AP Stats 1.6: Describing Distributions

Shape, center, spread, and unusual features

Topic 1.6 is about putting a quantitative distribution into words: the CED asks for shape, center, and variability (spread), plus any unusual features such as outliers, gaps, or clusters, always in context. Many students remember this as SOCS: shape, outliers, center, spread. Center and spread get their numbers in topic 1.7; here the focus is reading a graph and describing what you see. Every part of the description should be tied to the variable and its units, because a shape or a center with no context is only half an answer.

Naming the shape

Shape has two parts, skew and modes.

• A distribution is skewed right (positively skewed) if the right tail, toward larger values, is longer than the left. It is skewed left (negatively skewed) if the left tail is longer. It is approximately symmetric if the left half is roughly the mirror image of the right.
• A distribution with one main peak is unimodal, one with two prominent peaks is bimodal, and one where every frequency is about the same with no prominent peak is approximately uniform. Real data are rarely perfectly symmetric, so approximately is the right word for most shapes you will meet.

The direction of skew is named for the long tail, not the tall side, which is the reversal students most often get backward. See skewed left vs skewed right.

Outliers, gaps, and clusters

Three unusual features are worth naming. An outlier is a value that is unusually small or large relative to the rest of the data. A gap is a region in the distribution with no observed values. A cluster is a concentration of values, usually separated from other values by gaps.

Describe every feature in context, naming the variable and its units, and tie the description to the question being asked, since a graph of a quantitative variable may reveal information you can use to justify a claim. The most common lost point here is stopping after center: a full description also states shape, spread, and any outlier. Outliers are worth a second look because they can point to a data-entry error, a special case, or a genuinely unusual individual, and each of those leads to a different next step. Build the habit of covering every feature in the descriptive statistics sandbox.