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Standard Deviation Calculator
Your numbers
Example: 2, 4, 4, 4, 5, 5, 7, 9 → mean 5, population SD 2.00, sample SD 2.14.
Behind the numbers
How standard deviation is calculated
Standard deviation measures how spread out numbers are around their mean. First find the mean by adding all the numbers and dividing by how many there are. Then subtract the mean from each number, square each difference, and add the squares. For the population standard deviation, divide that sum by the number of values (N) and take the square root. For the sample standard deviation, divide by N − 1 instead, then take the square root.
Worked example — data 2, 4, 4, 4, 5, 5, 7, 9. Mean = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5. Squared differences from the mean: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32. Population SD = √(32 ÷ 8) = √4 = 2.00. Sample SD = √(32 ÷ 7) = √4.57 = 2.14.
Examples
Examples
| Data | Mean | Population SD | Sample SD |
|---|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 2.00 | 2.14 |
| 10, 20, 30 | 20 | 8.16 | 10.00 |
| 1, 2, 3, 4, 5 | 3 | 1.41 | 1.58 |
| 100, 110, 120, 130 | 115 | 11.18 | 12.91 |
| 5, 5, 5, 5 | 5 | 0.00 | 0.00 |
Tips before you use it
Use population SD when your numbers are the whole group you care about; use sample SD when they are a sample taken from a larger group.
Sample SD is always a little larger than population SD for the same data, because it divides by N − 1.
A standard deviation of 0 means every number is the same — there is no spread at all.
Standard deviation uses the same units as your data: if your numbers are in centimeters, the SD is in centimeters too.
Check for typing errors first — one wrong number, such as 90 instead of 9, changes the SD a lot.
Good to know
Frequently asked questions
What is the difference between population and sample standard deviation?
Population SD divides by N and describes a complete data set. Sample SD divides by N − 1, which corrects for the fact that a sample tends to understate the spread of the full population. For large data sets the two values are almost the same.
What does a standard deviation of 2 mean?
It means the numbers typically sit about 2 units away from the mean. A small SD means the numbers cluster close to the mean; a large SD means they are widely spread out.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is always zero or positive. A value of zero means all the numbers are identical.
What is the difference between standard deviation and variance?
Variance is the average of the squared differences from the mean — the step just before the square root. Standard deviation is the square root of the variance, which puts the result back in the original units of the data.