What Is Average and Standard Deviation?
An average describes the centre of your data and the standard deviation describes its spread. The word average usually means the mean, but the median, mode and midrange are averages too.
Enter your numbers to get all four averages, both standard deviations, a note on whether the data look symmetric or skewed and a suggestion for the best average to quote. A comparison table explains when each average is appropriate.
Below are the formulas, a worked example with an outlier and answers on sample versus population SD.
How to Use the Average and Standard Deviation Calculator
- Enter your data.
- Read each average.
- Read the standard deviation.
- Use the guidance to choose the best average.
Average and Standard Deviation Formula
Mean
Sample SD
Average and Standard Deviation Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Mean | |
| n | Number of values | |
| s | Sample standard deviation | |
| σ | Population standard deviation |
Average and Standard Deviation Worked Example
For 14, 18, 21, 21, 25, 27, 30, 34, 52 the mean is 26.9 and the median is 25. The mean is higher because 52 pulls it up, so the median is the safer summary of a typical value.
With the example values loaded in the calculator above, the steps are:
- Mean = 242 ÷ 9 = 26.888889.
- Median (middle of the sorted list) = 25; midrange = (min + max) ÷ 2 = 33.
- Sample SD = √(Σ(x − mean)² ÷ (n − 1)) = √(1008.8889 ÷ 8) = 11.22992.
- Population SD divides by n = 9 instead: 10.58767.
Result: Average (mean) = 26.888889; Standard deviation (sample) = 11.22992.
Where Average and Standard Deviation Is Used
Choosing an average
See why the median can beat the mean.
Reports
Quote centre and spread together.
Homework
Check all the averages at once.
Skewed data
Spot a tail that pulls the mean.
Common Average and Standard Deviation Mistakes to Avoid
- Compare mean and median.
- Use the sample SD for sample data.
- Report both centre and spread.
Average and Standard Deviation Terms and Definitions
- Midrange
- The average of the smallest and largest values.
Average and Standard Deviation FAQs
What is the average and standard deviation?
The average describes the centre of the data and the standard deviation describes how far values typically lie from the mean.
Which average should I use?
Use the mean for symmetric data and the median when there are outliers or skew.
Sample or population SD?
Use the sample SD unless you have every member of the group.
What is the midrange?
The mean of the minimum and maximum values.