How to use the standard Deviation Calculator
- Type or paste into the Numbers box.
- Choose the data is a from the list.
- The result updates instantly as you type. There is no button to press.
- Use Copy result to copy the figures, or Copy link to share a link that reopens the standard Deviation Calculator with the same inputs.
How it works
Standard deviation measures how spread out numbers are around their mean. A small standard deviation means values cluster tightly around the average. A large one means they are widely scattered. Two classes might both average 70% on a test, but if one class's scores range from 65 to 75 and the other's from 30 to 100, their standard deviations will be very different.
To calculate it, find the mean, subtract it from each value, square those deviations so negatives don't cancel positives, average the squares to get the variance, then take the square root to return to the original units. The only question is how to "average" the squared deviations.
If your numbers are the whole population you care about, such as every employee in a company, divide by n. If they are a sample used to estimate a larger population, such as 50 customers surveyed out of thousands, divide by n − 1. This adjustment, called Bessel's correction, compensates for a sample's tendency to underestimate the true spread. When in doubt, use the sample version, which is what most statistics software reports by default. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two.
Formula
Variance is the square of the standard deviation. The coefficient of variation is SD ÷ mean, a unit-free measure of relative spread.
Example
For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which sum to 32. The population variance is 32 ÷ 8 = 4, so the population SD is 2. Treated as a sample, the variance is 32 ÷ 7 ≈ 4.571 and the sample SD is about 2.138.
Frequently asked questions
Should I use sample or population standard deviation?
Use population only when your data includes every member of the group you are describing. If the data is a subset used to draw conclusions about a larger group, use the sample standard deviation.
What is the difference between variance and standard deviation?
Variance is the average squared deviation, so it is in squared units. Standard deviation is its square root, which is in the same units as the data and easier to interpret.
What is a 'good' standard deviation?
There is no universal good value. Compare it with the mean (the coefficient of variation) or with other data sets measured in the same way.