Standard Deviation Calculator
Enter a data set to calculate standard deviation, variance, and mean for population or sample.
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Standard Deviation Explained
Standard deviation (SD) measures how spread out values are from the mean. It is one of the most fundamental statistics in data analysis, used in everything from quality control to finance to scientific research.
The Formula
Population SD: σ = √(Σ(xᵢ − μ)² / n). Sample SD: s = √(Σ(xᵢ − x̄)² / (n−1)). The steps are: (1) calculate the mean, (2) subtract the mean from each value and square the result, (3) sum those squared differences, (4) divide by n (population) or n−1 (sample), (5) take the square root.
Population vs. Sample
Use population SD (σ) when you have data for every member of the group. Use sample SD (s) when your data is a sample from a larger population — the n−1 denominator (Bessel's correction) compensates for the fact that a sample tends to underestimate the population variance.
The 68-95-99.7 Rule
For normally distributed data: 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This rule makes SD an intuitive measure of spread — if test scores have a mean of 70 and SD of 10, about 95% of scores fall between 50 and 90.
Coefficient of Variation
CV = (SD / mean) × 100%. This relative measure lets you compare spread across data sets with different units or scales. A data set with mean 100 and SD 10 (CV=10%) is proportionally less variable than one with mean 5 and SD 1 (CV=20%), even though the absolute SD is higher.
Frequently Asked Questions
Standard deviation measures spread around the mean. A low SD means values cluster close to the mean; a high SD means they are widely spread. It's always ≥ 0, and equals 0 only when all values are identical.
Population SD (σ) divides the sum of squared deviations by n (all data points). Sample SD (s) divides by n−1. Use sample SD when your data represents a subset of a larger population — it gives a less biased estimate of the true population spread.
Variance = SD². It's the average of squared deviations from the mean. SD is easier to interpret (same units as data), but variance appears in many statistical formulas and is additive for independent variables.
For a normal distribution: 68% of data falls within ±1 SD, 95% within ±2 SDs, 99.7% within ±3 SDs. Values more than 3 SDs from the mean are considered extreme outliers.
Context matters. A SD of 5°F in daily temperatures is small. A SD of $500 in monthly rent could be large or small depending on the city. The coefficient of variation (SD/mean) helps compare SD across different scales.