Standard Deviation Calculator
Calculate standard deviation and variance for a set of numbers — for a whole population or for a sample. Enter your numbers separated by commas or spaces.
How to use the standard deviation calculator
- Enter your numbers in the field, separated by commas or spaces.
- Choose whether the numbers represent a whole population or a sample.
- The calculator automatically works out standard deviation, variance and average as you type.
How standard deviation is calculated
Standard deviation measures how much the values in a set of numbers spread out around the average. A low standard deviation means the values are tightly clustered around the average, while a high standard deviation means they're spread over a larger range.
Standard deviation = √Variance
Worked example: the numbers 4, 8, 6, 5, 3, 9, 7 have a mean of 6. The squared deviations come to 4 + 4 + 0 + 1 + 9 + 9 + 1 = 28. For a sample this is divided by n − 1 = 6, giving a variance of 4.67 and a standard deviation of √4.67 = 2.16. Treated as a population it is divided by 7 instead, and the answer is exactly 2.
The difference between population and sample lies in the denominator: for a population (where the number set is the entire dataset you're interested in) you divide by n. For a sample (where the number set is a subset drawn from a larger population) you divide by n − 1, known as Bessel's correction. This adjusts for the tendency of a sample to underestimate the actual spread of the whole population.
Frequently asked questions
What does standard deviation tell me?
Standard deviation tells you how much the values in a dataset typically deviate from the average. A low standard deviation means little spread, while a high standard deviation means large spread in the data.
What's the difference between population and sample?
Population means the number set is the entire dataset you're interested in. Sample means the number set is only a subset drawn from a larger population. The sample calculation is adjusted to give a more reliable estimate of the spread in the whole population.
Why do you divide by n − 1 for a sample?
Dividing by n − 1 (Bessel's correction) compensates for the fact that a sample usually underestimates the actual spread in the whole population. This gives a more accurate estimate of the population's standard deviation when you only have a sample available.