Shape of a Distribution vs Skewness

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.

Skewness is a statement about the tail. A distribution is skewed right when the values thin out gradually toward the high end, and skewed left when they thin out toward the low end. The name follows the tail, never the peak, so a right-skewed graph has its tall bars on the left and its thin straggle on the right.

Take 1, 2, 3, 4, 5, 6, 21. The bulk sits between 1 and 6 with one value far above, so the shape is skewed right. The median is 4, the fourth of seven ordered values. The mean is $42/7 = 6$, and only the 21 sits above it. The long tail pulled the mean toward it and left the median alone, because the median counts only how many values lie on each side of it.

The sentence to unlearn is "it is skewed right because most of the data are on the right." That describes a left skew. The pile shows where values are common; the skew is named for where they run out. Trace the tail with a finger and read off the direction it points.

The mean landing above the median is a consequence of right skew, not its definition, and it is a strong tendency rather than a theorem: statisticians have built right-skewed distributions whose mean falls below their median. Use the comparison to confirm a shape you have already read off a graph, not to decide the shape without one.

Two boundaries. Skew is a word for single-peaked quantitative distributions, so a graph with two clear peaks gets called bimodal instead, and one distant point in an otherwise balanced set is usually reported as roughly symmetric with an outlier, since skew describes a tail that thins out gradually rather than a lone gap. And skew means nothing for categorical data: the bars of a bar graph can be reordered at will, so any lopsidedness you see there is an artifact of the order you chose. Shape vocabulary is Unit 1 topic 1.6.