Mean

The mean is the sum of values divided by their count — the balance point of a distribution. Two of its properties explain nearly every practical choice between it and the median.

• It adds. The mean of a total is the sum of the means of its parts. Totals, budgets, and capacity all rest on this, and no other summary has it.
• It moves with every value, including the extremes. One customer at a hundred times the typical spend shifts the mean and leaves the median alone.

It is what squared error estimates

The mean is not just one summary among several — it is the answer to a specific optimization, and that connection is what makes metric choices consequential. As the forecasting literature puts it, “a forecast method that minimizes the MAE will lead to forecasts of the median, while minimizing the RMSE will lead to forecasts of the mean.”

So choosing RMSE, or squared error in a model’s loss, chooses the mean as the thing being estimated — whether or not anyone said so. The same source notes the cost of that choice: “consequently, the RMSE is also widely used, despite being more difficult to interpret.” See forecast error and MSE.

It is also the quantity most inference machinery is built around. For independent normal observations with known population standard deviation \(\sigma\), a confidence interval for a population mean is \(\bar{Y} \pm z_{1-\alpha/2}\sigma/\sqrt{N}\). With unknown variance estimated by \(s^2\), the normal-sample interval uses a t critical value with \(N-1\) degrees of freedom instead. Large-sample approximations require appropriate sampling conditions. The estimated standard error of an iid sample mean is \(s/\sqrt{n}\) — see sample mean for the estimate as distinct from the population quantity, true mean for the target.

When reporting it alone misleads

For a symmetric distribution the mean and median coincide and the choice is academic. For a skewed one they can differ enormously, and business metrics are usually skewed: revenue per customer, session duration, latency, order value.

Mean revenue per user can rise while most users spend less, because a handful of large accounts moved. Mean latency can look acceptable while a tenth of requests are unusable. In both cases the mean is arithmetically correct and answers a question nobody asked. The rule that avoids it: report a mean with at least one other number — a median, a high percentile, or a spread. See quantile level.

Two further cautions. A mean over unequal-sized groups is not the mean of their means unless you weight by size — averaging per-country conversion rates without weighting describes an equally weighted country average, rather than the overall user conversion rate. And a mean is not a target: a value negotiated to match a plan is a commitment, and comparing it with an outcome measures the negotiation.

A note on units, since it bears on which summaries are meaningful at all: percentage-based measures “assume the unit of measurement has a meaningful zero,” which is why a percentage change in a temperature reading is not interpretable. That condition is stricter than a mean needs: the average of Celsius readings is still meaningful because differences on that scale are, even though ratios of the readings are not. For indexed or scored quantities, whether the mean is interpretable depends on how the scale is defined.

References

• Hyndman & Athanasopoulos, Forecasting: Principles and Practice (3rd ed.), §5.8 “Evaluating point forecast accuracy”
• §5.5 “Distributional forecasts and prediction intervals”