Distribution Shape: Skewness and Kurtosis

Centre and spread are two numbers. But two columns can share both and still look nothing alike: one a neat hump, the other a pile of ordinary days with a few wild ones. The shape of a distribution tells you which, and it decides whether the mean and standard deviation can be trusted at all.

The rule

Skew points to the long tail; kurtosis weighs the tails.

Skewness describes symmetry.

• Symmetric (skewness near 0): the two sides mirror each other. Mean ≈ median.
• Right-skewed, or positively skewed: most values are low, with a long tail stretching to the right, towards high values. The tail drags the mean above the median. Delivery times, incomes and the café's daily sales all tend to look like this.
• Left-skewed, or negatively skewed: the long tail stretches left, towards low values, and the mean sits below the median. Scores on an easy test often look like this: most people score high, a few score very low.

A common sign of right skew: mode < median < mean. For left skew the order usually reverses. It is a rule of thumb, not a law, but it holds for most everyday data.

Kurtosis describes the tails: how often extreme values happen compared with a normal, bell-shaped curve. Software usually reports excess kurtosis, which is 0 for a normal curve.

• Positive excess kurtosis (heavy tails): extreme values turn up more often than a bell curve predicts.
• Negative excess kurtosis (light tails): values stay within a narrower band, with few surprises.

For the week (36, 38, 38, 40, 43, 46, 95): mode 38 < median 40 < mean 48, so it is right-skewed. Excel's =SKEW(B2:B8) gives about 2.5 and =KURT(B2:B8) about 6.4, both strongly positive: one far-out day in a short list.

The shortcut

You rarely need the exact skewness number. Compare the mean with the median, and look at which side the stragglers are on. The tail is where the stragglers are; the skew is named after the tail.

The trap is naming skew after the hump. In a right-skewed column the bulk of values sits on the left; the name follows the tail.

Drills

Answer fast — trust the rule.

Remember

• Skew points to the long tail; kurtosis weighs the tails.
• Shortcut: mean above median means right skew; name the skew after the stragglers.
• Trap: naming skew after the hump; the bulk of a right-skewed column sits on the left.