Standard Deviation Calculator
Enter a list of numbers to get the standard deviation and variance, for a sample or a whole population, along with the mean and count.
Last updated November 2, 2026
Separate numbers with commas, spaces or new lines.
Result
Standard deviation = 2.13809
Sample standard deviation (n − 1)
| Standard deviation | 2.13809 |
|---|---|
| Variance | 4.571429 |
| Mean | 5 |
| Count | 8 |
| Sum | 40 |
Show the work
Formula: standard deviation = square root of (sum of squared distances from the mean / n − 1)
- Mean: 40 / 8 = 5
- Square each distance from the mean: (2 − 5)² = 9; (4 − 5)² = 1; (4 − 5)² = 1; (4 − 5)² = 1; (5 − 5)² = 0; (5 − 5)² = 0; (7 − 5)² = 4; (9 − 5)² = 16
- Add the squares: 32
- Sample variance: 32 / 7 = 4.571429
- Square root of the variance: √4.571429 = 2.13809
Result: Standard deviation = 2.13809
How standard deviation is calculated
s = √( Σ(x − mean)² ÷ (n − 1) ) for a sample
σ = √( Σ(x − mean)² ÷ n ) for a population
Standard deviation measures how spread out numbers are around their mean. Find the mean, subtract it from each number, square each difference, add the squares, divide, and take the square root. The result is in the same units as the data.
Divide by n (the count) when your numbers are the whole population, and by n − 1 when they are a sample drawn from a larger group. The n − 1 version is the sample standard deviation and is slightly larger. The variance is the same calculation without the final square root.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Z-Score Calculator, Probability Calculator and Sample Size Calculator.
Sample or population?
Use population when the list is every value you care about, such as the scores of all students in one class. Use sample when the list is only part of a larger group, such as 30 customers out of thousands. If you are unsure, sample is the usual choice.
A sample standard deviation needs at least two numbers. The population standard deviation of a single number is 0.
What the number tells you
The standard deviation only makes sense next to the mean. Two lists can share a mean and still differ in spread. Scores of 70, 75, 80, 85 and 90 and scores of 50, 70, 80, 90 and 110 both average 80. The first has a sample standard deviation of about 7.9057 and the second about 22.3607, so the second group is nearly three times as spread out.
Because the result is in the same units as your data, it can be read directly: a standard deviation of 7.9 on test scores means a typical score sits roughly 8 points from the average.
How one extreme value changes it
Every difference is squared, so a value far from the mean has an outsized effect. The list 10, 12, 11, 13 and 9 has a mean of 11 and a population standard deviation of about 1.4142. Replace the 9 with 60 and the mean jumps to 21.2, while the population standard deviation grows to about 19.4258, more than thirteen times larger, although four of the five numbers did not change.
If a result looks surprisingly large, check the list for a typing error such as an extra digit or a misplaced decimal point before drawing conclusions. The median, which barely moves in the example above (11 before, 12 after), is a useful companion check.
Standard deviations from the mean
A z-score counts how many standard deviations a value lies from the mean: z = (value − mean) ÷ standard deviation. With a mean of 100 and a standard deviation of 15, a value of 85 has z = (85 − 100) ÷ 15 = −1, one standard deviation below average.
For data that forms a roughly bell-shaped curve, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. This rule of thumb does not hold for lopsided data, so treat it as a guide rather than a guarantee.
How to use the Standard Deviation Calculator
- Type or paste your numbers, separated by commas, spaces or new lines.
- Choose sample or population.
- Read the standard deviation, variance, mean and count.
- Use Copy, Print or Share to keep the result.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 ÷ 8 = 5. The squared differences from 5 are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. For a population, the variance is 32 ÷ 8 = 4 and the standard deviation is √4 = 2. For a sample, the variance is 32 ÷ 7 ≈ 4.5714 and the standard deviation is about 2.1381.
| Numbers | Population σ | Sample s |
|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | 2 | 2.1381 |
| 1, 2, 3, 4, 5 | 1.4142 | 1.5811 |
| 2, 4, 6, 8 | 2.2361 | 2.582 |
| 5, 15 | 5 | 7.0711 |
| 10, 10, 10 | 0 | 0 |
Frequently asked questions
What is the difference between sample and population standard deviation?
Population divides the squared differences by n, sample divides by n − 1. Use population when you have every value, and sample when your numbers are only part of a larger group.
What does a standard deviation of 0 mean?
Every number in the list is the same, so there is no spread.
How is variance related to standard deviation?
Standard deviation is the square root of the variance, so the variance is the standard deviation squared.
Why does the sample version divide by n − 1?
A sample tends to understate the spread of the full group, and dividing by n − 1 corrects for that.
Are my numbers stored or sent anywhere?
No. The calculation runs in your browser and nothing you enter is saved or uploaded.