared value above 0.8 with 30 cases per variable, there isn’t much shrinkage. Lower estimates of R-square, however, experience considerable shrinkage.

You can’t control the magnitude of the relationship between a dependent variable and a set of independent variables, and often, you won’t have total control over the number of samples and variables either. So, you have to be aware that R-squared will be overestimated and treat your regression models with some skepticism.

No Intercept

Almost all software that performs regression analysis provides an option to not include an intercept term in the model. This sounds convenient, especially for relationships that presume a one-to-one relationship between the dependent and independent variables. But when an intercept is excluded from the model, it’s not omitted from the analysis; it is set to zero. Look at any regression model with “no intercept” and you’ll see that the regression line goes through the origin of the axes.

With the regression line nailed down on one end at the origin, you might expect that the value of R-squared would be diminished because the line wouldn’t necessarily travel through the data in a way that minimizes the differences between the data points and the regression line, called the errors or residuals. Instead, R-squared is artificially inflated because when the correction provided by the intercept is removed, the total variation in the model increases. But, the ratio of the variability attributable to the model compared to the total variability also increases, hence the increase in R-squared.

The solution is simple. Always have an intercept term in the model unless there is a compelling theoretical reason not to include it. In that case, don’t put all your trust in R-square (or the F-tests).

Stepwise Regression

Stepwise regression is a data analyst’s dream. Throw all the variables into a hopper, grab a cup of coffee, and the silicon chips will tell you which variables yield the best model. That irritates hard-core statisticians who don’t like amateurs messing around with their numbers. You can bet, though, that at least some of them go home at night, throw all the food in their cupboard into a crock pot, and expect to get a meal out of it.

The cause of some statistician’s consternation is that stepwise regression will select the variables that are best for the data set, but not necessarily the population. Model test probabilities are optimistic because they don’t account for the stepwise procedure’s ability to capitalize on chance. Moreover, adding new variables will always increase R-squared, so you have to have some good ways to decide how many variables is too many. There are ways to do this. So using stepwise regression alone isn’t a fatal flaw. Like with guns, drugs, and fast food, you have to be careful how you use it.

If you use stepwise regression, be sure to look at the diagnostic statistics for the model. Also, verify your results using a different data set by splitting the data set before you do any analysis, by randomly extracting observations from the original data set to create new data sets, or by collecting new samples.


Outliers are a special irritant for data analysts. They’re not really that tough to identify but they cause a variety of problems that data analysts have to deal with. The first problem is convincing reviewers not familiar with the data that the outliers are in fact outliers. Second, the data analysts have to convince all reviewers that what they want to do with them, delete or include or whatever, is the appropriate thing to do. One way or another, though, outliers will wreak havoc with R-squared.

Consider this figure, which comes from an analysis of slug tests to estimate the hydraulic conductivity of an aquifer. The red circles show the relationship between rising-head and falling-head slug tests performed on groundwater monitoring wells. The model for this relationship has an R-square of 0.90. The blue diamond is an outlier along the trend (same regression equation) about 60% greater than the next highest value. The R-squared of this equation is 0.95. The green square is an outlier perpendicular to the trend. The R-squared of this equation is 0.42. Those are fairly sizable differences to have been caused by a single data point.

How should you deal with outliers? I usually delete them because I’m usually looking to model trends and other patterns. But outliers are great thought provokers. Sometimes they tell you things the patterns don’t. If you’re not comfortable deciding what to do with an outlier, run the analysis both with and without outliers, a time consuming and expensive approach. The other approach would be to get the reviewer, an interested stakeholder, or an independent expert involved in the decision. That approach is time consuming and expensive too. Pick your poison.

Non-linear relationships

Linear regression assumes that the relationship between a dependent variable and a set of independent variables are additive, or linear. If the relationship is actually nonlinear, the R-squared for the linear model will be lower than it would be for a better fitting nonlinear model.

This figure shows the relationship between the number of employed individuals and the number of individuals not in the U.S. work force between 1980 and 2009. The linear model has a respectable R-squared value of 0.84, but the polynomial model fits the data much better with an R-squared value of 0.95.

Non-linear relationships are a relatively simple problem to fix, or at least acknowledge, once you know what to look for. Graph your data and go from there.


Overfitting involves building a statistical model solely by optimizing statistical parameters, and usually involves using a large number of variables and transformations of the variables. The resulting model may fit the data almost perfectly but will produce erroneous results when applied to another sample from the population.

The concern about overfitting may be somewhat overstated. Overfitting is like becoming too muscular from weight training. It doesn’t happen suddenly or simply. If you know what overfitting is, you’re not likely to become a victim. It’s not something that happens in a keystroke. It takes a lot of work fine tuning variables and what not. It’s also usually easy to identify overfitting in other people’s models. Simply look for a conglomeration of manual numerical adjustments, mathematical functions, and variable combinations.


Misspecification involves including terms in a model that make the model look great statistically even though the model is problematical. Often, misspecification involves placing the same or very similar variable on both sides of the equation.

Consider this example from economics. A model for the U.S. Gross Domestic Product (GDP) was developed using data on government spending and unemployment from 1947 to 1997. The model:

GDP = (121*Spending) – (3.5*Spending2) + (136*Time) – (61*Unemployment) – 566

had an R-squared value of 0.9994. Such a high R-squared value is a signal that something is amiss. R-squared values that high are usually only seen in models involving equipment calibration, and certainly not anything involving capricious human behavior. A closer look at the study indicated that the model term involving spending were an index of the government’s outlays relative to the economy. Usually, indexing a variable to a baseline or standard is a good thing to do. In this case, though, the spending index was the proportion of government outlays per the GDP. Thus, the model was:

GDP = (121*Outlays/GDP) – (3.5* (Outlays/GDP)2) + (136*Time) – (61*Unemployment) – 566

GDP appears on both sides of the equation, thus accounting for the near perfect correlation. This is a case in which an index, at least one involving the dependent variable, should not have been used.

Another misspecification involves creating a prediction model having independent variables that are more difficult, time consuming, or expensive to generate than the dependent variable. You might as well just measure the dependent variable when you need to know its value. Similarly with forecasting (prediction of the future) models, if you need to forecast something a year in advance, don’t use predictors that are measured less than a year in advance.


Multicollinearity occurs when a model has two or more independent variables that are highly correlated with each other. The consequences are that the model will look fine, but predictions from the model will be erratic. It’s like a football team. The players perform well together but you can’t necessarily tell how good individual players are. The team wins, yet in some situations, the cornerback or offensive tackle will get beat on most every play.

If you ever tried to use independent variables that add to a constant, you’ve seen multicollinearity in action. In the case of perfect correlations, such as these, statistical software will crash because it won’t be able to perform the matrix mathemagics of regression. Most instances of multicollinearity involve weaker correlations that allow statistical software to function, yet the predictions of the model will still be erratic.

Multicollinearity occurs often in the social sciences and other fields of study in which many variables are measured in the process of model building. Diagnosis of the problem is simple if you have access to the data. Look at correlations between the independent variables. You can also look at the variance inflation factors, reciprocals of one minus the R-squared values for the independent variables and the dependent variable. VIFs are measures of how much the model’s coefficients change because of multicollinearity. The VIF for a variable should be less than 10 and ideally near 1.

If you suspect multicollinearity, don’t worry about the model but don’t believe any of the predictions.


Regression, and practically all parametric statistics, requires that the variances in the model residuals be equal at every value of the dependent variable. This assumption is called equal variances, homogeneity of variances, or coolest of all, homoscedasticity. Violate the assumption and you have heteroscedasticity.

Heteroscedasticity is assessed much more commonly in analysis of variance models than in regression models. This is probably because the dependent variable in ANOVA is measured on a categorical scale while the dependent variable in regression is measured on a continuous scale. The solution to this is fairly simple. Break the dependent variable scale into intervals, like in a histogram, and calculate the variance for each interval. The variances don’t have to be precisely equal, but variances different by a factor of five are problematical. Unequal variances will wreak havoc on any tests or confidence limits calculated for model predictions.


Autocorrelation involves a variable being correlated with itself. It is the correlation between data points with the previously listed data points (termed a lag). Usually, autocorrelation involves time-series data or spatial data, but it can also involve the order in which data are collected. The terms autocorrelation and serial correlation are often used interchangeably. If the data points are collected at a constant time interval, the term autocorrelation is more typically used.

If the residuals of a model are autocorrelated, it’s a sure bet that the variances will also be unequal. That means, again, that tests or confidence limits calculated from variances should be suspect.

To check a variable or residuals from a model for autocorrelation, you can conduct a Durban-Watson test. The Durban-Watson test statistic ranges from 0 to 4. If the statistic is close to 2.0, then serial correlation is not a problem. Most statistical software will allow you to conduct this test as part of a regression analysis.


Most software that calculates regression parameters also allows you to weight the data points. You might want to do this for several reasons. Weighting is used to make more reliable or relevant data points more important in model building. It’s also used when each data point represents more than one value. The issue with weighting is that it will change the degrees of freedom, and hence, the results of statistical tests. Usually this is OK, a necessary change to accommodate the realities of the model. However, if you ever come upon a weighted least squares regression model in which the weightings are arbitrary, perhaps done by an analyst who doesn’t understand the consequence, don’t believe the test results.

Is Your Regression Model Telling the Truth?

There are many technologies we use in our lives without really understanding how they work. Television. Computers. Cell phones. Microwave ovens. Cars. Even many things about the human body are not well understood. But I don’t mean how to use these mechanisms. Everyone knows how to use these things. I mean understanding them well enough to fix them when they break. Regression analysis is like that too. Only with regression analysis, sometimes you can’t even tell if there’s something wrong without consulting an expert.

Here are some tips for troubleshooting regression models.


You may know how to use regression analysis, but unless you’re an expert, you may not know about some of the more subtle pitfalls you may encounter. The biggest red flag that something is amiss is the TGTBT, too good to be true. If you encounter an R-squared value above 0.9, especially unexpectedly, there’s probably something wrong. Another red flag is inconsistency. If estimates of the model’s parameters change between data sets, there’s probably something wrong. And if predictions from the model are less accurate or precise than you expected, there’s probably something wrong. Here are some guidelines for troubleshooting a model you developed.

Your Model Identification Correction
Not Enough Samples If you have fewer than 10 observations for each independent variable you want to put in a model, you don’t have enough samples. Collect more samples. 100 observations per variable is a good target to shoot for although more is usually better.
No Intercept You’ll know it if you do it. Put in an intercept and see if the model changes.
Stepwise Regression You’ll know it if you do it. Don’t abdicate model building decisions to software alone. What’s the fun in that?
Outliers Plot the dependent variable against each independent variable. If more than about 5% of the data pairs plot noticeable apart from the rest of the data points, you may have outliers. Conduct a test on the aberrant data points to determine if they are statistical anomalies. Use diagnostic statistics like leverage to evaluate the effects of suspected outliers. Evaluate the metadata of the samples to determine if they are representative of the population being modeled. If so, retain the outlier as an influential observation (AKA leverage point).
Non-linear relationships Plot the dependent variable against each independent variable. Look for nonlinear patterns in the data Find an appropriate transformation of the independent variable.
Overfitting If you have a large number of independent variables, especially if they use a variety of transformation and don’t contribute much to the accuracy and precision of the model, you may have overfit the model. Keep the model as simple as possible. Make sure the ratio of observations to independent variables is large. Use diagnostic statistics like AIC and BIC to help select an appropriate number of variables.
Misspecification Look for any variants of the dependent variable in the independent variables. Assess whether the model meets the objectives of the effort. Remove any elements of the dependent variable from the independent variables. Remove at least one component of variables describing mixtures. Ensure the model meets the objectives of the effort with the desired accuracy and precision..
Multicollinearity Calculate correlation coefficients and plot the relationships between all the independent variables in the model. Look for high correlations. Use diagnostic statistics like VIF to evaluate the effects of suspected multicollinearity. Remove intercorrelated independent variables from the model.
Heteroscedasticity Plot the variance at each level of an ordinal-scale dependent variable or appropriate ranges of a continuous-scale dependent variable. Look for any differences in the variances of more than about five times. Try to find an appropriate Box-Cox transformation or consider nonparametric regression or data mining methods.
Autocorrelation Plot the data over time, location or the order of sample collection. Calculate a Durbin–Watson statistic for serial correlation. If the autocorrelation is related to time, develop a correlogram and a partial correlogram. If the autocorrelation is spatial, develop a variogram. If the autocorrelation is related to the order of sample collection, examine metadata to try to identify a cause.
Weighting You’ll know it if you do it. Compare the weighted model with the corresponding unweighted model to assess the effects of weighting. Consider the validity of weighting; seek expert advice if needed.

Sometimes the model you are skeptical about isn’t one you developed; it is models that are developed by other data analysts. The major difference is that with other analysts’ models, you won’t have access to all their diagnostic statistics and plots, let alone their data. If you have been retained to review another analyst’s work, you can always ask for the information you need. If, however, you’re reading about a model in a journal article, book, or website, you’ve probably got all the information you’re ever going to get. You have to be a statistical detective. Here are some clues you might look for.

Another Analyst’s Model Identification
Not Enough Samples If the analyst reported the number of samples used, look for at least 10 observations for each independent variable in the model. If not, you may be able to estimate the number from a scatterplot.
No Intercept If the analyst reported the actual model (some don’t), look for a constant term.
Stepwise Regression Unless another approach is reported, assume the analyst used some form of stepwise regression.
Outliers Assuming the analyst did not provide plots of the dependent variable versus the independent variables, look for R-squared values that are much higher or lower than expected.
Non-linear relationships Assuming the analyst did not provide plots of the dependent variable versus the independent variables, look for a lower-than-expected R-squared value from a linear model. If there are non-linear terms in the model, this is probably not an issue.
Overfitting Look for a large number of independent variables in the model, especially if they use different types of transformation
Misspecification Look for any variants of the dependent variable in the independent variables. Assess whether the model meets the objectives of the effort.
Multicollinearity Assuming relevant plots and diagnostic statistics are not available, there may not be any way to identify multicollinearity.
Heteroscedasticity Assuming relevant plots and diagnostic statistics are not available, there may not be any way to identify heteroscedasticity.
Autocorrelation Assuming relevant plots and diagnostic statistics are not available, there may not be any way to identify serial correlation.
Weighting Compare the reported number of samples to the degrees of freedom. More DF than samples is usually attributable to weighting.

Follow-up Care

So there are some ways you can identify and evaluate eleven reasons for doubting a regression model. Remember when evaluating other analyst’s models that not everyone is an expert and that even experts make mistakes. Try to be helpful in your critiques, but at a minimum, be professional.


Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at, or other online booksellers.

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How to Write Data Analysis Reports in Six Easy Lessons

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In every data analysis, putting the analysis and the results into a comprehensible report is the final, and for some, the biggest hurdle. The goal of a technical report is to communicate information. However, the technical information is difficult to understand because it is complicated and not readily known. Add math anxiety and the all too prevalent notion that anything can be proven with statistics and you can understand why reporting on a data analysis is a challenge.

The ability to write effective reports on a data analysis shouldn’t be assumed. It’s not the same as writing a report for a class project that only the instructor will read. It’s not uncommon for data analysts to receive little or no training in this style of technical writing. Some data analysts have never done it, and they fear the process. Some haven’t done it much, and they think every report is pretty much the same. Some learned under different conditions, like writing company newsletters, and figure they know everything there is to know about it. And worst of all, some have done it without guidance and have developed bad habits, but don’t know it.

It’s a pretty safe bet that if you haven’t taken college classes or professional development courses, haven’t been mentored on the job, and haven’t done some independent reading, you have a bit to learn about writing technical reports. Report writing is like any other skill, you get better by learning more about the process and by practicing. Here are four things you can try to improve your skills.READ JC 561445_460814310610242_552613593_n

  • Educate yourself. Learn what other people think about technical writing. Visit websites on “statistical analysis reports” and “technical writing,” there are millions of them. Take online or local classes. Read books and manuals. Join Internet groups, such as through Yahoo, Google, or LinkedIn. Immerse yourself in the topic as you did when you were in school.
  • Understand criticism. Over the course of your career, you’ll give and receive a lot of criticism on technical reports. Not all criticism is created equal. First, consider the source. Some critics have never written a report on a data analysis and some have never even analyzed data. Still, if the critic is the one paying the bills you have to deal with it. For your part, you should learn how to provide constructivecriticism. Unless a report you are reviewing is a complete mess, respect the report writer’s discretion for structure and format. Focus on content. Be nice.
  • Download examples. Search the internet for examples of data analysis reports (Hint: adding pdf and download to the search might help). Critique them. Who’s the audience? What’s the message? What’s good and bad about each report? Which reports do you think are good examples? What do they do that you might want to do yourself in the future?
  • Find what’s right for you. When you search the Internet for advice on technical writing or take a few classes from knowledgeable instructors, you’ll hear some different opinions. Everyone will talk about audience and content but most will have more limited views of report organization, writing style, and how you work at writing. Ignore what the experts tell you to do if it doesn’t feel right. Just be sure that the path you eventually choose works for you and the audiences who will read your reports.

If you’ve done all that, it’s just a matter of practice. You’ll learn something from each report you write. If you are new to the process of reporting on a data analysis, consider these six easy lessons:

  • Lesson 1—Know your content
  • Lesson 2—Know your audience
  • Lesson 3—Know your route
  • Lesson 4—Get their attention
  • Lesson 5—Get it done
  • Lesson 6—Get acceptance.

Lesson 1—Know your Content

CONTENT busy-cat-19Start with what you know best. In writing a data analysis report, what you know best would be the statistics, graphing, and modeling you did.

You should be able to describe how you characterized the population, how you generated the data or the sources that provided them, what problems you found in the data during your exploratory analysis, how you scrubbed the data, what you did to treat outliers, what transformations you applied, what you did about dropouts and replicates, and what you did with violations of assumptions and non-significant results.

From that, you’ll need to determine what’s important, and then, what’s important to the reader. Unless you’re writing the report to your Professor in college or your peers in a group of professional data analysts, you can be pretty sure that no one will want to hear about all the issues you had to deal with, the techniques you used, or how hard you worked on the analysis. No one will care if your results came from Excel or an R program you wrote. They’ll just want to hear your conclusions. So, what’s the message you want to deliver? That’s the most important thing you’ll have to keep in mind while writing.

Once you work out your message, write an overview to the report so you’ll know where you’re going. It will help you stay on track. Your summary might take one of three forms:

  • Executive Summary. Aimed at decision makers and people with not enough time or patience to read more than 400 words. Limit your summary to less than one-page, do not use any jargon, and provide only the result the decision maker needs to know to take an appropriate action (i.e., the message you want to convey).
  • Overview. Aimed at most people, whether they would read the report or not. An overview is an abridged version of what is in the report, with a focus on the message you wBLOCK fs-cat-birthday-card-2ant to convey. The overview shouldn’t be more than a few pages.
  • Abstract.  Aimed at peers and other people who understand data analysis. An abstract summarizes in a page or less everything of importance that you did, from defining the population through assessing effect sizes. Abstracts are most often used in academic articles.

Once you understand who your audience is, you can rewrite the summary to catch the attention of your readers.


Lesson 2—Know Your Audience

AUDIENCE many-cat-cats-islandEvery self-help article about technical writing starts by telling readers to consider their audience. Even so, probably few report writers do.

In a statistical analysis, you usually start by considering the characteristics of the population about which you want to make inferences. Similarly, when you begin to write a report on an analysis, you usually start by considering the characteristics of the audience with which you want to communicate. You have to think about the who, what, why, where, when, and how of the key people who will be reading your report. Here are some things to consider about your audience.


Audience is often defined by the role a reader plays relative to the report. Some readers will use the report to make decisions. Some will learn new information from the report. Others will critique the report in terms of what they already know. Thus, the audience for a statistical report is often defined as decision makers, stakeholders, reviewers, or generally interested individuals.

Some reports are read by only a single individual but most are read by many. All kinds of people may read your report. As a consequent, there can be primary, secondary, and even more levels of audience participation. This is problematical; you can’t please everyone. So in defining your audience, focus first on the most important people to receive your message and second on the largest group of people in the audience.


Once you define who you are targeting with your report, you should try to understand their characteristics. Perhaps the most important audience characteristic for a technical report writer is the audience’s understanding of both the subject matter of the report and the statistical techniques being described. You may not be able to do much about their subject matter knowledge but you can adjust how you present statistical information. For example, audiences a data analyst might encounter include:

  • Mathphobes. Fear numbers but may listen to concepts. Don’t use any statistical jargon. Don’t show formulas. Use numbers sparingly. For example, substitute “about half” for any percentage around 50%. The extra precision won’t be important to a Mathphobe.
  • Bypassers. Understand some but have little interest. Don’t worry about Bypassers, they won’t read past the summary. Be sure to make the summary pithy and highlight the most important finding otherwise they might key on something relatively inconsequential.
  • Tourists. Understand some and are interested. Be gentle. Use only essential jargon that you define clearly. Using numbers is fine just don’t use too many in a single table. Round off values so you’re not implying false precision. Stick with nothing more sophisticated than pie charts, bar graphs, and maybe an occasional scatter chart. Don’t use any formulas.
  • Hot Dogs. Know less than they think and want to show it. Using jargon is fine so long as you define what you mean. Even a Hot Dog may learn something. In the same vein, using numbers, statistical graphics, and formulas is fine so long as you clearly explain their meanings. Hot Dogs may come to erroneous conclusions if not guided.
  • Associates. Other analysts who understand the basic jargon. Anything is fine so long as you clearly explain what you mean.
  • Peers. Other data analysts who understand all the jargon. Anything goes.

The audience characteristics provide guidance for report length and writing tone and style


Are readers likely to be very interested in your report or just curious about it (if they have no interest, they won’t be readers)? Be honest with yourself. Why would anyone be interested in reading your report? What is the objective of the who you defined as your audience? What will they do with your findings? Will they get informed? Will they make a decision or take an action? Is this a big thing for them or just something they have to tune in to?

audience Man-Built-a-Sanctuary-for-Homeless-Cats-5Where

Is the report aimed at a finite, confined group, like the organization the analysis was conducted for, or will anyone be able to read it? Is the report aimed at the upper levels of the organization or the rank-and-file (i.e., bottom up or top down)? Are there any concerns for security or confidentiality, either on the individual or organizational levels?


When does the population need to see your report? Who has to review the report and how long might they take before the report is released? How firm are the deadlines? How much time does this leave you to write the report? Will there be enough time to think through what you need to write? Will there be time to conduct additional analyses needed to fill in gaps in the report outline? Will you be outraged when the time taken to review your report is twice as long as the time you took to write it?

Here’s some advice you should take to heart. Never, never, never submit a draft report for review that isn’t your fully complete, edited, masterpiece. I tell myself to follow this rule with every report I write. Unfortunately, like most people, I don’t listen to what I say.


Finally, consider how the report should be presented so that the audience will get the most out of it. Here are five considerations:

  • Package. How will your writing be packaged (i.e., assembled into product for distribution)? Will it be a short letter report, a  comprehensive report, a blog or an Internet article, a professional journal article, a white paper, or will your writing be included as part of another document?
  • Format. Will your report be distributed as an electronic file of as a paper document? If it will be an electronic document, will it be available on the Internet? Will it be editable? Will it be restricted somehow, such as with a password?
  • Appearance. Will the report be limited to black-and-white or will color be included? What will be the ratio of graphics to text? Will the report be conventional or glitzy, like a marketing brochure? Will there be 11”x17” foldout pages or oversized inserts like maps.
  • Specialty items. Will you need to provide some items apart from the report, such as electronic data files, analysis scripts or program codes, and outputs? Will you have to create a presentation from the contents of the report? Will your graphics be used for courtroom or public presentations?
  • Accessibility. Do you need to follow the guidelines of Section 508 of the Rehabilitation Act of 1973, which may affect your use of headings, tables, graphic objects, and special characters? Should you account for common forms of color blindness in your color graphics?

X images (5)Take a Few Moments

You won’t have to address all of these details in evaluating your audience and many will only require a few moments of thought. But, if you think through these considerations, you’ll have a much better idea of who you are writing the report for and how you should write it.


Lesson 3—Know Your Route

ROUTE Cat in a maze

You’ve been taught since high school to start with an outline. Nothing has  changed with that. However, there are many possible outlines you can follow depending on your audience and what they expect. The first thing you have to decide is what the packaged report will look like.

Will your report be an executive brief (not to be confused with a legal brief), a letter report, a summary report, a comprehensive report, an Internet article or blog, a professional journal article, or a white paper to name a few. Each has its own types of audience, content, and whiting style. Here’s a summary of the differences.

Report tableWriting a report is like taking a trip. The message is the asset you want to deliver to the ultimate destination, the audience. The package is the vehicle that holds the message. Now you need a map for how to reach your destination. That’s the outline.

Just as there are several possible routes you could take with a map, there are several possible outline strategies you could use to write your report. Here are six.

  • The Whatever-Feels-Right Approach. This is what inexperienced report writers do when they have no guidelines. They do what they might have done in college or just make it up as they go along. This might work out just fine or be as confusing as The Maury Show on Father’s Day. Considering that the report involves statistics, you can guess which it would be.
  • The Historical Approach. This is another approach that inexperienced report writers use. They do what was done the last time a similar report was produced. This also might work out fine. Then again, the last report may have been a failure, ineffective in communicating its message.
  • The “Standard” Approach. Sometimes companies or organizations have standard guidelines for all their reports, even requiring the completion of a formal review process before the report is released. Many academic and professional journals use such a prescriptive approach. The results may or may not be good, but at least they look like all the other reports.
  • The Military Approach. You tell ‘em what you’re going to tell ‘em, you tell ‘em, and then you tell ‘em what you told ‘em. The military approach may be redundant and boring, but some professions live by it. It works well if you have a critical message that can get lost in details.
  • Cat-on-a-MapThe Follow-the-Data Approach. If you have a very structured data analysis it can be advantageous to report on each piece of data in sequence. Surveys often fall into this category. This approach makes it easy to write the report because sections can be segregated and doled out to other people to write, before being reassembled in the original order. The disadvantage is that there usually is no overall synthesis of the results. Readers are left on their own to figure out what it all means.
  • The Tell-a-Story Approach. This approach assumes that reading a statistical report shouldn’t be as monotonous as mowing the lawn. Instead, you should pique the reader’s curiosity by exposing the findings like a murder mystery, piece by piece, so that everything fits together when you announce the conclusion. This is almost the opposite of the follow-the-data approach. In the tell-a-story approach, the report starts with the simplest data analyses and builds, section by section, to the great climax—the message of the analysis. Analyses that are not relevant to the message are omitted. There are usually arcs, in which a previously introduced analytical result is reiterated in subsequent sections to show how it supports the story line. Graphics are critical in this approach; outlines are more like storyboards. There may be the equivalent of one page of graphics for every page of text. Telling a story usually takes longer to write than the other approaches but the results are more memorable if your audience has the patience to read everything (i.e., don’t try to tell a story to a Bypasser.)

So59502, be sure that you have an appropriate outline but don’t let it constrain you. Having a map doesn’t mean you can’t change your route along the way, you just need to get to the destination. In building the outline, try to balance sections so the reader has periodic resting points. Within each section, though, make the lengths of subsections correspond to their importance.

Lesson 4—Get Their Attention

If you’re writing a report about statistics, you have to expect that many readers will lose interest after a while, if they even had it to begin with. So, in writing the report, think about how you might engage your audience. Here are five ideas.

  • Find Common Ground.  Every relationship begins with having something in common. Fighting a common foe or solving a common problem can form the strongest and longest lasting of bonds. So the first thing you should try to establish in your report is that common ground. This isn’t so difficult if you are working on an analysis at the behest of a client. The client is already immersed in the data and has invested in you to help solve the problem. Establishing common ground is not so easy if you are proffering an uninvited message. Some people, perhaps subconsciously, don’t really want the message you are offering, especially when you’re analyzing data in their area of expertise. Try to establish common ground in other areas. Perhaps your analysis touches on a similar or analogous issue the reader might have. Maybe the analysis procedure could be used on a different problem the reader might have.
  • Clear the Decks. Get rid of everything that doesn’t add to the progression of the report. That doesn’t necessarily mean you have to omit the content. You can relegate it to an appendix, which is pretty much the same thing. Unless required to be in the body of the report, things like the data, data collection surveys and forms, and scrubbing and analysis procedures should all be put in an appendix.
  • the Tone. Your writing style can either add to or detract from the readability of your report. A formal tone, with strict adherence to grammar rules, complex sentence structures, use of third-person point-of-view and passive voice, and plentiful jargon, is appropriate for most data analysis reports. Formal tones are good for describing details, specifications, and step-by-step instructions. However, formal tones can be more difficult to understand, especially for individuals not accustomed to reading technical reports. An informal tone, with simple grammar and vocabulary, colloquialisms, contractions, analogies, and humor, works well for blogs. Informal tones are good for discussing ideas and concepts, and for inspiring readers or communicating a vision. They are more engaging and tend to be easier for most individuals to understand. If you’re being paid to write the report, a formal tone is usually more appropriate. This is problematical, of course, because formal writing is usually harder to read and maintain an interest in.
  • Add Mind Candy. A Harry Potter novel consisting of page-after-page of text will keep readers, young and old, transfixed for hours. A data analysis report consisting of page-after-page of text will put readers into a coma faster than a handful of barbiturates taken with a glass of warm milk in a tub of hot water while meditating. The difference is that the novel engages readers with mental images. Data analysis reports need to use visual imagery, which for the most part means good graphics. Granted, most readers won’t understand anything more complicated than a pie chart or a bar chart, but don’t add to the confusion. Three-dimensions are a no-no. Avoid graphing data in more than a few categories to avoid making the slices and bars uninterpretable. And most importantly, make sure they add to the analysis. You can do more, too. Break up the text with subheadings and bullets. Reiterate information nuggets in boxes instead of just letting them get lost in the text. Use tables for explaining differences in data groups and not just for number buckets. Add footnotes or hyperlinks to explain collateral concepts.

July 22 2013 028

  • Make it Better. Just when you think you’re done writing, you’re not. That’s the time when you have to do even more to make the report better. First, take some time off if you can. Then, read it through again making improvements along the way. Read it aloud if you need to, even record it when you read it aloud and then play it back so you can engage both your vision and hearing. Consider getting a second opinion, especially if you can’t distance yourself from the report by setting it aside for a few days. A second opinion may come from a data analysis peer, but don’t ignore nontechnical editors. A good editor can help with spelling, grammar, punctuation, word choice, style and tone, formatting, references, and accessibility. It’s usually worth the effort. This is the time to go for purrfection.

Lesson 5—Get It Done

Source: and  309 other sources.Perhaps the hardest part of writing a data analysis report is just getting it completed. It takes discipline and persistence to stay on track. Even so, it’s easy to get distracted. Sometimes the problem is that the story of the analysis hasn’t been thought all the way through. Sometimes there are gaps in the analysis that necessitate stopping to complete more calculations. Sometimes there are too many interruptions and distractions to maintain focus. Sometimes, the process of writing becomes boring and requires a great effort to continue.

Writer’s block is an impediment experienced by all writers. Writer’s block might be attributable to not knowing what to write next, trying to write text that is perfect, or fear of failure. Any of these reasons may be applicable to the report writer. Here are ten ways to fight off writer’s block.

1. Stick with a routine. Keep writing even if you are dissatisfied with what you’ve written. You can, and should, edit your draft after you’re done. Try to identify your productivity tipping point. For some people, accomplishing a specific goal by a certain time in a day helps ensure the rest of your day is productive. For example, my productivity tipping point is beginning to write by 8AM. If I do, I’ll be writing productively all day.

2. Visualize. If you’ve never used visualization techniques before, now is a good time to develop the skill. The idea is to close your eyes, get relaxed, and think about what you want to do or see. Start by visualizing what the next few sentences you have to write might look and sound like. Eventually, you’ll be able to visualize what paragraphs, sections, and even the entire final product will look like.

Source: and 33 other sites.3. Eschew perfection. If it’s not perfect the first time you write it, leave it alone. Let it age while you write the rest of the report. You can reevaluate and rewrite it later when you know more about the rest of the report.

4. Write in parallel. Some parts of reports, like introductions and summaries, and descriptions of variables and other details, are almost formulaic. Write all the similar parts at the same time. Set up a second file in your word processing software to serve as a staging area for the repeated parts. Then, copy and paste the standardized parts to your report and edit the text as appropriate.

5. Grow the outline. Instead of trying to write the report section by section, try using the outline as a template rather than a map. Add key phrases, instructions, notes, sentences, and even paragraphs to the template-outline. You can skip around the template-outline as you come up with ideas for what to write. Eventually, you can consolidate these ideas into paragraphs and then sections. Continue to expand the template-outline until it ultimately becomes the complete report.

6. Tiptoe through the tables. Create all or most of your graphics (i.e., tables and figures) before starting to write. Lay the graphics out in your word processing software and write the text that would go with each graphic. Then, go back and fill in the gaps between graphics. Continue joining the pieces until the report is complete.

7. Chunk it up. Don’t try to write the entire report by yourself. Break it up into pieces and get help.

8. Set deadlines. Sometimes it helps to be able to work towards an interim goal. Set deadlines for sections or other tasks you have to accomplish. Make them challenging but achievable.

9. Give it a restSource: Absence makes the mind grow sharper. Consider taking some time off from report writing, but make sure you use the time productively. Schedule that colonoscopy you’ve been putting off. Clean the garage and paint the house. Visit your in-laws. Don’t just play video games or watch Netflix.

10. Do something different. If your routine isn’t working, try doing something different. If you can’t get anywhere because you’re pressing, work on something else or take some time off. If you can’t get anywhere because you’re slacking, try researching. If you can’t get anywhere because you’re stuck on writing, pull together graphics or the appendices. If you can’t get anywhere because you’re procrastinating, ask yourself why.


Lesson 6—Get Acceptance

Source: and 1,196 other sites.Data analysis reports have to go through one more hurdle after they are completely written. They have to be approved for acceptance by a gatekeeper. The approval for acceptance may involve allowing report distribution, starting the publishing process, issuing payment for your services, or just acknowledging that your work is done. The gatekeeper may be your client, your supervisor, your publisher, or for blog writers, you. To get that approval, formal reports usually have to be reviewed by reviewers. Reviewers are usually individuals the gatekeeper chooses based on their technical background or role in the gatekeeper’s organization. Sometimes, reviewers are individuals the gatekeeper is forced to listen to, like regulatory reviewers. In academic publishing, you may not even know who the peer reviewers are.

Logically, the acceptance review shouldn’t take too long compared to the time you took to analyze the data and write the report. After all, the reviewers only have to read it. In practice, though, reviews take far longer than report preparation. The report you wrote in a month may take six months to be reviewed. Don’t panic. It’s just the way things seem to happen.

The number of comments you get from the reviewers is inconsequential. Great reports can get dozens of highly critical comments. Again, don’t panic. The only review you should be concerned about is the one that provides no comments. That usually signals a lack of interest by the reviewers and the gatekeeper.

Six Tips for Responding to Reviewers from Editage Insights

When the review is complete, be sure to get the comments in writing. If you don’t, some comments may be forgotten or misunderstood. If there is more than one reviewer, compile all the comments together. This is essential because sometimes reviewers provide conflicting comments. The gatekeeper may compile the comments for you if he or she wants to control the process. The comments should be placed in the order they correspond to in the report. Be sure to identify the source of each comment. If a single comment has many parts, break the comment apart so you can respond to each part individually.

Then comes the challenging part—you must respond to each comment separately. Create a new document listing all the compiled comments. For each comment in this document, either describe what you’ll do in response or explain why you won’t make any changes. Start with the easy comments, such as those involving grammar and spelling. As you describe your response to a comment in the document, make the associated change in the report. Proceed through increasingly more difficult comments until you are done. For very complex comments, try to parse the ideas and respond to each separately. If a particular comment is very difficult to address, you may have to conduct additional analyses or information research. Cite information sources if appropriate.

When you’re done, reread both the response document and the changes in the report. Be sure all the changes were made in the report and that they are consistent with the rest of the report. Also, make sure the tone of your response is even; be stoic.

Nine Tips for Responding to Criticism by Alain Briot

If you’ve written an informal piece, like a blog, you don’t have to go through the grueling process of responding to formal comments from an acceptance review. Since you are the gatekeeper, you can release your blog whenever you feel it is done. But after you release the blog, you may well get comments. That’s good because it shows that people are reading your blog. Furthermore, there’s no pressure to compile these comments and document your responses. Unfortunately, at least some of the comments will come from spammers, trolls, 13-year-olds, head cases, angry arguers, and other individuals who won’t be providing constructive criticism. Therefore, first consider the source of each comment. In some cases, you won’t have to respond to any of them. Your blogging software will allow you to delete unwelcome comments. Beware of the overly gracious comments, too. Sometimes malicious commenters use addresses that link to spam or malware. If you don’t trust your instincts, just delete the comment.

Source: and 2 other sites.Don’t get upset by reviewers pointing out flaws in your report. That’s what they’re supposed to do. Having been on both sides of the writer/reviewer divide, I can tell you that creating a report takes a hundred times more knowledge, creativity, effort, and time than reviewing a report. Providing constructive criticism on a report requires a hundred times more experience, situational awareness, and interpersonal sensitivity than creating a report. Good writing combined with constructive reviewing makes a data analysis report the best it can be.

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