Standard Deviation Calculator
Find the standard deviation of a list of numbers. Paste or type your data, choose whether it's a sample or a whole population, and see the standard deviation, variance, and mean.
Standard deviation
2.1381
Check the highlighted fields to see your result.
- Variance
- 4.5714
- Mean
- 5
- Count
- 8
Results are estimates for information only and do not constitute financial, medical, or legal advice.
How it's calculated
First, find the mean. Then subtract the mean from each value and square the result, so negative and positive differences don't cancel out. Add up those squares and divide by n for a population or by n − 1 for a sample: that's the variance. The standard deviation is its square root, which brings it back to the same units as your data.
The formulas
Population: σ = √(Σ(x − x̄)² ÷ n)
Sample: s = √(Σ(x − x̄)² ÷ (n − 1))
Worked example
Data: 2, 4, 4, 4, 5, 5, 7, 9.
- Mean: 40 ÷ 8 = 5.
- Squared differences: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
- Population: √(32 ÷ 8) = √4 = 2.
- Sample: √(32 ÷ 7) ≈ 2.1381.
Frequently asked questions
Should I use the sample or the population formula?
Use the population formula when your data includes every member of the group you care about, like the scores of everyone in one class. Use the sample formula when your data is part of a larger group and you want to estimate the spread of that group, like a survey of 100 voters.
Why does the sample formula divide by n − 1?
A sample's values tend to sit closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 (Bessel's correction) makes up for that.
What does the standard deviation tell me?
How spread out the values are around the mean. A small standard deviation means most values are close to the mean; a large one means they are more scattered. In a bell-shaped (normal) distribution, about 68% of values fall within one standard deviation of the mean.