About this calculator
Standard deviation measures how spread out a set of numbers is. A small value means the numbers cluster near the average; a large one means they are scattered.
Paste your numbers separated by commas or spaces and choose sample (your numbers are a selection from a bigger group, the usual case) or population (you have every value). Both versions are always shown.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
Classic set: 2, 4, 4, 4, 5, 5, 7, 9
- Standard deviation (sample)
- 2.13809
- Standard deviation (population)
- 2
- Mean
- 5
- Variance (sample)
- 4.571429
- Variance (population)
- 4
- Count
- 8
- Standard error of the mean
- 0.755929
- Coefficient of variation
- 42.761799%
The sample standard deviation of 8 values is 2.13809 around a mean of 5.
Show the working
- Mean = sum ÷ count = 40 ÷ 8 = 5 (values: 2, 4, 4, 4, 5, 5, 7, 9).
- Add the squared distances from the mean: Σ(x − mean)² = 32.
- Sample variance = 32 ÷ (8 − 1) = 4.571429.
- Standard deviation = √variance = √4.571429 = 2.13809.
Same numbers, as a whole population
- Standard deviation (population)
- 2
- Standard deviation (sample)
- 2.13809
- Mean
- 5
- Variance (sample)
- 4.571429
- Variance (population)
- 4
- Count
- 8
- Standard error of the mean
- 0.755929
- Coefficient of variation
- 42.761799%
The population standard deviation of 8 values is 2 around a mean of 5.
Show the working
- Mean = sum ÷ count = 40 ÷ 8 = 5 (values: 2, 4, 4, 4, 5, 5, 7, 9).
- Add the squared distances from the mean: Σ(x − mean)² = 32.
- Population variance = 32 ÷ 8 = 4.
- Standard deviation = √variance = √4 = 2.
Test scores: 72, 85, 90, 64, 78
- Standard deviation (sample)
- 10.305338
- Standard deviation (population)
- 9.217375
- Mean
- 77.8
- Variance (sample)
- 106.2
- Variance (population)
- 84.96
- Count
- 5
- Standard error of the mean
- 4.608687
- Coefficient of variation
- 13.245936%
The sample standard deviation of 5 values is 10.305338 around a mean of 77.8.
Show the working
- Mean = sum ÷ count = 389 ÷ 5 = 77.8 (values: 72, 85, 90, 64, 78).
- Add the squared distances from the mean: Σ(x − mean)² = 424.8.
- Sample variance = 424.8 ÷ (5 − 1) = 106.2.
- Standard deviation = √variance = √106.2 = 10.305338.
The formulas
- Mean: add the values and divide by how many there are
- Population variance: σ² = Σ(x − mean)² ÷ n
- Sample variance: s² = Σ(x − mean)² ÷ (n − 1)
- Standard deviation: the square root of the variance
Sample or population?
Use population only when your list is everything: every student in the class, every item in the batch. Use sample when your list is a selection used to estimate a bigger group, which is most real-world data. Dividing by n − 1 corrects for the fact that a sample tends to understate the spread.
Reading the result
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two. The coefficient of variation (standard deviation as a percentage of the mean) lets you compare spread between data sets with different units or sizes.
Frequently asked questions
What is standard deviation?
The typical distance of values from the average. It is the square root of the variance.
What is the difference between sample and population standard deviation?
Population divides by n; sample divides by n − 1 to correct for estimating from a selection.
Which one should I use?
Sample, unless you have every value in the group you care about.
What is variance?
The average of the squared distances from the mean. The standard deviation is its square root, so it is in the same units as your data.
Can the standard deviation be zero?
Yes, when every value is identical.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.