Math

Standard Deviation Calculator

Calculate population standard deviation from a comma-separated list of numbers — a measure of how spread out values are around the mean.

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Updates as you type

Result

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The formula

Population standard deviation is the square root of the average squared distance from the mean. This calculator divides by N (population form). Sample standard deviation typically divides by N−1 (Bessel’s correction).

σ = √(Σ(x − μ)² / N)

Worked example

  1. Data: 2, 4, 4, 4, 5, 5, 7, 9 (N = 8)
  2. Mean μ = (2+4+4+4+5+5+7+9)/8 = 5
  3. Variance = average of squared deviations = 4; σ = √4 = 2

Result: σ = 2

How standard deviation is calculated

SD answers “how dispersed is this set?” in the same units as the data.

Population vs sample

Divide by N when the data are the entire population of interest. Divide by N−1 when the data are a sample used to estimate a larger population’s SD. Spreadsheet STDEV.P vs STDEV.S mirrors that choice. This page uses population SD (÷ N).

Variance relationship

Variance is σ² (or s²). SD is easier to interpret because it returns to the original units (dollars, seconds, scores).

Normal-data rule of thumb

For roughly bell-shaped data, about 68% of values lie within 1 SD of the mean, 95% within 2, and 99.7% within 3. Heavy tails and outliers break that rule — check plots, not only σ.

Interesting facts

Central to normal distributions

Standard deviation parameterizes the Gaussian distribution used across science, engineering, and finance.

Population vs sample

Dividing by n estimates population SD; dividing by n−1 (Bessel’s correction) is the usual sample estimator.

Same units as the data

Unlike variance, SD is in the original units — easier to interpret next to the mean.

68–95–99.7 rule

For roughly normal data, about 68% of values lie within 1 SD of the mean, 95% within 2, and 99.7% within 3.

Outliers inflate SD

A single extreme value can stretch SD sharply. Median and IQR help when tails are heavy.

Frequently asked questions

Find the mean, average the squared deviations from the mean (divide by N here), then take the square root. Paste your numbers separated by commas to compute it automatically.

Population SD (divide by N). For a sample estimate of a larger population, use ÷ (N−1) instead.

It depends on context and the mean. Compare σ to the size of typical measurements, or use the coefficient of variation (σ/μ) when scales differ.

Not always. In investing, higher SD often means higher volatility (risk). In manufacturing, higher SD can mean poorer process control. Interpret in domain context.

References

  1. NIST/SEMATECH e-Handbook — measures of scale — National Institute of Standards and Technology Authoritative overview of standard deviation and related scale measures.
  2. Statistics definitions (SD) — Australian Bureau of Statistics Plain-language government explanation of SD.