What standard deviation tells you
Standard deviation measures how spread out values are around their mean. A small standard deviation means the values sit close to the average; a large one means they are scattered. Two classes can both average 70% on a test — one where everyone scored 65–75 (low spread) and one with scores from 30 to 100 (high spread).
Sample or population?
Use population (divide by n) when your data includes every member of the group you care about — for example, the heights of all 25 students in one class when that class is all you are describing. Use sample (divide by n − 1) when your data is a subset used to estimate a larger group — a survey of 500 voters, or 30 measurements from a production line. Dividing by n − 1 (Bessel's correction) compensates for a sample's tendency to underestimate the spread. In school and research problems, sample is the usual choice unless the question says otherwise.
The formula, step by step
- Find the mean: add the values and divide by n.
- Subtract the mean from each value to get its deviation.
- Square each deviation (this makes them all positive).
- Add the squares: the sum of squared deviations.
- Divide by n (population) or n − 1 (sample). This is the variance.
- Take the square root to get the standard deviation, which is back in the original units.
Worked example
Data: 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5. The deviations are −3, −1, −1, −1, 0, 0, 2, 4, and their squares are 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32.
- Population: variance = 32 ÷ 8 = 4, standard deviation = √4 = 2.
- Sample: variance = 32 ÷ 7 ≈ 4.571, standard deviation ≈ 2.138.
The 68–95–99.7 rule
For data that follows a roughly normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. If adult heights in a group average 170 cm with a standard deviation of 7 cm, about 95% of people are between 156 and 184 cm. The rule does not apply to strongly skewed data such as incomes.
Other results shown
The standard error of the mean (standard deviation ÷ √n) estimates how much a sample mean would vary between samples and is used in confidence intervals. The coefficient of variation (standard deviation ÷ mean) expresses spread as a percentage, which lets you compare variability between data sets with different units or scales. For the mean, median and mode on their own, use the average calculator.
Frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations from the mean; standard deviation is its square root. Standard deviation is easier to interpret because it is in the same units as the data.
Why does the sample standard deviation divide by n − 1?
A sample's values are on average closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 (Bessel's correction) removes that bias from the variance.
Can standard deviation be negative?
No. It is the square root of a sum of squares, so it is zero when all values are identical and positive otherwise.
What is a high standard deviation?
It depends on the scale of the data. Compare it with the mean: the coefficient of variation (standard deviation ÷ mean) puts spread on a common percentage scale, where values above about 30% usually indicate widely scattered data.
Last updated: October 9, 2026