Everything is worked out in your browser and saved on this device for next time.
About the standard deviation calculator
The standard deviation tells you how spread out your numbers are. A small value means the numbers sit close to the average. A big value means they are spread far apart.
This calculator shows:
- standard deviation (sample, s) — the usual one
- standard deviation (population, σ)
- variance for both (the standard deviation squared)
- The mean and how many numbers
- The standard error of the mean
Below the result is a table with each number, its distance from the mean (x − x̄) and that distance squared. So you can follow every step, or copy it for homework.
Features
- Sample and population — both standard deviations at once, so you do not have to choose first
- More results — both variances, the mean, the count and the standard error of the mean
- Step table — each number, its distance from the mean and that distance squared, plus the sum
- Paste any list — commas, spaces, semicolons or new lines all work
- Up to 5,000 numbers — the step table shows the first 40
- Live update — the result changes while you type
- Copy — copy the result with one click
- Private — it works in your browser, nothing is uploaded, and your numbers are saved on this device
How to use it
- Type or paste your numbers in the box Your numbers — any separator, or one per line
- Use commas, spaces, semicolons or new lines — all work
- You need at least two numbers
- Read the sample value if your numbers are a sample, or the population value if they are the whole group
- Press Copy to copy the result, or Start again to clear it
How it is worked out
- Find the mean: add the numbers and divide by the count (n)
- For each number, subtract the mean and square the result
- Add those squares (the sum of squares)
- Divide by n − 1 for the sample variance, or by n for the population variance
- Take the square root to get the standard deviation
Example
The tool opens with 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5.
| x | x − x̄ | (x − x̄)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 |
| sum | 32 |
| Result | Value |
|---|---|
| Sample variance | 32 ÷ 7 = 4.5714 |
| Sample standard deviation (s) | 2.1381 |
| Population variance | 32 ÷ 8 = 4 |
| Population standard deviation (σ) | 2 |
| Standard error of the mean | 0.7559 |
More examples
| Numbers | Mean | s (sample) | σ (population) |
|---|---|---|---|
| 60, 70, 80 | 70 | 10 | 8.165 |
| 10, 12, 23, 23, 16, 23, 21, 16 | 18 | 5.2372 | 4.899 |
Frequently asked questions
Sample or population — which one do I use?
Use the sample value when your numbers are only part of a bigger group, like 30 people from a whole town. That is the usual case. Use the population value when you have every member of the group.
Why divide by n − 1 for a sample?
A sample tends to sit a little closer to its own mean than the whole group does. Dividing by n − 1 instead of n corrects for that, so the result is not too small.
What is variance?
Variance is the standard deviation squared. It is in squared units, so the standard deviation is easier to read.
What is the standard error of the mean?
It shows how much the mean of a sample would change from one sample to the next. It is the sample standard deviation divided by the square root of n.
What is a good or bad standard deviation?
There is no fixed rule. Compare it with the mean: a standard deviation of 2 is small for numbers around 100, but large for numbers around 5.
How many numbers can I enter?
Up to 5,000. The step table shows the first 40 rows.
Good to know
- Everything is worked out in your browser — nothing is sent anywhere
- Your numbers are saved on this device for next time
- Results show up to 4 decimal places
- For roughly bell-shaped data, about 68 % of the values fall within one standard deviation of the mean
Related tools
- Average Calculator — mean, median, mode and range
- Statistics Calculator — more statistics in one place
- Random Number Generator — make test numbers
- Percentage Calculator — work with percentages
- Scientific Calculator — do the steps by hand
