Standard deviation calculator
Paste a list of numbers to get the sample or population standard deviation, with the mean, sum of squares and variance worked out step by step, as exact values and decimals.
Result
Population SD (σ): 2
Step by step
- Count and add the values: n = 8, Σx = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
- Mean: x̄ = Σx / n = 40 / 8 = 5
- Subtract the mean from each value and square the result:
x x − x̄ (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 Σx = 40 0 32 - Sum of squares: SS = Σ(x − x̄)² = 32
- Variance: s² = SS / n − 1 = 32 / 7 = 32/7 ≈ 4.5714
- Standard deviation: s = √s² = √(32/7) = 4√14/7 ≈ 2.1381
How it works
Standard deviation measures how far the values sit from their mean, on average. The calculator finds the mean, measures each value’s distance from it, squares those distances, adds them up, divides, and takes a square root. Here is the default data set, 2, 4, 4, 4, 5, 5, 7, 9, done by hand.
- There are 8 values and they add to 40, so the mean is 40 / 8 = 5.
- The deviations from 5 are −3, −1, −1, −1, 0, 0, 2, 4. Squared, they are 9, 1, 1, 1, 0, 0, 4, 16.
- Those squares add to the sum of squares, SS = 32.
- As a population, divide by N = 8: the variance is 32 / 8 = 4 and the standard deviation is √4 = 2.
- As a sample, divide by n − 1 = 7: the variance is 32/7 ≈ 4.5714 and the standard deviation is √(32/7) = 4√14/7 ≈ 2.1381.
A quick check on your own working: the deviations in step 2 always add to 0. If yours don’t, the mean is off. The calculator keeps every value as an exact fraction until the end, so 0.1, 0.2, 0.3 gives a mean of exactly 1/5 rather than a rounding error.
If the mean itself is new to you, start with the guide to mean, median, mode and range. Once you have a mean and a standard deviation, you can turn any value into a z-score, which is how the normal distribution and z-scores guide finds percentiles and probabilities.
Frequently asked questions
Should I use sample or population standard deviation?
Use population when your list is every value you care about, such as the test scores of all 25 students in one class. Use sample when the list is a subset drawn to estimate a bigger group, such as 25 students picked from a whole school. Most homework and lab data is a sample, which is why this calculator starts on sample.
Why does the sample formula divide by n − 1?
A sample sits closer to its own mean than to the true population mean, so the squared deviations come out a little too small. Dividing by n − 1 instead of n makes the variance slightly bigger and corrects that bias on average. The fix is called Bessel’s correction, and it matters most when n is small.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean, so its units are squared (cm², dollars²). Standard deviation is the square root of the variance, which puts it back in the same units as the data. That makes the SD easier to read: a class with mean 70 and SD 8 has most scores within about 8 points of 70.
Can the standard deviation be zero or negative?
It can be zero, but never negative. Every squared deviation is zero or positive, so their sum is too, and a square root of that is never below zero. The SD is exactly 0 only when every value is the same, like 5, 5, 5. If you get a negative answer by hand, a sign slipped somewhere in the working.
How do I enter my data, and how do share links work?
Separate numbers with commas, spaces, semicolons, tabs or new lines, so you can paste a column straight from a spreadsheet. Decimals and negatives work; scientific notation does not. The Copy share link button copies a URL holding your numbers and the sample or population choice, so anyone who opens it sees the same result.
Check your statistics homework
Work the problem yourself first, then take a photo of it. Solver AI on iPhone and iPad shows each step, so you can see where your answer went wrong.
