Population standard deviation
For 2, 4, 4, 4, 5, 5, 7, and 9, the population mean is 5, the variance is 4, and σ = 2.
Measure the spread of a number list
Standard deviation calculator
Calculate population or sample standard deviation, variance, mean, count, and sum from one list of values.
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Formula
Population and sample standard deviation formulas
Population: σ = √(Σ(x − μ)² / N); Sample: s = √(Σ(x − x̄)² / (n − 1))
Standard deviation measures the typical distance of values from their mean. Squaring deviations prevents positive and negative differences from canceling and gives more weight to larger distances.
Use the population formula when the values are the entire group of interest. Use the sample formula when the values are a sample used to estimate a larger population; dividing by n − 1 corrects the sample variance estimate.
How to use the formula
Choose population or sample
Decide whether the entered values are the complete population or a sample from a larger group.
Calculate the mean
Add every value and divide by the number of values.
Square and total the deviations
Subtract the mean from each value, square each difference, and add the squared differences.
Find variance and its square root
Divide by N for a population or n − 1 for a sample, then take the square root to get standard deviation.
Worked examples
Standard deviation examples
Sample standard deviation
For 10, 12, and 14, the sample mean is 12. The squared deviations total 8, so s = √(8 / 2) = 2.
Common questions
Standard deviation FAQ
Should I choose population or sample standard deviation?
Choose population when your data contains every member of the group you want to describe. Choose sample when the data is only part of a larger population you want to estimate.
What does a standard deviation of zero mean?
It means every value in the list is identical, so no value differs from the mean.
What units does standard deviation use?
Standard deviation uses the same units as the original data. Variance uses squared units because it averages squared deviations.
Is my data list uploaded or saved?
Your data list is not saved by default or uploaded. You can opt into Remember on this device or download a local JSON backup.