Читать книгу Applied Univariate, Bivariate, and Multivariate Statistics - Daniel J. Denis - Страница 79

Writing out a formula and plugging in numbers, unfortunately, does not necessarily give us a feeling for what the formula actually means. This is especially true with regard to Cohen's d. We now discuss the statistic in a bit more detail, pointing out why it is usually interpreted as the standardized difference between means.

Imagine you have two independent samples of laboratory rats. To one sample, you provide normal feeding and observe their weight over the next 30 days. To the other sample, you also feed normally, but also give them regular doses of a weight‐loss drug. You are interested in learning whether your weight‐loss drug works or not. Suppose that after 30 days, on average, a mean difference of 0.2 pounds is observed between groups. How big is a difference of 0.2 pounds for these groups? If the average difference in weight among rats in the population were very large, say, 0.8 pounds, then a mean difference of 0.2 pounds is not that impressive. After all, if rats weigh very differently from one rat to the next, then really, finding a mean difference of 0.2 between groups cannot be that exciting. However, if the average weight difference between rats were equal to 0.1 pounds, then all of a sudden, a mean difference of 0.2 pounds seems more impressive, because that size of difference is atypical relative to the population. What is “typical?” This is exactly what the standard deviation reveals. Hence, when we are computing Cohen's d, we are in actuality producing a ratio of one deviation relative to another, similar to how when we compute a z‐score, we are comparing the deviation of y − μ with the standard deviation σ. The extent to which observed differences are large relative to “average” differences will be the extent to which d will be large in magnitude.