Shapes of distributions

The 25 weekend gaming times in an OpenStax table have a median of 17.5 hours and a mean of 15.55: a long tail toward 0 pulls the mean down.

Once a distribution is drawn, the first thing to describe is its shape. Statisticians use a small vocabulary for this: symmetric or skewed, with one peak or two, with or without gaps and outliers. Shape matters because it decides which summaries make sense. The mean is a fine centre for a symmetric pile of values and a misleading one for a lopsided pile.

Symmetric

A symmetric distribution looks roughly the same on both sides of its centre. Quetelet's 5,738 chest measurements of Scottish soldiers are a classic example: they peak at 39 and 40 inches and fall away on both sides. Their mean is 39.83 inches and their median is 40, almost the same number, which is what symmetry predicts. The 100 OpenStax soccer heights are close to symmetric too, with mean 67.26 and median 67 inches.

Skewed left and skewed right

A skewed distribution has a long tail on one side, and the skew is named after the tail, not the peak. The 22 New York restaurants in OpenStax have between 15 and 42 employees; half of them, 11, have 20 to 29, and the larger ones, up to 42, stretch the right tail. That is skewed right, and the mean, 28.09, sits above the median, 26.5. The weekend gaming times of 25 classmates go the other way. Most spent 15 to 24 hours, a handful spent very little, and one spent 0. The tail points left, and the mean, 15.55, sits below the median, 17.5. The mean follows the tail because every value pulls on it, while the median only cares about the middle position.

Uniform and bimodal

A uniform distribution has no peak: every value is about equally common, like the faces of a fair die after many rolls. A bimodal distribution has two peaks, which is often a sign that two different groups have been mixed. Heights measured in a room that holds both adults and young children would show one hump for each group, and summarising them with one mean would describe nobody in the room. When you see two peaks, ask whether the data should be split.

Describing a distribution in one sentence

A good description names four things: shape, centre, spread and anything unusual. For the 100 OpenStax heights it might read: roughly symmetric with one peak, centred near 67 inches (mean 67.26, median 67), spread from 60 to 74 inches with a standard deviation of about 2.89, and four short players (60, 60.5, 61 and 61 inches) below the lower outlier fence of 61.125 from the averages course. For the gaming hours: skewed left, median 17.5 hours, spread from 0 to 24, with the 0 about 2.2 standard deviations below the mean but inside the fences. A sentence like that carries more than a single average ever can.