What is the standard deviation?
The standard deviation (SD, s or σ) measures how far the values typically are from the mean. A small standard deviation means the numbers are close together; a large one means they are widely spread. Two classes can both average 75% on a test, while in one class everyone scores between 70 and 80 and in the other the scores range from 40 to 100. The mean is the same; the standard deviation is not.
The formula in four steps
- Calculate the mean x̄: add all values and divide by the count (n).
- Subtract the mean from each value and square the result: (x − x̄)².
- Add up those squares (the sum of squares) and divide by n (population) or n − 1 (sample). This is the variance.
- Take the square root of the variance. That is the standard deviation.
σ = √( Σ(x − x̄)² / n ) for a population
>
s = √( Σ(x − x̄)² / (n − 1) ) for a sample
Worked example
Take the numbers 2, 4, 4, 4, 5, 5, 7 and 9.
- The mean is 40 / 8 = 5.
- The deviations are −3, −1, −1, −1, 0, 0, 2 and 4; the squares are 9, 1, 1, 1, 0, 0, 4 and 16.
- The sum of squares is 32.
- Population: 32 / 8 = 4, and √4 = σ = 2.
- Sample: 32 / 7 ≈ 4.571, and √4.571 ≈ s = 2.138.
The calculator shows the same steps in a table, so you can check your own homework.
Sample or population?
Use population (n) when your data covers every case you want to describe, for example the scores of every student in one class when you only describe that class. Use sample (n − 1) when your data is part of a larger group, such as a survey of 500 adults. Dividing by n − 1 (Bessel's correction) prevents the spread in the larger group from being systematically underestimated. Research and spreadsheets mostly use the sample version (Excel's STDEV.S versus STDEV.P). With large datasets the difference is small.
The 68–95–99.7 rule
If the data is roughly normally distributed (a bell curve), about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. If adult heights in a group average 175 cm with a standard deviation of 7 cm, about 95% are between 161 and 189 cm.
Other results
- Variance: the square of the standard deviation, in squared units of the original data.
- Standard error of the mean: s / √n. It tells you how precise the mean of a sample is.
- Coefficient of variation: the standard deviation as a percentage of the mean, useful to compare the spread of different quantities.
For the mean, median and mode, see the average calculator; for chances and odds, the probability calculator.
Where is the standard deviation used?
In education, test analyses use it: a small spread in scores can mean a test did not distinguish well between students. Factories monitor whether products stay within a few standard deviations of the target size. Investors use the standard deviation of returns as a measure of risk or volatility, and medical studies often report measurements as mean ± standard deviation. Weather forecasters and sports analysts use it to describe how unusual a value is compared with the long-term average.
Frequently asked questions
How do I calculate the standard deviation?
Find the mean, subtract it from each value, square the results, add them up, divide by n or n − 1 and take the square root. The calculator does every step for you.
What is the difference between sample and population standard deviation?
For a population you divide by n, for a sample by n − 1. The sample version is slightly larger and corrects for having measured only part of the group.
What is the difference between variance and standard deviation?
The standard deviation is the square root of the variance. It has the same unit as your data, which makes it easier to interpret.
What counts as a high standard deviation?
It depends on the context. Compare it with the mean: the coefficient of variation expresses the spread as a percentage.
Can the standard deviation be negative?
No. It is always 0 or more, and only 0 when all values are equal.
Which Excel function should I use?
STDEV.S for a sample and STDEV.P for a population. Google Sheets also accepts STDEV and STDEVP.
Last reviewed: 2026-10-06. Results are estimates for information only.