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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Result
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).
Worked example
- Data: 2, 4, 4, 4, 5, 5, 7, 9 (N = 8)
- Mean μ = (2+4+4+4+5+5+7+9)/8 = 5
- 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.
Find the mean, subtract it from each value, square each difference, average the squared differences (÷N for population), then take the square root. For 2,4,4,4,5,5,7,9 that gives σ = 2.
Variance is the average of squared deviations from the mean; standard deviation is its square root. SD is easier to interpret because it is expressed in the same units as the original data.
No. Because it is a square root of squared values, standard deviation is always zero or positive. A value of 0 means every number in the set is identical.
References
- NIST/SEMATECH e-Handbook — measures of scale Authoritative overview of standard deviation and related scale measures.
- Statistics definitions (SD) Plain-language government explanation of SD.