Standard Deviation Calculator
Population or sample standard deviation, variance, mean and standard error for up to 10 values — with the full deviation table so you can see every step.
Standard Deviation Calculator: with the default inputs, standard deviation is 2.1381.
Only the first N value boxes are used. Values 6–10 live under More options.
Use Sample when your values are a subset of a bigger group you want to describe; Population when they are the whole group.
- Variance
- 4.5714
- Mean
- 5
- Sum of squared deviations
- 32
- Standard error of the mean
- 0.7559SD ÷ √n — how much the sample mean is expected to wander from the true mean.
- Coefficient of variation
- 42.76%SD as a percentage of the mean. Undefined when the mean is 0.
Assumptions
- Sample SD uses Bessel's correction (n − 1).
- Values are treated as an unweighted list; frequencies are not supported — repeat a value to weight it.
| Value | Value − mean | (Value − mean)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
Squared deviations sum to 32; divided by 7 that is the variance, and its square root is the standard deviation.
How this is worked out
The formula
Mean: x̄ = Σx ÷ n Population variance: σ² = Σ(x − x̄)² ÷ n σ = √σ² Sample variance: s² = Σ(x − x̄)² ÷ (n − 1) s = √s² Standard error of the mean: SEM = s ÷ √n Coefficient of variation: CV = s ÷ |x̄| × 100%
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- How many values?
- Only the first N value boxes are used. Values 6–10 live under More options.from 2 to 10 · whole numbers only · defaults to 8
- Data type
- Use Sample when your values are a subset of a bigger group you want to describe; Population when they are the whole group.Sample (divide by n − 1) · Population (divide by n)
- Value 1
- A number.defaults to 2
- Value 2
- A number.defaults to 4
- Value 3
- A number.defaults to 4
- Value 4
- A number.defaults to 4
- Value 5
- A number.defaults to 5
- Value 6(under More options)
- A number.defaults to 5
- Value 7(under More options)
- A number.defaults to 7
- Value 8(under More options)
- A number.defaults to 9
- Value 9(under More options)
- A number.defaults to 0
- Value 10(under More options)
- A number.defaults to 0
What you get back
- Standard deviationmain answer
- Variance
- Mean
- Sum of squared deviations
- Standard error of the mean
- SD ÷ √n — how much the sample mean is expected to wander from the true mean.
- Coefficient of variation
- SD as a percentage of the mean. Undefined when the mean is 0.
What this assumes
- Sample SD uses Bessel's correction (n − 1).
- Values are treated as an unweighted list; frequencies are not supported — repeat a value to weight it.
About this calculator
Standard deviation answers one question: how far, typically, do the values sit from their mean? Two classes can both average 75 on a test, but a class with a standard deviation of 3 is tightly bunched while one with a standard deviation of 15 has students everywhere from failing to perfect. Because it's in the same units as the data, it's the most readable measure of spread there is.
How to use it
Set how many values you have, type them in (values 6–10 are under More options), and choose Sample or Population. The deviation table shows each value's distance from the mean and that distance squared, which is exactly the working a statistics course asks for.
Why we square, then square-root
Deviations from the mean always add up to zero — positives cancel negatives — so averaging them tells you nothing. Squaring makes every deviation positive and gives big misses extra weight. The average squared deviation is the variance, but it's in squared units (points², dollars²), so we take the square root to get back to something you can picture. That's the standard deviation.
Sample vs. population: the n − 1 question
If your values are the entire group you care about (every student in one class), divide by n. If they're a sample used to estimate a larger group (30 customers standing in for all customers), divide by n − 1. A sample's values cluster around their own mean more tightly than around the true mean, so dividing by n would underestimate the spread; n − 1 (Bessel's correction) fixes that on average. Spreadsheets default to the sample version (STDEV.S), which is what most people need.
Reading the other numbers
- Standard error of the mean (SD ÷ √n) tells you how precisely your sample mean pins down the true mean. Quadruple the sample and it halves.
- Coefficient of variation expresses the SD as a percentage of the mean, so you can compare the spread of quantities on different scales.
Common mistakes
- Forgetting to square before summing (or square-rooting the wrong thing).
- Using population SD on a sample — it makes your data look more consistent than it is.
- Comparing standard deviations of variables measured in different units; use the coefficient of variation instead.
Frequently asked questions
▸What is a standard deviation, in plain words?
It's the typical distance between a value and the average. A small SD means the values are bunched together; a large one means they're spread out.
▸Should I use sample or population standard deviation?
Population if your numbers are the whole group you're describing; sample if they're a subset used to estimate a bigger group. When in doubt, sample — it's the conservative choice and what Excel's STDEV.S uses.
▸Why divide by n − 1 for a sample?
A sample's values are closer to their own mean than to the true population mean, so dividing by n underestimates the spread. Dividing by n − 1 corrects that bias.
▸What's the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. SD is easier to interpret because it's in the same units as the data.
▸What does a standard deviation of 0 mean?
Every value is identical. There is no spread at all.
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