Statistics for researchers · Lesson 1 of 3

Population, sample, and the shape of data

A mean alone hides the shape. Ask for the median, the spread, and ideally the picture.

8 min read · Free — no account needed to read

A population is everyone or everything you want to say something about — not necessarily people: it might be all transactions, all documents, all hospitals. A sample is the part you actually observe. A parameter is a true value in the population; a statistic is your estimate of it from the sample. The entire enterprise of inference is reasoning from statistic to parameter, carefully.

A distribution is how values are spread. This matters more than most summaries admit, because the summary you were given may hide the shape entirely.

The mean is the arithmetic average and it is pulled by extreme values. The median is the middle value and is not. For income, wealth, response times, company size and almost anything with a long tail, the median describes the typical case and the mean does not. When someone reports a mean for that kind of data without a median, that choice is worth noticing.

Variance and standard deviation describe spread. Two groups with identical means can be completely different — one tightly clustered, one wildly variable — and the difference is often the finding. Percentiles describe position and are frequently the most honest summary available.

The practical habit: whenever you are given an average, ask for the spread and the shape. A mean on its own is a single number standing in for a distribution you were not shown.

See it happen

A report states that the average time to resolve a support ticket is 4 hours.

The read that arrives on its own

Four hours is reasonable, so resolution times are in good shape.

The same situation, with this idea

You ask for the distribution. It turns out most tickets are resolved in under 30 minutes, and a small number take several days — which drags the mean to four hours. The median is 25 minutes. Both numbers are correct and they describe different realities: there is no general slowness problem, there is a specific category of ticket that gets stuck. The mean concealed exactly the thing worth acting on.

After this, you can

ask what the distribution looks like whenever you are handed an average.

The takeaway

A mean alone hides the shape. Ask for the median, the spread, and ideally the picture.

Easy to take the wrong way

That the median is always better. It is more robust for skewed data; for a total budget or a sum, the mean is what you want.

Try this

Find a mean quoted in your field and ask whether the underlying data is likely skewed. If it is, ask what the median would be.