In school, you probably had to line up by height now and then. That wasn’t too difficult. There weren’t too many individuals being lined up and they were all in the same place at the same time. An individual’s place in line was decided by comparing his or her height to the heights of other individuals. The comparisons were visual; no measurements were made. Everyone made the same decisions about the height comparisons. You didn’t need statistics to solve the problem. So why might you ever need statistics to compare heights?
Statistics are primarily concerned about groups of individuals or items, especially those having some fundamental commonalities. These groups are called populations. Populations are more difficult to compare than pairs of individuals because you have to define the population and measure the characteristics of the phenomenon that you want to compare. Statistical comparisons, or statistical tests as they are usually called, can involve one population (comparing a population phenomenon to a constant), two populations (comparing a population phenomenon the phenomenon in another population), or three or more populations . You can also compare just one phenomenon (called univariate tests) or two or more phenomena (called multivariate tests). You can test if the phenomena are equal (called a nondirectional or two-sided test) or less then/greater then (called a directional or one-sided test).
For example, you might want to compare the heights of male high school freshmen in two different school districts. There would be a two-population test – male high school freshmen in school district 1 and male high school freshmen in school district 2. The phenomenon you want to compare is the height of the two populations. But, it’s not as easy as just visually comparing the heights of pairs of individuals because they are not located in the same place. You have to measure at least some of the heights of the individuals in the two populations.
Fortunately, you don’t have to measure every individual in the population so long as you measure a representative sample of the individuals in the populations. You can improve your chances of getting a representative sample by using the three Rs of variance control — Reference, Replication, and Randomization.
How many samples should you have? No, the answer isn’t as many as possible. Some people think the answer is 30 samples, but that’s a myth based on a misunderstood tradition. Like potato chips and middle managers, too many can be as bad as not enough. It’s a matter of resolution.
If you were comparing two individuals, you would only be concerned with whether one height is greater than, equal to, or less than the other height. When you’re comparing populations, there’s not just one height but many, and you only know what some of the heights are (hopefully a representative sample of them). That’s where distributions come in.
In statistical testing, a frequency distribution refers to the number of times each value of the measured phenomena appears in the population. A bar chart of these values, with the values on the horizontal axis and their frequencies on the vertical axis, is called a histogram. Histograms often looks like a bell, which is why they are called bell curves.
Measured phenomena that have a histogram that looks like a bell curve have many values located at the middle of the distribution and fewer values farther away from the center, called the tails. The center of the distribution of values is estimated by the average. The variability of the values, how far they stretch along the horizontal axis, is estimated by the variance or the standard deviation, the square root of the variance.
A bell curve is usually assumed to represent a Normal distribution. The average and the variance of the values are called parameters of the distribution because they are in the mathematical formula that defines the form of the distribution.
Having a mathematical equation that you can use as a model of the frequency of phenomenon values in the population is advantageous because you can use the distribution model to represent the characteristics of the population.
Once you have data on the phenomenon from the representative sample of the population, you calculate descriptive statistics for the population. Statistical comparisons consider both the accuracy (i.e., the difference between the measured heights and the true heights in the population of individuals) and the precision (i.e., how consistent or variable are the heights) of the measurements of the population. Formulas for statistical tests usually involve some measure of accuracy divided by some measure of precision.
Statistical tests that compare the distributions of population characteristics are called parametric tests. They are usually based on the Normal distribution and involve using averages as measures of the center of the population distribution and standard deviations as measures of the variability of the distribution. (This is not always the case but is true most of the time.) The average and standard deviation are called test parameters. You can still test whether population means are equal (called non-directional or two-sided tests because differences in both tails of the Normal distribution are considered) or less then/greater then (called directional or one-sided tests because the difference in only one tail of the Normal distribution is considered).
Statistical tests that don’t rely on the distributions of the phenomenon in the populations are called nonparametric tests. Nonparametric tests usually involve converting the data to ranks and analyzing the ranks using the median and the range.
And, there is still a lot more to know about statistical comparisons … more to come
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