Standard Deviation Calculator
Paste a list of numbers to get the standard deviation and variance, with every deviation squared and summed in a table you can check by hand.
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- No sign-up
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Paste or type numbers separated by commas, spaces or new lines (e.g. a column copied from a spreadsheet).
Choose sample unless you have every member of the group.
- Count
- 8
- Mean
- 5
- Sum of squared deviations
- 32
- Variance (sยฒ)
- 4.571429
- Sample SD (s)
- 2.13809
- Population SD (ฯ)
- 2
- Standard error of the mean
- 0.755929
- Coefficient of variation
- 42.76%
- Min / max
- 2 / 9
- Range
- 7
| x | x โ mean | (x โ mean)ยฒ | z-score |
|---|---|---|---|
| 2 | โ3 | 9 | โ1.403 |
| 4 | โ1 | 1 | โ0.468 |
| 4 | โ1 | 1 | โ0.468 |
| 4 | โ1 | 1 | โ0.468 |
| 5 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 7 | 2 | 4 | 0.935 |
| 9 | 4 | 16 | 1.871 |
| Range | From | To | Values inside | Expected if normal |
|---|---|---|---|---|
| mean ยฑ 1s | 2.8619 | 7.1381 | 6 of 8 (75%) | 68% |
| mean ยฑ 2s | 0.7238 | 9.2762 | 8 of 8 (100%) | 95% |
| mean ยฑ 3s | โ1.4143 | 11.4143 | 8 of 8 (100%) | 99.7% |
How this was calculated
- Count the values: n = 8.
- Find the mean: 40 รท 8 = 5.
- Subtract the mean from each value and square the result (see the table).
- Add the squares: ฮฃ(x โ mean)ยฒ = 32.
- Divide by n โ 1 = 7 to get the sample variance: 32 รท 7 = 4.571429.
- Take the square root: s = โ4.571429 = 2.13809.
How this standard deviation calculator works
Standard deviation measures how far values typically sit from their mean. A small standard deviation means the numbers cluster tightly; a large one means they are spread out. This standard deviation calculator computes it the way you would by hand: find the mean, subtract it from every value, square those deviations, add them up, divide, and take the square root. The deviation table shows every intermediate number so you can check your homework line by line.
The only choice is what to divide by. For a sample, s = โ(ฮฃ(x โ xฬ)ยฒ รท (n โ 1)). For a whole population, ฯ = โ(ฮฃ(x โ ฮผ)ยฒ รท N). Spreadsheets make the same split: STDEV.S is the sample formula and STDEV.P the population formula.
Worked example
Take 2, 4, 4, 4, 5, 5, 7, 9. The sum is 40 and there are 8 values, so the mean is 5. The deviations are โ3, โ1, โ1, โ1, 0, 0, 2 and 4; squared, they are 9, 1, 1, 1, 0, 0, 4 and 16, which add to 32. Treated as a population, the variance is 32 รท 8 = 4 and the standard deviation is โ4 = 2. Treated as a sample, the variance is 32 รท 7 = 4.571 and the standard deviation is 2.138.
Sample or population?
Use the sample formula when your data is a subset you are using to learn about a larger group: a survey, a lab experiment, last month's sales as a guide to every month. That covers nearly all homework and research. Use the population formula only when the list is the entire group you care about, such as the final scores of every student in one class when you only want to describe that class. Dividing by n โ 1 (Bessel's correction) makes the sample variance slightly larger, compensating for the fact that a sample sits closer to its own mean than to the true one. The difference shrinks as n grows.
Common mistakes
- Forgetting to square. The raw deviations always add to zero, which is why they are squared first.
- Forgetting the square root. That gives the variance, which is in squared units (dollarsยฒ, cmยฒ) and is not the standard deviation.
- Using the wrong divisor. If your answer is slightly off from the textbook, try the other mode.
When standard deviation is misleading
The 68โ95โ99.7 rule (about 68% of values within one standard deviation of the mean, 95% within two, 99.7% within three) only holds for roughly bell-shaped data. Incomes, house prices and wait times are skewed, and a single outlier can inflate the standard deviation enormously. Compare the "values inside" column of the spread table with the expected percentages to see how normal your data looks. For skewed data, the median and range from the mean, median, mode calculator often describe it better.
Frequently asked questions
How do you calculate standard deviation?
Find the mean, subtract it from each value and square the result, add the squares, divide by n โ 1 for a sample (or N for a population), and take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the population standard deviation is 2.
Should I use sample or population standard deviation?
Use the sample version (divide by n โ 1) when your numbers are a subset of a larger group you want to describe, which is most homework and research data. Use the population version (divide by N) only when you have every member of the group.
What is the difference between variance and standard deviation?
Variance is the average squared deviation from the mean; standard deviation is its square root. Standard deviation is in the same units as the data, so it is easier to interpret.
What does a standard deviation tell you?
How spread out the values are around the mean. For roughly bell-shaped data about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
Why divide by n โ 1 for a sample?
A sample's values sit closer to their own mean than to the true population mean, so dividing by n underestimates the spread. Dividing by n โ 1 (Bessel's correction) removes that bias from the variance.
Learn more
Key terms
Glossary โ- Standard Deviation
- Standard deviation is a measure of spread equal to the square root of the variance; it shows how far values typically fall from the mean, in the same units as the data.
- Variance
- Variance is a measure of spread equal to the average squared distance of values from their mean; its square root is the standard deviation, and it is expressed in squared units.