How it works
Standard deviation measures how spread out values are around the mean. A small standard deviation means the values cluster tightly; a large one means they are widely scattered. Variance is the standard deviation squared.
Choose population when your list contains every member of the group you care about, and sample when it is a subset used to estimate a larger group. The sample formula divides by n − 1 (Bessel’s correction) instead of n, which makes the variance estimate unbiased.
The table shows each value’s deviation from the mean and its square, so you can follow the calculation by hand.
Formula
s = √( Σ(x − x̄)² ÷ (n − 1) ) for a sample · σ = √( Σ(x − μ)² ÷ N ) for a population
Questions people ask
Why does the sample formula divide by n − 1?
A sample’s values sit closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 corrects that bias.
What is a good standard deviation?
There is no universal good value; it depends on the units and the mean. The coefficient of variation (standard deviation ÷ mean) helps compare spread between data sets on different scales.
What is the difference between standard deviation and standard error?
Standard deviation describes spread of individual values. Standard error (s ÷ √n) describes how precisely the sample mean estimates the population mean, and it shrinks as the sample grows.