This standard deviation calculator finds the mean, variance and standard deviation of any set of numbers — and shows every step of the working. Paste or type your data below, choose population or sample, and get the results with a full step table.
| x | x − mean | (x − mean)² |
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What Is Standard Deviation?
Standard deviation measures how spread out numbers are around their mean. A small standard deviation means values cluster close to the average; a large one means they are scattered widely. It is the most common measure of variability in statistics — used for test scores, quality control, finance risk, science experiments and weather data. This page is part of our Statistics Calculators collection; for the center of the data itself, see the Mean Median Mode Calculator.
How to Calculate Standard Deviation (Steps)
- Step 1: Find the mean (average) of the data.
- Step 2: Subtract the mean from every value (the deviations).
- Step 3: Square each deviation.
- Step 4: Add the squared deviations together.
- Step 5: Divide by n for a population, or n − 1 for a sample — that is the variance.
- Step 6: Take the square root — that is the standard deviation.
The calculator above prints this exact table for your data, which makes it ideal for homework checking.
Worked Example (Verified)
The classic textbook data set 2, 4, 4, 4, 5, 5, 7, 9: mean = 5. Squared deviations sum to 32.
- Population: variance = 32 ÷ 8 = 4 → SD = 2.000
- Sample: variance = 32 ÷ 7 = 4.571 → SD = 2.138
Population vs. Sample Standard Deviation
Use population when your data is the complete group you care about — every student in one class, every day of last year. Use sample when your data is a subset meant to represent a bigger group — 200 surveyed voters standing in for millions. Dividing by n − 1 (Bessel’s correction) compensates for the sample underestimating the true spread. When in doubt for research and surveys, sample is the standard choice.
Once you have a mean and standard deviation, the Z-Score Calculator shows where any single value sits in the distribution, and the Probability Calculator handles event probabilities. Averages alone are in the Average Calculator.
Related Tools
- Z-Score Calculator — position of a value in a distribution
- Probability Calculator — single and multiple events
- Mean Median Mode Calculator
- Average Calculator
Frequently Asked Questions
What is a good standard deviation?
There is no universal “good” value — it depends on the units and context. What matters is the size relative to the mean: an SD of 2 on test scores averaging 80 is tight consistency; an SD of 2 on scores averaging 10 is wide spread.
Why do we square the deviations?
Raw deviations always sum to zero (positives cancel negatives). Squaring makes them all positive and penalizes large deviations more, which is what “spread” should reflect.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root, back in the original units. Report standard deviation when you want the number to be directly comparable with the data.
Can standard deviation be zero?
Yes — when every value is identical, there is no spread at all.
How do I calculate standard deviation in Excel?
Use =STDEV.P for a population or =STDEV.S for a sample. This calculator matches both, and shows the steps Excel hides.
Is this calculator free?
Yes — free, instant, no sign-up.