Standard Deviation Calculator
Calculate sample and population standard deviation, variance, mean, range, and coefficient of variation for any dataset. Free statistics calculator.
Reviewed for accuracy by Marcus Webb and the Blueprint Dynamics editorial team (last updated July 2026). Our calculators use primary-source formulas and are cross-checked against IRS publications, Fannie Mae guidelines, and Federal Reserve data. Learn more about our methodology.
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What standard deviation actually measures
Standard deviation measures how spread out values are around the mean. A low standard deviation means most values cluster close to the average; a high standard deviation means values are widely dispersed. For a set of test scores with a mean of 75: a standard deviation of 5 means most students scored between 70–80. A standard deviation of 20 means scores ranged widely, perhaps from 35 to 100+. Standard deviation is the square root of variance, and it's expressed in the same units as the original data — making it directly interpretable in a way that variance (which is in squared units) is not.
Sample vs. population standard deviation
Use population standard deviation (σ, divides by N) only when your dataset is the complete population — every member of the group you're analyzing. Use sample standard deviation (s, divides by N−1) when your data is a sample drawn from a larger population and you want to estimate the population's standard deviation. The N−1 denominator (Bessel's correction) compensates for the systematic underestimation bias that occurs when using a sample to estimate population spread. Most practical uses — analyzing survey responses, financial returns, test scores — call for sample standard deviation.
The 68-95-99.7 rule for normal distributions
In a normally distributed dataset, approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This is called the empirical rule or 68-95-99.7 rule. For a manufacturing process with a mean output of 100mm and standard deviation of 2mm: 68% of parts fall between 98–102mm; 95% between 96–104mm; 99.7% between 94–106mm. This rule is the basis of quality control (Six Sigma targets ±3σ defect rates), IQ scoring (mean 100, SD 15), and many statistical tests.
Coefficient of variation — comparing across different scales
The coefficient of variation (CV) expresses standard deviation as a percentage of the mean: CV = (σ / mean) × 100. This normalizes spread so you can compare variability across datasets with different units or scales. A portfolio with annual returns averaging 8% and standard deviation of 4% has CV = 50%. Another with average returns of 15% and standard deviation of 10% has CV = 67% — more variable relative to its return. Without CV, you can't compare variability meaningfully across datasets with different means.