Standard Deviation Calculator — with Variance, Mean & Steps

Standard deviation measures how far a set of numbers typically sits from their average. The calculator follows the definitions in the NIST/SEMATECH e-Handbook of Statistical Methods and OpenStax Introductory Statistics, computing everything from a single pass over your data. Let the values be x₁ … xₙ.

1. Count and sum. n is the number of valid values; Sum = Σxᵢ.
2. Mean. x̄ = Sum ÷ n — the arithmetic average, and the point every deviation is measured from.
3. Deviations and squares. For each value compute the deviation (xᵢ − x̄), then square it. Squaring removes the sign so values above and below the mean both add to the spread.
4. Sum of squares. SS = Σ(xᵢ − x̄)². This is the single most important intermediate figure and it is printed on its own line in the working panel.
5. Variance. Population variance σ² = SS ÷ N. Sample variance s² = SS ÷ (n − 1). The n − 1 divisor is Bessel's correction: because a sample's mean sits closer to its own points than the true population mean does, dividing by n − 1 rather than n makes s² an unbiased estimate of the population variance (NIST/SEMATECH §1.3.5.6).
6. Standard deviation. The positive square root of the variance: σ = √σ² or s = √s². Taking the root returns the figure to the same units as the original data, which is why it is reported more often than variance.
7. Median, mode and range. The median is the middle value once sorted (or the average of the two middle values for an even count); the mode is the most frequent value or values; the range is max − min.
8. Standard error of the mean. SEM = s ÷ √n estimates how precisely the sample mean pins down the true population mean (NIST/SEMATECH §1.3.5.2). It shrinks as the sample grows.
9. Coefficient of variation. CV = SD ÷ x̄ × 100% expresses spread relative to the average so you can compare data sets with different units; it is shown only when the mean is not zero.

The sum of squares is computed from the direct definition Σ(xᵢ − x̄)² rather than the algebraically-equivalent Σxᵢ² − (Σxᵢ)²/n shortcut. Both give the same answer for ordinary data, but the shortcut loses precision on large-magnitude values (a phenomenon called catastrophic cancellation), so the direct form keeps results accurate even for figures in the billions. All arithmetic runs client-side in double precision — your data never leaves the browser.

• NIST/SEMATECH e-Handbook of Statistical Methods §1.3.5.6 — Measures of Scale (variance, standard deviation, n−1 estimator)
• NIST/SEMATECH e-Handbook §1.3.5.2 — Confidence Limits for the Mean (standard error SE = s/√n)
• OpenStax Introductory Statistics §2.7 — Measures of the Spread of the Data

These are stable mathematical definitions, not rates that change. The formulas and worked examples on this page were last cross-checked against the cited sources on 2026-07-16.