Standard Deviation Calculator

Find the standard deviation for your data set by entering the numbers in the calculator below. Keep reading to learn how to calculate standard deviation and the formula.

Separate numbers using a comma (,)
Separate numbers using a comma (,)

Results:

Standard Deviation (σ):
 
Variance (σ²):
 
Sum of Squares (SS):
 
Population Size (N):
 
Mean (μ):
 

Steps to Solve

Standard Deviation Formula

graphic showing the standard deviation formula where sigma is equal to the square root of the sum of squares divided by N

Step One: Find the Mean

Mean (μ) = sumpopulation size (N)

Step Two: Find the Sum of Squares

Sum of Squares (SS) = ∑(xi - μ)²

Step Three: Calculate the Standard Deviation

Standard Deviation (σ) = √SSN



How to Find Standard Deviation

In statistics, the standard deviation is a measure of the distribution or variance between numbers in a data set. A smaller standard deviation value indicates that the data are relatively close to the mean, while a higher value suggests that the data are more widely spread out.

Standard deviation is sometimes shortened to SD, but is often represented using the symbol σ (the Greek letter sigma).

Standard Deviation Formula

The standard deviation is the square root of the variance, so the formula is:

graphic showing the standard deviation formula where sigma is equal to the square root of the sum of squares divided by N

Using the formula above, you can find the standard deviation in a few simple steps.

Find the Mean

The first step to finding the standard deviation is to find the mean of the data set.

To calculate the mean, sum each value in the data set, then divide the result by the number of elements in the set.

mean = sumcount

The mean formula looks more like this:

μ = (x1 + x2 + … + xi) ÷ N
μ = ∑xi ÷ N

Find the Sum of Squares

The next step is to find the sum of squares. You can use the sum of squares formula to calculate it.

SS = ∑(xi – μ)²

The sum of squares SS is equal to the sum of each value xi minus the mean μ, squared.

Find the Standard Deviation

Finally, using the sum of squares, you can find the variance and then take its root to find the standard deviation.

σ = √SSN

Thus, the standard deviation σ is equal to the square root of the sum of squares SS minus the number of elements in the data set N.

How to Find Sample Standard Deviation

You can use the formula above to calculate the standard deviation for a data set that represents a population, but the formula to find the standard deviation of a sample is slightly different.

graphic showing the standard deviation formula for a sample where s is equal to the square root of the sum of squares divided by n minus 1

The standard deviation of a sample s is equal to the square root of the sum of squares divided by the number of elements in the sample n minus 1.

If you know the standard deviation of the population then you can also use the central limit theorem to find the sample standard deviation.