Standard Deviation Calculator
This standard deviation calculator measures how spread out your numbers are. Paste a list, choose sample or population, and you'll get the standard deviation along with the variance, the mean, and a chart that shows the spread around the average. Every result comes with the steps, so it's just as good for learning the method as it is for getting a quick answer.
- Sample and population
- Variance and mean
- Paste any list
- Chart with spread band
- Steps shown
Last updated June 16, 2026 Sample (n − 1) and population (n) Reviewed by the Calcowa math team
Separate with commas, spaces, or new lines. Decimals and negatives are fine.
Enter at least two numbers to see the standard deviation.
√( Σ(x − mean)² ÷ (n − 1) )
What is standard deviation?
Standard deviation tells you how spread out a set of numbers is around their average. A small standard deviation means the values sit close to the mean, so they're consistent. A large one means they're scattered far from the mean. It's the most common way to describe spread, because it comes back in the same units as the data, unlike variance.
How do you calculate standard deviation?
Here's the method, the same one the calculator runs:
- 1
Find the meanAdd the numbers and divide by how many there are.
- 2
Square each differenceTake each value minus the mean, then square it so it turns positive.
- 3
Average the squaresDivide the total by n for a population, or by n minus 1 for a sample. That is the variance.
- 4
Take the square rootThe square root of the variance is the standard deviation.
Sample vs population standard deviation
The only difference is the number you divide by. If your numbers are the whole group you care about, that's a population, and you divide by n. If they're a sample drawn from a bigger group, you divide by n minus 1, which nudges the answer up a little to fix the way a sample understates the real spread. Most real-world data is a sample, so when you aren't sure, the sample version is the safer pick. This tool shows both, so you don't have to choose blind.
Variance and the formula
Variance is the step right before the standard deviation: it's the average of the squared differences from the mean, and the standard deviation is its square root. So the standard deviation formula is the square root of Σ(x − mean)² ÷ n for a population, or the same thing over n − 1 for a sample. Because variance is in squared units, like dollars squared, people usually report the standard deviation, which lands back in plain dollars. The result above lists the variance next to the standard deviation, so you've got both at a glance.
Standard deviation and z-scores
Once you have the mean and standard deviation, you can say how unusual a single value is with a z-score, which counts how many standard deviations a value sits from the mean. A z-score of 2 means a value's two standard deviations above average. It's a handy way to compare scores from different scales, and it leans directly on the standard deviation you work out here. The z-score calculator takes it from there, turning a value into a z-score, a percentile, and a probability.
Frequently asked questions
Why do you square the differences?
Squaring turns every difference positive, so values above and below the mean don't cancel out. It also weights bigger gaps more heavily. Taking the square root at the end brings the answer back to the original units.
Find the mean, subtract it from each value and square the result, average those squared differences to get the variance, then take the square root. For a sample you divide by one less than the count, and for a whole population you divide by the count. The calculator shows both and walks through the steps.
It's the number you divide by. Population standard deviation divides by the count, n, because you have every value. Sample standard deviation divides by n minus 1, which corrects for the way a sample usually understates the spread. When in doubt with real data from a sample, use the sample version.
Variance is the average of the squared differences from the mean, and the standard deviation is just its square root. They measure the same spread, but variance is in squared units while the standard deviation is back in the original units, which is why people usually quote the standard deviation.
For a population it's the square root of the sum of (each value minus the mean) squared, divided by n. For a sample, you divide by n minus 1 instead. In symbols that's the square root of Σ(x − mean)² ÷ n for a population, swapping n for n − 1 for a sample.
A low standard deviation means the values huddle close to the mean, so the data is consistent. A high one means they're spread far from the mean, so the data is more variable. A standard deviation of zero means every value is identical.
Yes, the calculator handles negatives and decimals fine. Squaring the differences makes everything positive along the way, so a negative value affects the spread the same as a positive one the same distance from the mean.
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