Describing a Distribution

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

Two thirds of Pantry's orders are below average

Pantry's average order value last week was $54.29.

Now sort the 25,079 orders and count how many came in below that. The answer is 16,652 of them, 66.4%.

That's not a typo and nothing is broken. It's a plain consequence of the shape of the data, and if you report "our average order is $54" to a room of people who hear "a typical order is $54", you've misled them while being arithmetically correct.

Checkout latency, on the right, behaves the way the textbook picture behaves: the mean and the median are both 1,840 ms, and 50.2% of checkouts fall below the average. Exactly what you'd expect.

Order value doesn't. Same summary statistics, same tools, and one of them is describing the data while the other is describing a number that almost no order is near.

Centre: three questions hiding inside "the average"

Ask for "the average order" and you could be asking any of three things.

Mean. Add everything up, divide by the count. $54.29. It's the balance point of the distribution: the place the data would sit level on a fulcrum.

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

Median. Sort the values, take the middle one. $37.83. Half the orders are above, half below, by definition.

Mode. The most common value. Pantry's busiest $5 band is $15 to $20, holding 2,166 orders, which is where a shopper doing a small top-up lands.

For checkout latency all three sit on top of each other. For order value they run from the high teens to $54.29, and the reason is the tail on the right: a handful of large orders pull the balance point up without moving the middle at all.

Here's the demonstration in one line. Add a single $50,000 catering order to the week:

One order out of 25,080 moves the mean by two dollars. The median doesn't notice it exists.

There's a fourth average worth knowing, because it comes up whenever data spans orders of magnitude. The geometric mean averages the logs and converts back:

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

On Pantry's orders it's $37.99, which lands on the median almost exactly. That's not luck, and choosing a distribution explains why.

The advice that stops one step too early

"The data is skewed, so use the median instead of the mean."

You'll hear this constantly, and it's the right instinct attached to an incomplete rule. The median is not a drop-in replacement, because there's one job it can't do.

Only the mean multiplies back up to the total. That's what it is: the total divided by the count. So the total is the mean times the count, exactly, always.

Hand a finance model the median and let somebody scale it to a quarter, and you've deleted 30% of the revenue. Nobody will catch it, because $37.83 looks like the more honest number.

So the rule isn't "skewed means median". It's this:

• Reporting what a typical unit looks like? Median.
• Anything that has to add up or scale, like revenue, capacity, cost, headcount? Mean.
• Comparing two groups on a skewed metric? Neither, on its own. See when assumptions fail.

Report both when you can. It costs one extra number and it's the fastest way to show a room that a metric is skewed.

Spread: the standard deviation is as big as the average

Centre without spread is half a description. Two teams with the same average latency are not the same team if one of them is stable and the other swings.

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

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

Squaring is doing two jobs there. It makes distances positive so they can't cancel, and it makes a point twice as far away count four times as much. That second one is why standard deviation reacts to a tail the way the mean does, and harder.

The $n-1$ instead of $n$ is called Bessel's correction. Using $\bar{x}$ instead of the true mean pulls the squared distances slightly too small, because $\bar{x}$ sits closer to your own data than $\mu$ does. Dividing by $n-1$ corrects for it exactly. At $n = 25{,}079$ it changes nothing; at $n = 6$ it matters, and estimators proves it.

Now the numbers:

Order value has a standard deviation the same size as its mean. That ratio has a name, the coefficient of variation, and a value near 1 tells you the spread is as large as the thing you're measuring. Averages of a metric like that move around a lot, which is the entire subject of sampling distributions.

Where the 68-95-99.7 rule actually comes from

Everyone learns that about two thirds of the data sits within one standard deviation of the mean. Check it:

Checkout latency matches to the decimal. Order value covers 89.3% inside one standard deviation, twenty-one points off.

That rule is not a fact about data. It's a fact about the normal distribution, and it holds exactly as well as your data is normal. Checkout latency was drawn from a normal, so it obeys. Order value wasn't, so it doesn't.

The quantile ladder describes both without assuming anything:

Read the two columns side by side. Latency's p99 is 32% above its median. Order value's p99 is more than seven times its median. Quantiles never lie about shape, which is why a latency SLO is written as p99 and not as "mean plus two SDs".

Takeaways

• "Below average" is not a minority position. 66.4% of Pantry's orders are below the mean order value, because a right tail pulls the balance point past most of the data.
• The mean is the only summary that scales. Mean times count reproduces $1,361,615 of revenue exactly; median times count is short by $412,876.
• Skew is why they diverge. Order value's skewness is 3.33 and its mean sits $16.46 above its median. Checkout latency's skewness is 0.00 and they're identical.
• Standard deviation is in the same units as the data, and reacts to tails harder than the mean does. Order value's SD is 100% of its mean; latency's is 14%.
• 68-95-99.7 is a property of the normal distribution, not of data. It covers 68.4% of checkout times and 89.3% of order values.
• Quantiles never assume a shape. When mean and median disagree, report p50, p90 and p99 and let the reader see the tail.

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, which rules out most of the options in this lesson.

Next, probability rules stops describing data you have and starts putting numbers on data you don't.