Standard Deviation Calculator
Enter a dataset to calculate mean, variance, standard deviation, and more — with a step-by-step breakdown.
Enter numbers separated by commas, spaces, or semicolons
Calculation type
Load example data
Population (σ)
2
Standard Deviation
Mean (x̄)
5
Variance
4
Median
4.5
Count (n)
8
Sum
40
Minimum
2
Maximum
9
Range
7
Step-by-Step Deviations
| Value (x) | Deviation (x − x̄) | Deviation² (x − x̄)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | +0 | 0 |
| 5 | +0 | 0 |
| 7 | +2 | 4 |
| 9 | +4 | 16 |
Frequently Asked Questions
Standard deviation (SD) measures how spread out numbers are from their mean (average). A low SD means values are clustered close to the mean; a high SD means they are spread widely. For example, two classes both averaging 70% on a test: if SD = 5, most scored 65–75%; if SD = 20, scores ranged from 50% to 90%. SD is used in statistics, finance (volatility), quality control, and science to quantify variability.
Population SD (σ) is used when your dataset IS the entire population — e.g., heights of all 30 students in one class. Sample SD (s) is used when your dataset is a sample drawn from a larger population — e.g., 30 students chosen from a whole school. The formula differs by one step: population divides by n, sample divides by n−1 (Bessel's correction). When in doubt, use sample SD if your data represents a subset.
Step 1: Find the mean (average) of all values. Step 2: For each value, subtract the mean to get the deviation. Step 3: Square each deviation. Step 4: Sum all squared deviations. Step 5: Divide by n (population) or n−1 (sample) to get the variance. Step 6: Take the square root of the variance — that is the standard deviation. This calculator shows all steps in the deviation table below the results.
Finance: SD measures stock price volatility — a volatile stock has high SD. Quality control: a manufacturing process with low SD produces consistently sized parts. Exam grading: teachers use SD to understand score distribution and set grade boundaries. Weather: daily temperature SD shows how much temperatures vary around the seasonal average. Research: SD is reported alongside the mean in almost every scientific study.