Describing a Distribution

By the end of this lesson, you should be able to: say which measure of centre a question needs, read variance and standard deviation as a spread rather than a formula, name the one job the median can't do, and tell when the 68-95-99.7 rule applies to your data.

Two distributions, two very different averages

In 2025, the average US household had an income of $126,700. The median household, the one in the middle when all 137 million are lined up from poorest to richest, had $87,460.1

The average is 45% higher than the middle. More than 60% of households earned less than the average: the Census Bureau's 60th percentile is $110,300, still below it.

Now take height. The National Center for Health Statistics measured 5,092 US men aged 20 and over between 2015 and 2018. Their mean height was 175.3 cm and their median 175.4 cm.2 The two are the same to within a millimetre.

Same two summaries, very different behaviour. The difference is the shape of the distribution: how the values are spread out, and whether they trail off further on one side than the other.

Centre: three questions inside "the average"

"The average" can mean three different things.

Mean. Add everything up and divide by the count. It's the balance point of the distribution.

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$

Median. Sort the values and take the middle one. Half the values are above it and half below, by definition.

Mode. The most common value, or for continuous data the most common range of values.

For height, all three sit in the same place, because the distribution is symmetric: it falls away at the same rate on both sides. For income, they spread apart, because the distribution is skewed to the right. A long tail of very high incomes pulls the balance point up without moving the middle household at all.

You can see why with one made-up household. Take 1,000 households, and add one with an income of $1 billion. The mean of the 1,001 rises by about $1 million. The median moves by at most one place in the sorted list. A single extreme value can drag the mean anywhere, and the median barely notices it.

The slider stretches the right tail of a distribution whose median is fixed at 100. With almost no tail, the mean is 100 and half the values are below it, just like height. At 0.86, the shape of US household income, the mean is 145 and 66.6% of values sit below it.

There's one more average worth knowing for skewed data. The geometric mean averages the logarithms and converts back:

$\bar{x}_{\text{geo}} = \exp\!\left(\frac{1}{n}\sum \ln x_i\right)$

For data like income, where values span several orders of magnitude, it lands close to the median. Choosing a distribution explains why.

The advice that stops one step too early

"The data is skewed, so use the median."

That's the right instinct with an incomplete rule. The median isn't a drop-in replacement for the mean, because there's one job it can't do.

Only the mean multiplies back up to the total. The mean is the total divided by the count, so the mean times the count gives back the total, exactly.

Use the median in a model that has to add up, and you lose almost a third of the money. Nobody catches it, because the median looks like the more honest number.

So the rule isn't "skewed means median". It depends on the question:

• What does a typical unit look like? Use the median.
• Anything that has to add up or scale (revenue, cost, capacity, headcount)? Use the mean.
• Comparing two groups on a skewed metric? Neither alone settles it. See when assumptions fail.

When you can, report both. It costs one extra number, and the gap between them tells the reader the data is skewed.

Spread: variance and standard deviation

Two products with the same average response time aren't the same if one is steady and the other swings. Centre without spread is half a description.

Variance is the average squared distance from the mean. Standard deviation is its square root, which puts it back into the units of the data.

$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 \qquad s = \sqrt{s^2}$

Squaring does two jobs. It makes every distance positive, so distances above and below the mean can't cancel. And it makes a value twice as far away count four times as much, which is why the standard deviation reacts to a long tail even more strongly than the mean does.

Why divide by $n-1$ and not $n$? The distances are measured from $\bar{x}$, which is computed from the same data, and $\bar{x}$ always sits closer to your data than the true population mean $\mu$ does. So the squared distances come out slightly too small. Dividing by $n-1$, called Bessel's correction, fixes that exactly. With thousands of values it makes no difference. With five, it changes the answer by 25%. Estimators proves it.

Dividing the standard deviation by the mean gives the coefficient of variation: the spread as a share of the typical value. Judging from the percentiles, men's heights have a standard deviation of about 7.5 cm, about 4% of the mean. For a skewed metric like income or spend per user, the standard deviation can be as large as the mean itself. The averages of a metric like that jump around from sample to sample, which is the subject of sampling distributions.

Where the 68-95-99.7 rule comes from

Most people learn that about 68% of data falls within one standard deviation of the mean, and 95% within two.

That rule isn't a fact about data. It's a fact about the normal distribution, the symmetric bell curve. It holds as well as your data is normal.

Go back to the explorer. With almost no tail, 68.3% of values fall within one standard deviation of the mean, exactly as the rule says. At the income-like setting of 0.86, it's 89.7%. The rule is off by more than 20 points, and nothing warns you.

Quantiles describe any shape without assuming one. A quantile, or percentile, is the value below which a given share of the data falls: the 90th percentile is the value 90% of units are below.

Height is symmetric: the 5th percentile is 12.6 cm below the median and the 95th is 12.0 cm above. Income isn't: the 10th percentile is $67,000 below the median, and the 90th is $174,000 above it.3 The 90th-percentile household has 13 times the income of the 10th.

Percentiles in practice: latency

Website and app response times are skewed the same way. Most requests are fast, and a few are very slow.

Google's site reliability engineers describe a service where a typical request takes about 50 milliseconds, but 5% of requests are 20 times slower.4 An average hides those slow requests, and it can stay flat over a day while the slowest requests get much worse. So Google states its targets as percentiles: the 50th percentile for the typical case, and the 99th or 99.9th for a plausible worst case. In the book's words, "we generally prefer to work with percentiles rather than the mean".

A quick check tells you which tools fit your data: compare the mean and the median. If they're close, the data is roughly symmetric and the standard deviation and the 68-95-99.7 rule are fine. If the mean is well above the median, describe the data with percentiles instead.

Takeaways

• The mean and the median answer different questions. US household income: mean $126,700, median $87,460. Men's height: 175.3 cm and 175.4 cm.
• Skew pulls the mean toward the long tail. For an income-like shape, two thirds of values sit below the mean.
• Only the mean scales to a total. Median times household count misses $5.4 trillion, 31% of US household income.
• Standard deviation is in the units of the data and reacts strongly to tails. Divide by $n-1$ for a sample.
• 68-95-99.7 is a property of the normal distribution, not of data. For an income-like shape, one standard deviation covers 89.7%, not 68%.
• When the mean and median disagree, use percentiles. That's why Google sets latency targets at the 50th and 99th percentiles.

Quick check

Practice

Take it further

Each of these turns on picking a summary that matches the decision.

• YouTube: watch time up, satisfaction down is a metric-choice problem where the mean and the tail disagree about whether a change helped.
• YouTube: average view duration is a ratio shows what happens when the thing you're averaging has a random denominator too.
• Amazon: go or no-go on a new device needs a number that scales to a total.