Standard Deviation Calculator
Find the standard deviation, variance, and mean of any data set: sample or population, with every step shown.
Details
Separate them with commas, spaces, or new lines.
Standard deviation (s)
2.1381
From 8 numbers, treated as a sample
Average (mean)
5
How many numbers
8
Total (sum)
40
Squared differences
32
This measures how spread out a set of numbers is. Paste or type your data, say whether it is a sample or the whole group, and it returns the standard deviation.
It also gives the mean, the variance and the standard error, and shows the working step by step so you can see where each figure came from.
What is standard deviation?
Standard deviation measures how spread out a set of numbers is. A small value means they cluster near the average; a large one means they are scattered.
It matters because the average alone can hide everything. Two classes can both average 60% in a test: one where nearly everyone scored between 55 and 65, and one where half scored 30 and half scored 90. Standard deviation is what separates them.
The answer comes out in the same units as your data. Test scores in marks give a standard deviation in marks, which makes it easy to read: it is roughly the typical distance from the average.
Squaring the gaps stops the negatives cancelling out the positives. Taking the square root at the end undoes that, returning the answer to the original units.
What to enter
- What do you want to find?
- Standard deviation, variance (the same thing before the square root), or standard error of the mean.
- Your numbers
- Type or paste your data, separated by commas, spaces, or new lines.
- Sample or whole group?
- The most important choice here. Pick 'whole group' only if your numbers cover every single member. If they are a sample standing in for a bigger group, pick sample.
What this assumes
Every value is counted equally. Weighted data needs a different calculation.
The numbers are the actual measurements, not already-averaged figures.
How to calculate standard deviation
Four steps, and the middle two are the ones worth understanding.
- mean
- The ordinary average: add them up, divide by how many
- how many
- Use n for a whole group, or n − 1 for a sample
Find the mean. Add all the numbers and divide by how many there are. The mean, median and mode calculator does this on its own if you need it.
Find each gap, and square it. Subtract the mean from each value. Squaring turns every gap positive, so numbers below the average do not cancel out the ones above.
Average those squares. Add them up and divide. This result is the variance.
Take the square root. That converts back to the original units and gives the standard deviation.
See a worked example: the numbers 2, 4, 4, 4, 5, 5, 7, 9
- Values
- 2, 4, 4, 4, 5, 5, 7, 9
- Count
- 8
Mean: they add to 40, and 40 ÷ 8 = 5.
Gaps from 5: −3, −1, −1, −1, 0, 0, 2, 4.
Squared: 9, 1, 1, 1, 0, 0, 4, 16. These add to 32.
Variance: 32 ÷ 8 = 4. Square root of 4 is 2.
Treated as a sample instead, you would divide by 7 rather than 8, giving 4.571 and a standard deviation of 2.138.
Standard deviation = 2 (whole group)
Frequently asked questions
Pick population only if your numbers include every member of the group you care about, such as the test scores of all 30 students in a class you are describing.
Pick sample if the numbers stand in for a larger group, such as 30 people surveyed to learn about a city. Most real data is a sample.
Because a sample's own mean sits slightly closer to its own values than the true mean of the whole group does, which makes the spread look smaller than it really is.
Dividing by one less corrects for that. On our example it lifts the answer from 2 to 2.138, and the smaller your sample, the bigger the correction.
Variance is the step before the square root. Standard deviation is the square root of the variance.
They tell you the same thing, but variance is in squared units, which is hard to interpret. If test marks give a variance of 4 'marks squared', the standard deviation of 2 marks is the one that actually means something.
There is no universal threshold, because it depends entirely on your units and scale. A standard deviation of 5 is huge for shoe sizes and tiny for house prices.
To compare across different things, divide the standard deviation by the mean, which the percentage calculator will do for you. That gives a relative measure that is not tied to units.
For data with a normal bell-shaped spread, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.
It is what makes the number useful in practice: with a mean of 100 and a standard deviation of 15, roughly two thirds of people fall between 85 and 115.
Problems people actually run into
Choosing population when you have a sample
This is the most common error, and it always understates the spread. It happens because 'population' sounds like the safer, more complete choice.
The test is simple: could you have collected more data that belongs to the same group? If yes, it is a sample. Survey results, measurements from a batch, and any experiment are all samples.
Comparing standard deviations from different scales
A standard deviation of 12 on a test marked out of 20 is enormous. The same 12 on a test marked out of 500 is negligible.
The number is only meaningful next to the mean and the range it came from, so never compare two directly unless they are measuring the same thing in the same units.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: August 29, 2026