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Standard Deviation Calculator
Enter a data set to get its standard deviation — both the sample version (dividing by n − 1) and the population version (dividing by n) — together with the mean, the variances, the sum of squared deviations and the standard error of the mean.
Divides by n − 1. Use when your data is a sample of a larger group.
- Population standard deviation (σ)
- 2
- Mean
- 5
- Count (n)
- 8
- Sample variance (s²)
- 4.5714
- Population variance (σ²)
- 4
- Sum of squared deviations
- 32
- Standard error of the mean
- 0.7559
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Your recent calculations
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Formula
How to use it
- Type or paste your data, separated by commas, spaces or new lines.
- Decide whether the data is a sample or a whole population (see below).
- Read the matching standard deviation.
Worked examples
The data set 2, 4, 4, 4, 5, 5, 7, 9
- Sample standard deviation (s)
- 2.1381
- Population standard deviation (σ)
- 2
- Mean
- 5
- Count (n)
- 8
- Sample variance (s²)
- 4.5714
- Population variance (σ²)
- 4
- Sum of squared deviations
- 32
The data set 10, 12, 23, 23, 16, 23, 21, 16
- Sample standard deviation (s)
- 5.2372
- Population standard deviation (σ)
- 4.899
- Mean
- 18
- Sum of squared deviations
- 192
Sample or population?
Use the population formula when your data is every member of the group you care about — the test scores of everyone in one class, say. Use the sample formula when the data is a subset used to estimate a larger group, such as 50 surveyed customers out of thousands.
Dividing by n − 1 instead of n (Bessel’s correction) makes up for the fact that a sample tends to be less spread out than the population it came from. If in doubt, the sample version is the usual default. In Excel and Google Sheets these are STDEV.S and STDEV.P.
A worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. Dividing by 8 gives a population variance of 4 and a standard deviation of 2. Dividing by 7 gives a sample variance of 4.571 and a standard deviation of 2.138.
Reading the number
For data that is roughly bell-shaped, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three.
Questions people ask
What is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
The population standard deviation is exactly 2; the sample standard deviation is 2.138.
Should I use sample or population standard deviation?
Use population (divide by n) only when you have data for the entire group. For a sample drawn from a larger group, use the sample formula (divide by n − 1).
Can standard deviation be negative?
No. It is a square root, so it is zero or positive. It is zero only when every value is the same.