Data made clearer
How to Calculate Standard Deviation
A readable introduction to standard deviation, variation, and why the average alone rarely tells the whole story in a dataset.
Standard deviation measures how spread out a list of numbers is. A low standard deviation means values sit close to the average. A high standard deviation means values vary more. Use the standard deviation calculator to check a dataset quickly.
The short version
For a population standard deviation:
- Find the mean.
- Subtract the mean from each value.
- Square each difference.
- Average the squared differences.
- Take the square root.
Sample standard deviation is similar, but divides by n - 1 instead of n.
Try it with your own numbers
Use the standard deviation calculator when you have a list of values. Use the average calculator if you only need mean, minimum, maximum, and count.
How the calculation works
Standard deviation starts with the mean because spread is measured around the centre of the data. Squaring the differences avoids positive and negative differences cancelling each other out. Taking the square root puts the result back into the original units.
Population standard deviation is used when your data is the whole group. Sample standard deviation is used when your data is a sample from a larger group.
A worked example
For the numbers 2, 4, 4, 4, 5, 5, 7, 9:
- Mean: 5
- Differences from mean: -3, -1, -1, -1, 0, 0, 2, 4
- Squared differences: 9, 1, 1, 1, 0, 0, 4, 16
- Average squared difference: 32 / 8 = 4
- Square root: 2
The population standard deviation is 2.
Watch-outs
- Mixing sample and population formulas.
- Forgetting to square the differences.
- Comparing standard deviations from datasets with very different units.
- Treating standard deviation as the full range.
- Ignoring outliers.
How to read the result
Use standard deviation to understand variation in scores, measurements, prices, times, or repeated observations. It is useful beside the average because two datasets can have the same mean but very different spread.
Tools mentioned in this article
Reader questions
Is a higher standard deviation bad?
Not automatically. It means more variation. Whether that is good or bad depends on the context.
What is the difference between variance and standard deviation?
Variance is the average squared difference. Standard deviation is the square root of variance.
Should I use sample or population standard deviation?
Use population when the data is the whole group. Use sample when the data represents a wider group.
Can standard deviation be negative?
No. Because it is based on squared differences and a square root, it cannot be negative.
