Descriptive Statistics

The descriptive-statistics calculator takes a list of values and returns the complete summary commonly reported in a statistics methods section: mean, sample standard deviation (n − 1, Bessel-corrected), population standard deviation (n), variance for both, median, min/max/range, and the sum of squared deviations Σ(xᵢ − x̄)².

It's upstream of the rest of the site: the σ that the test and CI calculators assume known often comes from this kind of descriptive summary. Use this calculator first when σ isn't given, then carry the result into the hypothesis-test or confidence-interval calculators.

When to use this calculator

Use this when you have a list of measurements and need any of the standard descriptive statistics. The input is a single textarea that accepts numbers separated by whitespace or commas — paste from a spreadsheet column directly. The parser validates each token and rejects partial numbers ("1.5e", "42abc") rather than silently dropping the bad character.

The hero shows sample SD by default — the educational and most-commonly-cited statistic. Population SD is reported in a sibling sub-card so the choice between them is explicit. When n = 1 (sample SD is mathematically undefined), the hero falls back to population SD.

Frequently Asked Questions

What's the difference between sample SD and population SD?

Sample SD divides by (n − 1) — the Bessel correction — to produce an unbiased estimator of the population standard deviation when computed from a sample. Population SD divides by n and is the true standard deviation of a complete, fully-enumerated population. Use sample SD when your data is a sample from a larger population (almost always in research). Use population SD when you have measurements on every member of the population.

Why does the mean differ from the median?

The mean is sensitive to extreme values; the median is robust to them. In symmetric distributions the two are close. In skewed distributions they diverge — income distributions, for example, typically have a mean noticeably higher than the median because a few high earners pull the mean up. Reporting both reveals the shape of the data.

Why is Σ(xᵢ − x̄)² reported?

Sum of squared deviations is the building block for both variance (divide by n or n − 1) and many downstream statistics including ANOVA's sums of squares. Reporting it explicitly lets you build other statistics by hand without re-summing from scratch.

How does this handle whitespace and commas mixed?

The parser splits on whitespace OR commas, so you can paste a Google Sheets column (newline-separated) or a CSV row (comma-separated) or a mix. Each token is validated independently. If any token is invalid, the calculator returns a parse error rather than silently dropping the bad token.

What's the largest input size?

The calculator handles thousands of values without performance issues. For very large datasets (10k+), browser memory and copy-paste become the practical limit.