Sample standard deviation

Sample standard deviation is the square root of sample variance:

s = √(Σ(xᵢ − x̄)² / (n − 1))

Where x̄ is the sample mean, n is the sample size, and the sum runs over all values. The n − 1 divisor is Bessel’s correction — it compensates for the fact that the sample mean is closer to the data than the (unknown) true population mean would be, which makes the raw sum of squared deviations under-estimate the true population variance.

Use sample SD when your dataset is drawn from a larger group you can’t measure exhaustively (which is almost always). Use population SD (divide by n) only when the dataset literally is the entire population — every employee in your company, every transaction in March.

At large sample sizes the difference is negligible (n vs n-1 is rounding noise). At small sample sizes — say, n < 30 — the correction matters meaningfully and you should prefer the sample form.

Our statistics calculator defaults to the sample form with a toggle to switch to population.

Why the square root reintroduces a small bias: Bessel’s correction makes the sample variance an unbiased estimator of population variance, but the square root operation is non-linear and Jensen’s inequality bites — the sample SD systematically under-estimates the true population SD, even after the N−1 correction. The bias is roughly (1/4n) for normal data, so 2.5% at n=10, 0.25% at n=100, and negligible past n=1000. Statistical packages mostly ignore this; the unbiased c4-correction estimator s × √((n−1)/2) × Γ((n−1)/2) / Γ(n/2) exists for applications where it matters (quality control with tiny sample sizes). Reference: NIST/SEMATECH e-Handbook — Standard Deviation.

Worked example

Five measurements of a chemical assay: 9.8, 10.1, 9.9, 10.3, 10.4. Mean x̄ = 10.10. Squared deviations: 0.09, 0.00, 0.04, 0.04, 0.09 — sum 0.26. Sample variance s² = 0.26 / 4 = 0.065; sample SD s ≈ 0.255. Population SD (divide by 5) would be 0.228 — a 12% understatement of the underlying process spread when treating a sample as a census. For a quality-control chart with control limits at x̄ ± 3s, that difference moves the upper limit from 10.78 to 10.87, materially changing which production runs would trigger an out-of-control alarm.

When it matters in practice

A/B testing, lab science, polling, and finance all draw inferences from samples and report uncertainty as ±s or as a confidence interval built on s/√n. Using the population formula on a sample understates uncertainty and inflates statistical significance — the cardinal sin in reproducibility-crisis papers. Spreadsheets reflect this distinction in their function names: Excel’s STDEV.S divides by n−1,STDEV.P by n; pandas’.std() defaults to ddof=1(sample), NumPy’s np.std() defaults to ddof=0(population). Mixing these is one of the most common silent numerical bugs in data pipelines. See also variance and Bessel’s correction.