Quick answer: For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations add up to 32, so the population SD is √(32 ÷ 8) = 2 and the sample SD is √(32 ÷ 7) = 2.138.
How it's calculated
Find the mean, subtract it from each value, square each difference and add the squares. That sum of squares divided by the count is the variance; the standard deviation is its square root.
Divide by N when your list is the whole population (every student in a class). Divide by n − 1 (Bessel's correction) when it is a sample used to estimate a larger population; this makes the sample variance an unbiased estimate.
The standard deviation is in the same units as the data. In roughly normal data about 68% of values lie within 1 SD of the mean and 95% within 2 SD. To place one value on that scale, use the z-score calculator.
Worked examples
- 2, 4, 4, 4, 5, 5, 7, 9 (population): σ = √(32 ÷ 8) = 2
- Same data as a sample: s = √(32 ÷ 7) = 2.138090
- 10, 12, 23, 23, 16, 23, 21, 16 (population): mean 18, Σ = 192, σ = √(192 ÷ 8) = 4.898979
Frequently asked questions
Should I use sample or population standard deviation?
Use population when you have every member of the group you care about. Use sample when the numbers are a subset you are using to describe a bigger group, which is the usual case in surveys and experiments.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root, so it is back in the original units and easier to interpret.
Can the standard deviation be negative?
No. It is a square root of a sum of squares, so it is 0 or more. It is 0 only when all values are equal.
What's the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
Exactly 2 if those eight values are the whole group, or about 2.14 if they're a sample from something bigger. The mean is 5 either way.
What does a high standard deviation tell me?
That the values are spread out. Two classes can both average 75, but with an SD of 3 the marks are bunched together, and with 15 they're all over the place.
Is standard deviation the same as standard error?
No. The SD describes how spread out the data are. The standard error, SD ÷ √n, describes how precisely you know the mean.