Distribution Shape and Outliers

Beyond averages and spread, data has a shape. A symmetric distribution is balanced, with a peak in the middle and the two sides roughly mirroring each other. A positively skewed distribution has a long tail stretching to the right, with most values bunched on the left. A negatively skewed one has its tail to the left. The naming catches people out, because skew is named after the tail, not the peak — so a positively skewed graph actually has its peak on the left. Real examples help: house prices and incomes are positively skewed, since most are modest and a few are enormous. An easy test that most students pass is negatively skewed.

Two more features are worth naming. A cluster is a group of values bunched together, and a gap is a stretch of the range where nothing appears. Look at ages at a school concert: one cluster around 8 to 12, another around 36 to 44, and a gap between them. Two clusters nearly always means two different groups have been mixed together — here students and their parents. And this is where a single average goes badly wrong: the mean age is about 24, which is an age nobody at the concert actually is. So when the data shows two clusters, describe both rather than quoting one average that describes neither.

An outlier is a value clearly separated from the rest of the data. In 12, 14, 15, 15, 16, 17 and 48, the six smaller values sit close together and 48 stands well apart. Look at what it does. The mean with the 48 is 137 over 7, about 19.6 — dragged above every single value except the outlier itself. Without it, the mean is 89 over 6, about 14.8, which describes the six values well. Now the median: with the 48 it is 15, and without it, still 15. Completely unchanged. The mean moved by nearly 5 while the median did not move at all, which is exactly why the median is preferred when outliers are present.

Finding an outlier does not tell you what to do with it — you have to investigate first. If it is an error, remove it: a height recorded as 1750 centimetres, an age of 200, or a typing slip of 48 for 4.8. Those values are impossible or obviously mistyped, and you should say that you removed one and why. But if it is genuine, keep it: one very tall student, one mansion in the street, one exceptional score. Those are unusual but real, and often the most interesting part of the data — the faulty machine, the record-breaking athlete. In that case use the median rather than the mean. What you must never do is delete a value just because it is inconvenient. That is dishonest, not maths.

When a question says 'describe the data', it wants four things every time. Centre: which average you used and its value. Spread: the range, and whether that means consistent or variable. Shape: symmetric or skewed, plus any clusters or gaps. And outliers: any values clearly apart, and what you did about them. Put together it reads like this — the median mark was 15 and the range 5 once the outlier was excluded; the data is roughly symmetric with most marks between 12 and 17; one student scored 48, well above the rest, and the median was used since it is not affected by that value. Centre, spread, shape, outliers, always in context.