Standard Deviation Calculator
Calculate mean, variance and standard deviation for any set of numbers. Population and sample formulas. Free.
How to use
Population vs Sample
Population SD (divide by n): use when you have ALL values in the group. Sample SD (divide by n-1): use when your values are a sample drawn from a larger population. Most real-world analysis uses sample SD. Excel's STDEV is sample SD; STDEVP is population SD.
What SD tells you
In a normal distribution: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. A high SD means data is spread out; a low SD means it's clustered around the mean. CV (Coefficient of Variation) = SD÷Mean expresses variability as a percentage, useful for comparing datasets with different scales.
Standard deviation was formalised by Francis Galton in the 1880s, though the concept existed earlier. The '68-95-99.7 rule' (68% of data within 1 SD, 95% within 2, 99.7% within 3) only applies to normal distributions — which many real-world datasets approximate but none perfectly match.
In data analysis, outliers affect standard deviation dramatically more than they affect median. If your SD is much larger than expected, check for data entry errors first — a misplaced decimal point in one row can inflate SD 10-fold while barely changing the mean.
Frequently asked questions
What is the difference between population and sample standard deviation?
Use population standard deviation when your data covers the whole group. Use sample standard deviation when your data is a sample of a larger group; it divides by n-1 instead of n to correct for bias.
What does standard deviation tell me?
It measures how spread out values are around the mean. A small standard deviation means values cluster close to the average; a large one means they are widely spread.
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value and square the differences, average those squares (dividing by n or n-1) to get the variance, then take the square root.