What standard deviation tells you
The mean tells you the centre of your data. The standard deviation tells you how spread out it is. A small standard deviation means most values sit close to the mean; a large one means they are scattered widely.
Two classes can both average 70% on a test. In one, everyone scored between 65 and 75. In the other, scores ran from 30 to 100. The averages match, but the standard deviations are very different, and that difference matters when you decide who needs help.
The formulas
Standard deviation is the square root of the variance. Variance is the average squared distance from the mean.
Here x is each value, μ or x̄ is the mean, N or n is the number of values, and Σ means “add them all up”. The only difference is the divisor: N for a population, n − 1 for a sample.
Step by step
- Add up the values and divide by how many there are to get the mean.
- Subtract the mean from each value. Some answers are negative; together they always add up to 0.
- Square each difference so that every one is positive.
- Add the squares. This is the sum of squares.
- Divide by n (population) or n − 1 (sample) to get the variance.
- Take the square root to get the standard deviation, back in the original units.
Worked example
Take the eight values 2, 4, 4, 4, 5, 5, 7, 9.
- Sum = 40, so the mean is 40 ÷ 8 = 5.
- Differences from the mean: −3, −1, −1, −1, 0, 0, 2, 4.
- Squares: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
| Value | x − mean | (x − mean)² |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
| Total 40 | 0 | 32 |
Population variance = 32 ÷ 8 = 4, so the population SD is 2. Sample variance = 32 ÷ 7 ≈ 4.571, so the sample SD is about 2.138. These are the numbers the calculator shows when it first loads.
Sample or population?
Use the population formula when your list is the whole group you care about: every employee’s salary in a small firm, or every score in one class when you only want to describe that class.
Use the sample formula when your list is a subset used to learn about a bigger group: 50 customers from a survey, or 20 parts tested from a factory batch. Dividing by n − 1 instead of n (called Bessel’s correction) makes the variance slightly bigger. That corrects for the fact that a sample’s values sit, on average, a little closer to their own mean than to the true mean of the whole group.
If you are not sure, the sample version is the safer default. Excel’s STDEV.S and Google Sheets’ STDEV use it; STDEV.P is the population version. The difference shrinks as n grows: with 8 values the two SDs differ by about 7%, with 100 values by only 0.5%.
Reading the result
For data shaped roughly like a bell curve, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. If adult heights in a group have a mean of 170 cm and an SD of 7 cm, roughly two in three people are between 163 and 177 cm.
The calculator also shows:
- Standard error of the mean (s ÷ √n): how much the mean itself would wobble between samples. Quadruple the sample size and it halves.
- Coefficient of variation (SD ÷ mean): spread as a percentage of the mean, handy for comparing data on different scales.
- Median and range: the middle value and the gap between the smallest and largest. The median is less affected by one extreme value than the mean.
Tips
- One outlier can inflate the SD a lot, because differences are squared. Check the step table for a value with a huge squared difference before drawing conclusions.
- The SD is never negative. It is 0 only when every value is identical.
- Paste straight from a spreadsheet column; new lines count as separators. Use dots for decimals, since commas separate values.
- The calculator uses a numerically stable method, so values like 1,000,000,004 and 1,000,000,007 still give an exact variance.
To turn a score into a grade, try the grade calculator; for percentage differences between two averages, use the percentage calculator. Need random test data? The random number generator can produce a list to paste in. Everything is calculated in your browser and nothing is uploaded.