Standard Deviation Calculator
Quick answer: Standard deviation measures how spread out numbers are around their mean. For the data 4, 8, 6, 5, 3, the mean is 5.2, the population standard deviation is 1.72 and the sample standard deviation is 1.92. Paste your numbers to get both.
Use the Standard Deviation Calculator
Standard Deviation formula and what each symbol means
Population: σ = √(Σ(x − μ)² ÷ N). Sample: s = √(Σ(x − x̄)² ÷ (n − 1)).
| Symbol | Meaning |
|---|---|
| x | Each value |
| μ or x̄ | Mean of the values |
| N or n | Number of values |
Standard Deviation examples
Data 4, 8, 6, 5, 3
Mean = 26 ÷ 5 = 5.2. Squared differences: 1.44, 7.84, 0.64, 0.04, 4.84; sum = 14.8. Population variance = 14.8 ÷ 5 = 2.96, so σ = 1.72. Sample variance = 14.8 ÷ 4 = 3.7, so s = 1.92.
How to calculate Standard Deviation step by step
- Find the mean.
- Subtract the mean from each value and square the result.
- Add the squares.
- Divide by N (population) or n − 1 (sample).
- Take the square root.
How to read your Standard Deviation result
A small standard deviation means values sit close to the mean. In roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean.
Common mistakes when calculating Standard Deviation
- Using the population formula on a sample.
- Forgetting to square-root the variance.
- Reading standard deviation without checking for outliers.
Standard Deviation FAQ
When should I use sample vs population standard deviation?
Use population when you have every value in the group. Use sample when your data is a subset used to estimate a larger group.
Why divide by n − 1 for a sample?
A sample tends to underestimate spread. Dividing by n − 1 corrects this bias, known as Bessel's correction.
How do I calculate standard deviation in Excel?
Use STDEV.S for a sample and STDEV.P for a population.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean. Standard deviation is its square root, so it has the same units as the data.
Method, review and disclaimer
Edited by: Hemant Kumar and Anjita Rai, founders of ToolsVault (SEO and content professional, not a licensed adviser). Method: this tool runs in your browser using the formula shown on this page; the numbers you enter are not sent to our servers.