Shape of a distribution

Shape has two parts, and an answer giving only one of them is half an answer. First, symmetry or skew: approximately symmetric when the left half roughly mirrors the right, skewed right when the tail toward larger values is longer, skewed left when the tail toward smaller values is longer. Second, the number of peaks: unimodal for one, bimodal for two prominent peaks, approximately uniform when no peak stands out. A complete shape therefore reads like "unimodal and skewed right".

Ten daily absence counts: 1, 1, 2, 2, 2, 3, 3, 4, 6, 11. The bulk sits at 1 to 3 and the values thin out toward 11, so the long tail points right and the shape is unimodal and skewed right. The mean is 3.5 and the median is 2.5, and it is that thin upper tail pulling the two apart.

"The tall bars are on the left, so it is skewed left." Skew is named for the tail, never for the bulk. The values pile up on the left in this set precisely because the long stretched-out end is on the right, so it is skewed right. Point along the thin end and read the direction off your finger. Positively skewed means the same as skewed right, and negatively skewed means skewed left.

The mean-above-median signal is a tendency, and the course framework words it that way: in a right-skewed distribution the mean is usually larger than the median. Usually is not always. In 3, 3, 3, 4, 12, 14, 16, 17, 30 the largest value sits 18 above the median while the smallest sits only 9 below it, and yet the mean of 11.33 falls below the median of 12. Name the shape from the graph and use the gap as a check on it, never as a replacement for it.

Shape is topic 1.6 of the Fall 2026 course and returns in topic 1.9 for comparing two distributions. It also picks your summaries: approximately symmetric with no outliers earns the mean and standard deviation, while anything skewed or carrying an outlier earns the median and IQR.