Shape, Skew, and Outliers

Look at the incomes in almost any country: most people earn a fairly modest amount, while a handful of very high earners stretch a long tail off to the right. That lopsided shape is exactly why the "average" income usually sits well above what a typical person actually earns — proof that the shape of data, not just its centre, changes the story it tells.

You already know how to find the centre of a data set and how spread out it is. But two data sets can share the same mean and the same spread and still tell completely different stories — because they have a different shape. Is it a neat symmetric bell, or lopsided with a long tail dribbling off to one side? Does it rise to a single peak, or two humps? Are there a few stray points marooned far from everyone else?

Shape is the last thing the summary numbers miss, and it matters intensely, because shape tells you which statistics you can trust. On a symmetric distribution the mean is a fine "typical" value; on a lopsided one it can be quietly misleading. A histogram shows shape directly. The simplest distinction is symmetry.

Symmetric — the two sides mirror each other; the mean and the median sit together in the middle.
Right-skewed (positively skewed) — a long tail stretches to the right, toward the high values.
Left-skewed (negatively skewed) — a long tail stretches to the left, toward the low values.

The direction of skew is named after the tail, not the bulk of the data.

Modality: one peak or two?

Shape isn't only about tails. Count the peaks:

Unimodal — a single hump. Most familiar data (heights, test scores) look like this.
Bimodal — two distinct humps, which almost always means you have two groups mixed together. Restaurant arrival times peak at lunch and again at dinner; adult shoe sizes have a women's hump and a men's hump.

Spotting bimodality is a genuine discovery: it is the histogram warning you that a single "average" is describing two different populations at once, and probably describing neither well.

Skew pulls the mean, but not the median

This is the key behaviour to internalise. The mean is a balance point, so it follows the long tail — a few far-away values drag it toward them. The median is a position (the middle value), so it barely moves no matter how far the tail stretches. Hence, reliably:

Right-skew: \text{mean} > \text{median}.
Left-skew: \text{mean} < \text{median}.
Symmetric: \text{mean} \approx \text{median}.

A handy memory hook: the mean is on the same side as the tail. Point in the direction the tail stretches and you are pointing at the mean.

Worked example: predict the order without computing

Nine houses on a street sell for these prices (in hundreds of thousands):

2,\; 2,\; 3,\; 3,\; 3,\; 4,\; 4,\; 5,\; 40.

You don't need to grind through the arithmetic to see the shape. Eight of the nine values huddle between 2 and 5; one lonely value at 40 forms a long tail off to the right. So the data are right-skewed, and we can predict \text{mean} > \text{median}.

Checking: the median is the 5th of 9 sorted values, = 3. The mean is \tfrac{2+2+3+3+3+4+4+5+40}{9} = \tfrac{66}{9} \approx 7.3. Sure enough the mean (7.3) is dragged well above the median (3) — and notice the mean, at 7.3, is a price no house is anywhere near. On skewed data the median is the honest "typical" value.

Worked example: flag the outlier with the 1.5 × IQR rule

An outlier is a point far from the bulk. To decide whether a value counts as one, use the IQR fences. Suppose a data set has Q_1 = 20 and Q_3 = 32, so \text{IQR} = 12. Is a reading of 55 an outlier?

\text{upper fence} = Q_3 + 1.5\,\text{IQR} = 32 + 1.5 \times 12 = 32 + 18 = 50.

Since 55 > 50, the reading sits beyond the fence and is flagged as a suspected outlier. A reading of 48, by contrast, would fall inside the fence and count as ordinary variation.

See it: one outlier, two reactions

Six points sit in a tidy cluster; a seventh (the highlighted one) starts among them. Drag it far to the right to create an outlier — an extreme, isolated value. Watch the two markers: the mean chases the runaway point off to the right, while the median stays put at 5. By the time the point is way out, the data are clearly right-skewed and the mean sits well above the median.

This is why a single outlier — a typo, a billionaire in an income survey — can make the mean deeply misleading, and why the median is the safer summary for skewed data.

Distributions are symmetric or skewed; skew is named after the long tail's direction.
The mean is pulled toward the tail; the median holds. So right-skew gives \text{mean} > \text{median}, left-skew \text{mean} < \text{median}.
An outlier shifts the mean a lot but the median barely at all.
For skewed data the median is the more representative centre.

Why income is the textbook case of skew

Incomes are famously right-skewed. Most people earn within a fairly narrow band, but a small number of very high earners form a long tail stretching far to the right — and there is no matching tail on the left, because you cannot earn less than nothing. A handful of billionaires can haul the mean income far above what any ordinary person makes, while the median — the earnings of the person standing exactly in the middle of the queue — barely notices them.

That is precisely why official statistics almost always report median household income, not the mean: the median answers the question people actually care about — "what does a typical household earn?" — while the mean answers a subtly different and easily-misread question.

Here is the trap. The word "average" usually means the mean, but on skewed data the mean and the median genuinely disagree — and someone choosing which to quote can paint two opposite pictures from the very same data.

Say a town's incomes are mostly modest, but one hedge-fund manager moves in. The mean ("average income") jumps, and a headline can now boast that the town is prosperous.
The median — the typical resident — hasn't moved at all. Most people are no better off.

Both numbers are correct; they answer different questions. Reporting the mean for skewed data, without saying so, is one of the oldest ways to mislead with true statistics. Whenever you meet an "average," ask: mean or median — and is this data skewed?

Outliers come in two very different flavours, and telling them apart is a real skill:

Sometimes an outlier is a mistake — a typo, a sensor glitch, a decimal in the wrong place, a survey respondent who wrote their age in months. These you investigate and usually fix or remove.
Sometimes the outlier is the discovery — the rare, real event that everything else was hiding.

The classic cautionary tale is the Antarctic ozone hole. For years, ozone readings so low that they looked "impossible" were reportedly treated as errors and set aside by automatic data-checking. It took scientists deliberately going back to the raw numbers to realise the impossibly-low values were real — a genuine, dramatic thinning of the ozone layer. The moral: never delete an outlier reflexively. First ask why it's there. The stray point might be the whole point.