Standard deviation calculator.
Measure how spread out your data is. Paste your numbers as a comma-separated list and the calculator returns the sample and population standard deviation, the variance and the mean.
Your data set
LiveSample standard deviation divides by n minus 1; population standard deviation divides by n. Use sample when your data is a subset of a larger group.
Sample standard deviation
13.49
s, dividing by n minus 1
Treats your entry as a complete data set of numbers. Calculations run in your browser; nothing you enter is stored.
How it works
Spread around the mean
Standard deviation tells you how far, on average, the values in a data set sit from their mean. A small standard deviation means the numbers cluster tightly; a large one means they are spread out.
You find it by taking each value’s distance from the mean, squaring it, averaging those squares to get the variance, and taking the square root. Whether you divide by n or by n minus 1 decides between population and sample standard deviation.
Reference
Standard deviation examples.
The sample standard deviation for a few small data sets.
| Data set | Sample SD |
|---|---|
| 2, 4, 6, 8, 10 | 3.16 |
| 1, 2, 3, 4, 5 | 1.58 |
| 5, 15, 25 | 10.00 |
| 10, 10, 10, 10 | 0.00 |
| 4, 8, 15, 16, 23, 42 | 13.49 |
The full guide
The complete guide to standard deviation.
What it measures, the difference between sample and population, and how to read it.
What standard deviation measures
Standard deviation is the typical distance between a value and the mean of its data set. It is in the same units as the data, which makes it easier to interpret than the variance, its squared cousin.
Two data sets can share a mean yet look completely different; the standard deviation captures that difference in spread.
Sample versus population
If your numbers are the entire group you care about, use the population standard deviation, which divides by n. If they are a sample drawn from a larger population, use the sample standard deviation, which divides by n minus 1 to correct for the bias of estimating from a sample.
In statistics you usually have a sample, so the sample value is the common default; the calculator shows both.
How it is calculated
Find the mean, subtract it from each value, and square each difference. Add those squares and divide by n for the population or n minus 1 for a sample to get the variance, then take the square root for the standard deviation.
The data set 4, 8, 15, 16, 23, 42 has a mean of 18 and a sample standard deviation of about 13.49.
Reading the result
About two thirds of values in a roughly normal data set fall within one standard deviation of the mean, and about 95% within two. That rule of thumb makes the standard deviation a quick guide to what counts as typical or unusual.
Outliers inflate it, so check for stray values if the number looks surprisingly large.
The formula
Square the
distances.
Variance is the mean squared distance from the mean; the standard deviation is its square root, dividing by n or by n minus 1.
Statistics calculator ›# Standard deviation
variance = sum( (x - mean)^2 ) / N
sd = sqrt(variance)
# N = n (population) or n-1 (sample)
mean(4,8,15,16,23,42)=18, sample sd=13.49Keep going
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Questions
Standard deviation questions.
How do I calculate standard deviation?
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Find the mean, square each value’s distance from it, average those squares to get the variance, and take the square root. Paste your numbers above and the calculator does it for both sample and population.
What is the difference between sample and population SD?
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Population standard deviation divides the squared differences by n; sample standard deviation divides by n minus 1. Use sample when your data is a subset of a larger group.
Why divide by n minus 1 for a sample?
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Dividing by n minus 1, known as Bessel’s correction, removes the downward bias you get when estimating spread from a sample rather than the whole population.
What does a high standard deviation mean?
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It means the values are spread widely around the mean. A low standard deviation means they cluster closely. It is always in the same units as your data.
Is this standard deviation calculator free?
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Yes. It is completely free with no sign-up, and every calculation runs locally in your browser, so nothing you enter is stored or sent anywhere.
About the developer
Jean Borg
Jean builds and maintains every calculator on freecalculators.pro from Malta, with a focus on tools that are fast, free and show their working. The standard deviation calculator reports both the sample and population values along with the variance and mean.