Standard Deviation Calculator

Enter your numbers (comma or space separated)

Population standard deviation (divide by n instead of n-1)

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Choose what the denominator represents

Enter at least two finite measurements separated by commas or spaces. The sample option uses squared deviations divided by n−1; the population option uses n. Both center the values on their entered mean.

The n−1 sample variance is unbiased for population variance under independent identically distributed sampling with finite variance. That does not mean sample SD itself is unbiased or that the estimate must equal the population value. Use the population option when summarizing the complete defined population, not merely because the sample is large.

Worked example

Enter 2,4,6. The mean is 4, deviations are −2,0,2, and squared deviations sum to 8. Sample variance is 8/2=4, and sample SD is 2. Population variance of these three values is 8/3≈2.6667, and population SD is approximately 1.6330.

The tool shows the deviation calculations so you can check the denominator. Variance has squared measurement units; SD has the original units.

Spread is different from uncertainty

SD describes variability among values. Under independent sampling, estimated SE of the mean is s/√n. For the example, SE≈1.1547. A mean interval uses this SE and a suitable reference cutoff; it does not simply use ±2 sample SD.

Normal-model proportions such as about 68% within one SD do not apply universally to arbitrary data. Standardizing or calculating SD does not make the distribution normal. Likewise, a high SD does not by itself identify erroneous observations.

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Read variance and SD and confidence intervals. Compare a histogram with the numerical summary before interpreting the distribution.