Standard deviation is a measure of how spread out a set of numbers is around its mean. A small standard deviation means the values cluster close to the average; a large one means they are widely scattered. It is in the same units as the data, which makes it easy to interpret.
What standard deviation tells you
The mean alone can hide a lot. Two classes can both average 60 marks, yet in one almost everyone scored between 55 and 65, and in the other the marks ranged from 20 to 100. Standard deviation captures that difference in a single number.
Roughly speaking, it is the typical distance of a value from the mean. Where the data follow a bell-shaped (normal) distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. That rule of thumb only applies to data that are roughly bell-shaped, so do not assume it for every data set.
If you only need the centre of the data, the mean, median and mode are covered by the Average, median and mode.
How to calculate it step by step
Take this data set: 2, 4, 4, 4, 5, 5, 7, 9.
- Find the mean. The total is 40 and there are 8 values, so the mean is 5.
- Subtract the mean from each value: −3, −1, −1, −1, 0, 0, 2, 4.
- Square each difference: 9, 1, 1, 1, 0, 0, 4, 16.
- Add the squares: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32. This is the sum of squared deviations.
- Divide to get the variance. Divide by n (8) for the population variance: 4. Divide by n − 1 (7) for the sample variance: about 4.5714.
- Take the square root. The population standard deviation is √4 = 2. The sample standard deviation is √4.5714, about 2.138.
The variance is the standard deviation squared. It is useful in formulas but is in squared units, which is why people usually quote the standard deviation instead.
Sample or population: which formula?
The two formulas differ only in the divisor.
| Population (σ) | Sample (s) | |
|---|---|---|
| When to use | You have every value in the group | Your values are a sample from a bigger group |
| Divisor | n | n − 1 |
| Example result | 2 | about 2.138 |
A sample tends to understate the true spread of the whole group, because its values are closer to their own mean than to the true mean. Dividing by n − 1 corrects for that. In practice most data are samples (survey answers, measurements, test results from some of the students), so the sample figure is the usual choice. With large data sets the two numbers become very close.
A single value has no sample standard deviation, because n − 1 would be zero. The Standard deviation calculator shows a dash in that case.
Standard error of the mean
The standard error is the sample standard deviation divided by the square root of n. For the example above, 2.138 ÷ √8 is about 0.756.
It is not a measure of spread in the data. It describes how precisely the sample mean estimates the mean of the wider group. It shrinks as the sample gets larger: more data means a more reliable mean, even if the data themselves are just as spread out.
A second worked example
Take eight values: 10, 12, 23, 23, 16, 23, 21, 16. The mean is 18 and the sum of squared deviations is 192.
- Population variance: 192 ÷ 8 = 24, so the standard deviation is about 4.899.
- Sample variance: 192 ÷ 7, about 27.43, so the sample standard deviation is about 5.237.
Here the two values are noticeably different because there are only eight numbers. Reporting which one you used matters.
Common mistakes
- Mixing up sample and population. State which you used, especially in coursework or reports.
- Forgetting to square. If you add the plain differences, they sum to zero by definition.
- Typing errors. One misplaced digit can inflate the result, so check your list.
- Comparing across different scales. A standard deviation of 5 means something different for heights in centimetres than for annual salaries. Compare relative spread only after considering the means.
For percentages and proportions in your data, see our guides on How to calculate percentages and Percentage change, increases and discounts.
Note: This guide is general information for learning, not statistical or professional advice. The calculator gives estimates based on the numbers you enter.
Try the calculator
Paste your numbers into the Standard deviation calculator to see the sample and population standard deviation, variance, mean and standard error together. For the median and mode, use the Average, median and mode.
Open the free Standard deviation calculator
Frequently asked questions
What is a good standard deviation?
There is no universally good value. It depends on the units and what you are measuring, so judge it against the mean and what is normal for the data.
Can standard deviation be negative?
No. It is the square root of an average of squared numbers, so it is zero or positive. It is zero only when every value is identical.
Should I use n or n − 1?
Use n − 1 (sample) when your numbers are a sample from a larger group, which is most of the time. Use n (population) only when you have all the values.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean, so it is in squared units. Standard deviation is its square root, back in the original units.