AP Stats 1.9: Comparing Distributions

Comparing two or more distributions

Topic 1.9 is about comparing two or more distributions of the same quantitative variable, and about z-scores for relative position. To compare distributions, use histograms, back-to-back stem-and-leaf plots, or dotplots, which let you compare center, variability, shape, outliers, clusters, and gaps. Boxplots compare center, variability, outliers, and skewness or symmetry, though they do not show clusters or gaps.

Any comparison can also cite numerical summaries such as the mean or standard deviation. The rule for full credit is to compare, not just describe: use words like higher, lower, or more spread out, and always name the context and the direction of the difference. Describing each distribution separately and stopping there is the most common way to lose these points.

z-scores for relative position

A z-score, or standardized score, measures how many standard deviations a value falls above or below the mean. With population parameters it is

where $\mu$ is the population mean and $\sigma$ is the population standard deviation. A positive z-score sits above the mean and a negative one below. When $\mu$ and $\sigma$ are unknown, you may use the sample mean and standard deviation instead.

Because a z-score strips away the units, it lets you compare relative positions within one distribution or across two different distributions. A value 1.5 standard deviations above its own mean ranks higher than one 0.8 above, even if the raw numbers are on different scales. See how to find a z-score.

Writing a good comparison

A strong comparison names a feature, gives the direction, and stays in context. Rather than describing each distribution on its own, use comparative words: one center is higher than the other, one distribution is more spread out, one is skewed while the other is symmetric. Support each point with a graph feature or a numerical summary such as a difference in medians.

z-scores add a second kind of comparison, of individual values rather than whole distributions. Standardizing puts two values from different distributions on one scale, so you can say which value ranks higher relative to its own group. Both tools answer the same question of relative position, one for distributions and one for single observations.