Descriptive statistics

Three questions describe most datasets: where is the centre, how spread out is it, and is anything unusual? Mean and median answer the first, standard deviation and IQR the second, Tukey's fences the third.

Sample or population — the n vs n−1 question

If your numbers are the entire group you care about, divide by n. If they are a sample you are using to estimate a larger population, divide by n − 1. Dividing by n understates the true spread, because the sample mean sits closer to your own data than the real mean does; n − 1 (Bessel's correction) removes that bias. When in doubt, use the sample figure.

sample sd = √( Σ(x − x̄)² / (n − 1) ) population sd = √( Σ(x − μ)² / n )

Mean versus median

The mean uses every value, so a single extreme number moves it. The median only cares about order, so it does not. When the two differ substantially, the distribution is skewed — which is exactly why incomes are reported as medians and test scores as means.

Outliers are not errors

The 1.5 × IQR rule flags points unusually far from the middle half of the data. That is a prompt to investigate, not permission to delete. An outlier may be a typo, or it may be the most interesting observation you have.