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.
Results:
| Standard Deviation (σ): | |
|---|---|
| Variance (σ²): | |
| Sum of Squares (SS): | |
| Population Size (N): | |
| Mean (μ): |
Steps to Solve
Standard Deviation Formula
Step One: Find the Mean
Mean (μ) = sum / population 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) = √SS / N
| Standard Deviation (s): | |
|---|---|
| Variance (s²): | |
| Sum of Squares (SS): | |
| Sample Size (n): | |
| Mean (x̄): |
Steps to Solve
Standard Deviation Formula for a Sample
Step One: Find the Mean
Mean (x̄) = sum / sample size (n)
Step Two: Find the Sum of Squares
Sum of Squares (SS) = ∑(xi - x̄)²
Step Three: Calculate the Standard Deviation
Standard Deviation (s) = √SS / n - 1
On this page:
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:
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 = sum / count
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 = √SS / N
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.
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.