Standard Deviation Calculator.
Paste or type a list of numbers to calculate mean, variance, and both population and sample standard deviation instantly.
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Population SD (σ)
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Sample Variance
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Mean ± SD
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Standard Deviation Formula
Standard deviation measures the average distance of each data point from the mean. A low SD means data clusters tightly; a high SD means data is spread out widely.
Internal Navigation
Standard Deviation: Population vs Sample Explained Methodology.
The Calculation Branch
Industrial Standards.
Bessel's correction (dividing by n−1 instead of n for sample SD) compensates for the fact that a sample mean is already optimized to minimize squared deviations from the sample data, creating downward bias in the variance estimate. This correction produces an unbiased estimate of the population variance. For large samples (n > 30), the difference between n and n−1 is negligible.
In-Depth Analysis & Reference Data
The empirical rule (68-95-99.7 rule) applies to normally distributed data: about 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This makes SD practically useful — if student test scores have a mean of 75 and SD of 10, you know about 95% of students scored between 55 and 95. Values more than 3 SDs from the mean are statistical outliers in most datasets.
Registry Questions & FAQ.
When should I use standard deviation vs variance?
Use standard deviation when you want a measure of spread in the same units as your data (e.g., dollars, centimeters, points). Variance is in squared units, which is harder to interpret intuitively. Statisticians often work with variance internally (it has nicer mathematical properties for combining distributions) but report standard deviation for communication. The relationship is simple: variance = SD².
What is coefficient of variation?
Coefficient of variation (CV) = SD ÷ Mean × 100%. It expresses standard deviation as a percentage of the mean, allowing comparison of variation between datasets with different scales or units. A CV of 15% means the standard deviation is 15% of the mean — useful for comparing variability between, say, the heights of children vs the heights of adults without the absolute values confusing the comparison.
Estimates for planning. Always confirm against an authoritative source.