A New View of Statistics

© 2000 Will G Hopkins

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Summarizing Data:

All the statistics we've met so far summarize a set of values of a single variable. But it's possible to have statistics describing the relationship between two or more sets of numbers. These are the statistics that really matter in research.

There is no agreed generic name for these statistics. I've seen measures of effect in the literature, so let's call them effect statistics. The effect refers to the idea that one variable has an effect on another. The main effect statistics are the difference in means (this page), the correlation coefficient (next page) and relative frequency (following page). Each of these effect statistics comes in several varieties, I close this section with a page on a scale of magnitudes for effect statistics

 Difference in Means
Sometimes you can express the finding of a study simply as a difference between the means of two groups, in the original raw units of measurement. For example, a group of elite male runners has a mean body mass of 66.4 kg, whereas a subelite group weighs in with an average of 68.5. The difference of 2.1 kg is the effect.

Depending on the variable, you often want to talk about the difference between means as a percent difference. In the above example, you could say that the subelite runners are 3.2% heavier than the elites (2.1/66.4 = 0.032 = 3.2%). Percent differences are a natural way to express differences in the mean of variables that need log transformation. Percent effects are particularly appropriate for measures of athletic performance.

Converting the difference to a percent is one way to make the difference dimensionless, and therefore more generic. Another important way is to express the difference as a fraction or multiple of a standard deviation. Work out the difference between the means, then divide it by the average standard deviation for the two groups. What you end up with is the standardized difference in the means, a number that represents "how many standard deviations" the two groups differ by. Look closely at the imaginary example in the figure and work out the effect size for the difference in body fat between boys and girls. Answer: one unit, or 1.0, or one standard deviation. Note that the unit of measurement for body fat is irrelevant. I've shown it as the usual % of body mass, but it could be kg or pounds--the effect-size statistic has the same value. The effect-size statistic is appropriate for studies of population health, where differences or changes in means that impact the average person are paramount.

The standardized difference in the means is sometimes known simply as the effect-size statistic, although this term confuses the concept with the magnitude of other kinds of effect. A page on this topic comes up shortly. Meanwhile, to get you thinking about it, how big is the effect shown in the above figure? This difference of one standard deviation has been regarded as large, although I now think it's only a moderate effect. Anything less than 0.2 standard deviations isn't worth worrying about.

The example above is for two groups of subjects, but you should also use the concept of effect size when looking at changes in the mean as a result of an experiment. For example, the above two bars could represent muscle mass before and after treatment with anabolic steroids. In this case you use the SD of the pre scores only to standardize the effect. (I'd figured this out years ago, but until Oct 06 I missed a mistaken assertion on this page that you average the pre and post SDs. Sorry about that.) Some people think mistakenly that you should use the SD of the change scores to standardize effects in experiments. If you have a control group as well, you use the SD of all the pre scores, and you subtract the change in the control group from the change in the experimental group to get the magnitude of the experimental effect.

When you understand effect sizes, you'll know why you should always show standard deviations rather than standard errors of the mean with means. See the page devoted this important issue.

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Last updated 2 Oct 06