Standard Deviation

 

What is Standard Deviation? | STDEV.P | STDEV.S

This page explains how to calculate the standard deviation based on the entire population using the STDEV.P function in Excel and how to estimate the standard deviation based on a sample using the STDEV.S function in Excel.

What is Standard Deviation?

Standard deviation is a number that tells you how far numbers are from their mean.

1. For example, the numbers below have a mean (average) of 10.

Standard Deviation of Zero in Excel

Explanation: the numbers are all the same which means there's no variation. As a result, the numbers have a standard deviation of zero. The STDEV function is an old function. Microsoft Excel recommends using the new STEDV.S function which produces the exact same result.

2. The numbers below also have a mean (average) of 10.

Low Standard Deviation in Excel

Explanation: the numbers are close to the mean. As a result, the numbers have a low standard deviation.

3. The numbers below also have a mean (average) of 10.

High Standard Deviation in Excel

Explanation: the numbers are spread out. As a result, the numbers have a high standard deviation.

STDEV.P

The STDEV.P function (the P stands for Population) in Excel calculates the standard deviation based on the entire population. For example, you're teaching a group of 5 students. You have the test scores of all students. The entire population consists of 5 data points. The STDEV.P function uses the following formula:

Formula of the Standard Deviation Based on the Entire Population

In this example, x1 = 5, x2 = 1, x3 = 4, x4 = 6, x5 = 9, μ = 5 (mean), N = 5 (number of data points).

1. Calculate the mean (μ).

Calculate the Mean

2. For each number, calculate the distance to the mean.

Distance to the Mean

3. For each number, square this distance.

Square of the Distance to the Mean

4. Sum (∑) these values.

Sum these Values

5. Divide by the number of data points (N = 5).

Divide by the Number of Data Points

6. Take the square root.

Standard Deviation Based on the Entire Population

7. Fortunately, the STDEV.P function in Excel can execute all these steps for you.

STDEV.P Function in Excel

STDEV.S

The STDEV.S function (the S stands for Sample) in Excel estimates the standard deviation based on a sample. For example, you're teaching a large group of students. You only have the test scores of 5 students. The sample size equals 5. The STDEV.S function uses the following formula:

Formula of the Standard Deviation Based on a Sample

In this example, x1=5, x2=1, x3=4, x4=6, x5=9 (same numbers as above), x̄=5 (sample mean), n=5 (sample size).

1. Repeat steps 1-5 above but at step 5 divide by n-1 instead of N.

Divide by n-1

2. Take the square root.

Standard Deviation Based on a Sample

3. Fortunately, the STDEV.S function in Excel can execute all these steps for you.

STDEV.S Function in Excel

Note: why do we divide by n - 1 instead of by n when we estimate the standard deviation based on a sample? Bessel's correction states that dividing by n-1 instead of by n gives a better estimation of the standard deviation.