Standard Deviation
Standard deviation calculator for a list of numbers: population σ and sample s side by side, both variances, plus mean, median, minimum, maximum, range and sum. How it works
Use the Standard Deviation
About the Standard Deviation
The standard deviation of 10, 12, 23, 23, 16, 23, 21, 16 is 4.899 as a population (σ) and 5.2372 as a sample (s). Find the mean, square each value’s distance from it, add the squares, divide by n for a population or by n − 1 for a sample, and take the square root. This standard deviation calculator shows both results for any list of numbers, along with the variances, mean and median, the steps worked with your numbers and a plot of the values.
Standard deviation by hand: 10, 12, 23, 23, 16, 23, 21, 16
| Value x | x − mean | (x − mean)² |
|---|---|---|
| 10 | -8 | 64 |
| 12 | -6 | 36 |
| 23 | 5 | 25 |
| 23 | 5 | 25 |
| 16 | -2 | 4 |
| 23 | 5 | 25 |
| 21 | 3 | 9 |
| 16 | -2 | 4 |
| Sum: 144 | 0 | 192 |
Mean = 144 ÷ 8 = 18. Population variance = 192 ÷ 8 = 24, so σ = √24 = 4.899. Sample variance = 192 ÷ 7 = 27.4286, so s = 5.2372.
Population and sample results for small data sets
| Data set | n | Mean | Population σ | Sample s | Population variance σ² | Sample variance s² |
|---|---|---|---|---|---|---|
| 10, 12, 23, 23, 16, 23, 21, 16 | 8 | 18 | 4.899 | 5.2372 | 24 | 27.4286 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 8 | 5 | 2 | 2.1381 | 4 | 4.5714 |
| 1, 2, 3, 4, 5 | 5 | 3 | 1.4142 | 1.5811 | 2 | 2.5 |
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | 10 | 5.5 | 2.8723 | 3.0277 | 8.25 | 9.1667 |
| 10, 20, 30 | 3 | 20 | 8.165 | 10 | 66.6667 | 100 |
| 85, 90, 78, 92, 88 | 5 | 86.6 | 4.8826 | 5.4589 | 23.84 | 29.8 |
| 1, 2, 3, 4, 100 | 5 | 22 | 39.0128 | 43.6177 | 1,522 | 1,902.5 |
| 5, 5, 5, 5 | 4 | 5 | 0 | 0 | 0 | 0 |
| 42 | 1 | 42 | 0 | — | 0 | — |
Rounded to 4 decimal places, as the calculator shows results of 1 or more. A dash means undefined: a sample needs at least two values.
How to use this tool
- Type or paste your numbers into the Data Set box, separated by commas, spaces or new lines.
- Read Pop. Std Dev (σ) if the numbers are the whole group, or Sample Std Dev (s) if they are a sample of a larger one.
- Open Show the steps for the mean, sum of squares, divisions and square roots with your numbers, and check the Deviation table: x, x − mean and (x − mean)² for each value.
- Read the distribution plot: one dot per value, with the band one standard deviation either side of the mean shaded.
- Use Copy results to take all eleven figures as text.
Frequently asked questions
- How do I calculate standard deviation by hand?
- Find the mean. Subtract it from each value and square the result. Add the squares. Divide by n for a population or by n − 1 for a sample, then take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squares add up to 32, and 32 ÷ 8 = 4, so σ = 2. Show the steps writes the same steps with your own numbers, here 32 ÷ 8 = 4 and σ = √4 = 2, and the Deviation table shows each column, so you can check a hand calculation line by line.
- What is the difference between population and sample standard deviation?
- The divisor. Population σ divides the sum of squares by n and describes exactly the numbers you entered. Sample s divides by n − 1, which makes it a little larger, because a sample tends to understate the spread of the group it was drawn from. For 1, 2, 3, 4, 5, σ is 1.4142 and s is 1.5811.
- Which one should I use, σ or s?
- Use σ when your data is every member of the group you care about, such as the scores of every student in one class. Use s when the data is a sample and you want to estimate the spread of a larger group. If a problem says "a sample of", it wants s.
- Which result matches STDEV.S and STDEV.P in Excel?
- Sample s matches STDEV.S, which Microsoft documents as using the "n-1" method. Population σ matches STDEV.P, which uses the "n" method. For 10, 20, 30 that is 10 and 8.165.
- What is the variance?
- The standard deviation squared: the average squared distance from the mean, before the square root is taken. For 10, 20, 30 the population variance is 66.6667 and the sample variance is 100.
- Can I paste data from Excel or Google Sheets?
- Yes. Copy a column or a row and paste it in: line breaks, tabs and spaces all work as separators. Write decimals with a point, because a comma is read as a separator and 3,5 is taken as 3 and 5.
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