Standard Deviation Calculator
Standard deviation, variance and mean for a list of numbers, sample or population.
Good to know. Two classes can have the same average grade and nothing else in common. The standard deviation tells them apart.
How it works
In plain words
Find the mean. Subtract it from each number and square the result. Add the squares up and divide by n − 1 for a sample, or by n for a whole population. The standard deviation is the square root of that variance.
The formula
mean = Σx ÷ n variance = Σ(x − mean)² ÷ (n − 1) (population: ÷ n) standard deviation = √variance
Worked example
The mean of 2, 4, 4, 4, 5, 5, 7, 9 is 5. The squared differences add up to 32. As a sample, 32 ÷ 7 = 4.5714, and the square root is 2.1381. As a population, 32 ÷ 8 = 4, so 2.
Sources
No outside data or rates: this is plain arithmetic.
3 automated tests check its answers on every change.
About the standard deviation calculator
The standard deviation measures how spread out numbers are around their average. A small standard deviation means most values are close to the mean; a large one means they are scattered. It is in the same units as the data, which makes it easier to read than the variance, its square.
Choose sample when your numbers are a selection from a larger group, such as 30 customers from all your customers or a few measurements from a production run. That is the usual case, and it divides by n − 1 (Bessel’s correction) to avoid underestimating the spread. Choose population when you have every value, such as the test scores of an entire class you are describing.
For data that follows a bell curve, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. So with a mean of 100 and a standard deviation of 15, as in IQ scores, about 95% of people score between 70 and 130.
The standard error, the standard deviation divided by the square root of the count, says how precisely the sample mean estimates the true mean. It shrinks as you collect more data.
Questions people ask
How do I calculate standard deviation?
Find the mean, square each value’s distance from it, add the squares, divide by n − 1 (or n for a population) and take the square root.
Should I use sample or population standard deviation?
Sample, unless your data includes every member of the group you are describing. Most real data sets are samples.
What is the difference between variance and standard deviation?
The variance is the average squared distance from the mean. The standard deviation is its square root, back in the original units.
What is a good standard deviation?
There is no good or bad value. It depends on the data: compare it with the mean, or with the spread you expect.