Most calculators offer both, and picking the wrong one is a common slip. The difference comes down to one question: do you have everyone, or only some of them?
Side by side
| Population | Sample | |
|---|---|---|
| Use when | Your data is the whole group | Your data is a subset used to estimate a larger group |
| Divide the squared differences by | n | n − 1 |
| Typical symbol | σ (sigma) | s |
| Needs at least | 1 value | 2 values |
The same numbers, two results
Take 2, 4, 4, 4, 5, 5, 7, 9. The squared differences from the mean of 5 total 32. As a population: 32 ÷ 8 = 4, so the standard deviation is 2. As a sample: 32 ÷ 7 = 4.571, so it is 2.138. The sample figure is larger, and the gap shrinks as the data set grows.
Try it on a small set of five test scores, 72, 85, 90, 64 and 79: the mean is 78 and the sample standard deviation is 10.32.
Why n − 1?
A sample tends to sit closer to its own mean than to the true mean of the whole group, so dividing by n would understate the spread. Dividing by n − 1 corrects that on average. It is known as Bessel’s correction.
Which to choose
If you measured every item that matters (every student in the class, every order this month), use population. If you measured some to say something about the rest (a survey, a quality check on a batch), use sample. When unsure, sample is the safer choice. Use the Standard Deviation Calculator to compare both, or read the step-by-step guide.
Frequently asked questions
Does it matter with a large data set?
Much less. With hundreds of values, n and n − 1 give almost the same result. The difference matters most for small data sets.