Standard deviation calculator
Standard deviation is the typical distance of a value from the mean. Paste data and watch each distance get squared, averaged and square-rooted.
Each bar is one value's distance from the mean (orange above, teal below). Standard deviation is a typical bar length.
Step by step
- Mean: x̄ = 1495 / 20 = 74.75
- Squared deviations (x − x̄)², added up: Σ = 1963.75
- Divide by n − 1 = 19: variance = 1963.75 / 19 = 103.3553
- Square root: standard deviation = √103.3553 = 10.1664
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 62 | −12.75 | 162.5625 |
| 71 | −3.75 | 14.0625 |
| 74 | −0.75 | 0.5625 |
| 58 | −16.75 | 280.5625 |
| 88 | 13.25 | 175.5625 |
| 91 | 16.25 | 264.0625 |
| 67 | −7.75 | 60.0625 |
| 73 | −1.75 | 3.0625 |
| 79 | 4.25 | 18.0625 |
| 84 | 9.25 | 85.5625 |
| 70 | −4.75 | 22.5625 |
| 65 | −9.75 | 95.0625 |
| 77 | 2.25 | 5.0625 |
| 81 | 6.25 | 39.0625 |
| 69 | −5.75 | 33.0625 |
| 95 | 20.25 | 410.0625 |
| 72 | −2.75 | 7.5625 |
| 76 | 1.25 | 1.5625 |
| 60 | −14.75 | 217.5625 |
| 83 | 8.25 | 68.0625 |
| Σ | 0 | 1963.75 |
What standard deviation measures
Standard deviation is roughly the typical distance between a value and the mean. Two classes can both average 72% on a test, but if one class has an SD of 4 points and the other 15, the first class scored very consistently and the second was all over the place.
s = √( Σ(x − x̄)² / (n − 1) ) σ = √( Σ(x − μ)² / N )
Why square and then square-root?
Deviations from the mean always add up to exactly zero (check the Σ row of the table above), so you cannot just average them. Squaring makes every deviation positive; the square root at the end puts the answer back in the original units. The variance calculator draws those squares literally.
Why n − 1 for a sample?
The sample mean is, by construction, the point that makes the squared deviations of that sample as small as possible, so they are smaller than the deviations from the true population mean would be. Dividing by n − 1 instead of n corrects this underestimate on average. This is called Bessel's correction; with large samples the difference is negligible.
Using it
If data are roughly bell-shaped, about 68% of values fall within one SD of the mean and 95% within two (the empirical rule). To see how far a single value sits from the mean in SD units, use the z-score calculator.
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Questions students ask
What is the difference between sample and population standard deviation?
Population SD divides the sum of squared deviations by N. Sample SD divides by n − 1 (Bessel's correction), because a sample's own mean sits closer to its data than the true mean does, which would otherwise make the spread look too small.
How do you calculate standard deviation by hand?
Find the mean, subtract it from each value, square each difference, add the squares, divide by n − 1 (sample) or N (population), then take the square root.
What does a standard deviation tell you?
How spread out the data are, in the same units as the data. A small SD means values cluster tightly around the mean; a large SD means they are widely scattered.
Which one should I use?
Use the sample SD when your numbers are a sample drawn from a bigger group you want to describe, which is most homework and research. Use the population SD only when you have every member of the group.