Spread around the mean, in the same units as your data
Standard deviation measures how far the values typically sit from their average. Square each value\'s distance from the mean, average those squares (that average is the variance), then take the square root to return to normal units. A small σ means the numbers cluster tightly; a large one means they scatter.
The one choice that changes the answer: n or n−1
Population and sample use the same steps and differ only in the divisor. For the list 2, 4, 6, 8, 10 (mean 6, squared distances summing to 40):
| Type | Divide by | Variance | Std deviation |
|---|---|---|---|
| Population (σ) | n = 5 | 8.0 | 2.83 |
| Sample (s) | n − 1 = 4 | 10.0 | 3.16 |
The gap shrinks as the list grows: it is about 10% at n = 10 but under 1% by n = 100. Dividing by n−1 (Bessel\'s correction) keeps a sample from understating the true spread.
Reading the result
- The 68–95–99.7 rule. If the data is bell-shaped, about 68% falls within one σ of the mean, 95% within two, and 99.7% within three. Heights averaging 170 cm with σ = 10 put most people between 150 and 190 cm.
- Compare across scales with CV. The coefficient of variation, σ ÷ mean, expresses spread as a percentage so you can compare data sets measured in different units.
- Outliers dominate. Because distances are squared, one extreme value inflates σ sharply. Check the list before trusting the number.
Common questions
Should I use sample or population standard deviation?
Use population when your list is the entire group you care about. Use sample when the list is a subset drawn from a larger group you want to describe. Sample divides by n minus 1 instead of n, which nudges the result up slightly to correct for the smaller data set.
What is the difference between variance and standard deviation?
Variance is the average of the squared distances from the mean. Standard deviation is its square root, which brings the number back to the original units. So standard deviation is the square root of variance, and variance is standard deviation squared.
Can standard deviation be negative?
No. It is a square root of squared distances, so it is always zero or positive. It equals zero only when every value in the list is identical.
