Skewed Distributions

The name follows the tail, not the hump

This is where nearly everyone goes wrong first. A right-skewed distribution has most of its data on the left and a thin tail reaching right; it looks left-heavy, and calling it left-skewed feels natural. The convention names the direction the tail points, because the tail is what makes the distribution unusual.

The alternative names help, because they attach a sign instead of a direction. Positive skew means the tail runs towards the larger, more positive values; negative skew means it runs towards the smaller ones. If you can remember which end of a number line is positive, you can recover the whole convention without remembering anything else.

What examiners actually ask

Questions rarely stop at naming the shape. They ask what the skew does to the averages, or hand you a mean and a median and ask you to deduce the shape — which is the same fact used backwards. Three cases cover it:

Read the middle case carefully. Mean ≈ median tells you the distribution is roughly symmetric; it does not tell you it is normal, or that it has one peak, or anything about how spread out it is. A class where half revised and half did not produces two separate humps, and the mean and median settle neatly between them, in a gap where almost nobody actually scored.

• Left, or negative, skew — long tail towards the low values: mean < median
• Roughly symmetric — the two tails match: mean ≈ median
• Right, or positive, skew — long tail towards the high values: mean > median

Why the mean moves and the median does not

The mean is built from the size of every value, so a value far out in a tail contributes its full distance and drags the mean towards it. The median is built only from position: the extreme value counts as one item at the end of the queue, and moving it further out does not change which item is in the middle. That asymmetry is the whole mechanism, and it is worth being able to say it in a sentence, because that is what an explanation mark is for.

One honest caveat. The mean-and-median ordering is reliable for the single-peaked distributions that school and SAT questions use, and it is what mark schemes expect, but it is a rule of thumb rather than a theorem — statisticians can construct data sets where a right-skewed shape has its mean below its median. Nothing you meet before university will be one of them.

A box plot shows the same information without a mean anywhere. If the median line sits nearer the bottom of the box and the upper whisker is the longer one, the data is right-skewed; if the median sits high in the box with a long lower whisker, it is left-skewed. A symmetric distribution puts the median in the middle with whiskers of similar length.