How to use the Standard Deviation Calculator
- Type or paste your numbers, separated by commas, spaces or semicolons.
- Choose sample if your numbers are a subset of a larger group, or population if they are the whole group.
- Read the standard deviation, variance, mean, sum and count.
- Change a number to see how the spread changes.
Formula used
Variance = Σ(x − mean)² ÷ (n − 1) for a sample, or ÷ n for a population. Standard deviation is the square root of the variance.
Worked example
Here is the Standard Deviation Calculator worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Numbers | 4, 8, 6, 5, 3, 7, 9, 5 |
| Data is a | Sample (divide by n − 1) |
| Result | |
| Standard deviation | 2.03 |
| Variance | 4.13 |
| Mean | 5.88 |
| Sum | 47 |
| Count | 8 |
The calculation, step by step
For the numbers 4, 8, 6, 5, 3, 7, 9, 5: the sum is 47, so the mean is 47 ÷ 8 = 5.875. Subtract the mean from each number, square the differences and add them up to get the sum of squares, 28.875.
For a sample, divide by n − 1 (7) to get the variance 4.125; the standard deviation is its square root, 2.031. For a population, divide by n (8) to get the variance 3.6094 and the standard deviation 1.8998.
Sample or population?
If your numbers are every member of the group you care about, such as the test scores of every student in a class when you only care about that class, use the population formula. If they are a smaller set used to learn about a larger group, such as a survey of 200 people from a city, use the sample formula.
The sample formula divides by n − 1 instead of n. This correction makes the estimate of spread from a sample slightly larger, to make up for the fact that a sample tends to understate the true spread. For large datasets the two results are almost identical.
What the number tells you
Standard deviation is in the same units as your data. A set with a mean of 5.9 and a standard deviation of 2 has most values within about two units of the mean. If the numbers were spread more widely, say 1, 12, 3, 9, the standard deviation would be 5.12.
For data that follows a bell-shaped (normal) distribution, about 68% of values fall within one standard deviation of the mean and about 95% within two. For other shapes this rule does not hold.
Where it is used
Standard deviation appears in statistics, science, quality control, finance and education. In finance it is a common measure of the volatility of returns, in manufacturing it shows how consistent a process is, and in grading it shows how spread out student scores are.
It is sensitive to outliers, because differences are squared. A single extreme value can raise it a lot, so look at the data as well as the number.
Frequently asked questions
Sample or population?
Use population if your numbers are the whole group you care about. Use sample if they are a subset used to estimate a larger group.
What does a high standard deviation mean?
The values are spread widely around the mean. A low value means they are clustered close to it.
What is variance?
The average of the squared differences from the mean. The standard deviation is its square root.
Why divide by n − 1 for a sample?
To correct for the fact that a sample tends to understate the spread of the whole group.
How many numbers can I enter?
As many as you can paste. Separate them with commas, spaces or semicolons.
Can the standard deviation be negative?
No. It is a square root, so it is zero or positive. It is zero only when all the numbers are equal.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.