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.
How to Read an Average and Standard Deviation
"Average" can mean the mean, median or mode. 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 description of a typical value.
The standard deviation, 11.2 for the sample, is the typical distance from the mean. The shape note on this page compares the mean and median, and the table shows each average with when it works best.
Where Average and Standard Deviation Is Used
Everyday summaries
Get the mean, median, mode and midrange of a list at once.
Choosing an average
See which average best describes skewed data.
Spread
Read the sample and population standard deviation.
Homework
Check several averages in one step.
Common Average and Standard Deviation Mistakes to Avoid
- Name the average you mean. "Average" alone is ambiguous.
- When mean and median differ a lot, quote the median.
- Use the sample SD for a sample and the population SD for a whole population.
- The midrange depends only on the two extremes, so it is a rough guide.
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, most often the mean, and the standard deviation describes how far values typically sit from it.
Which average should I use?
Use the mean for roughly symmetric data, the median for skewed data or data with outliers, and the mode for categories or the most common value.
Sample or population SD?
Use the sample SD, with n − 1, when your data are a sample from a larger group. Use the population SD only when you have the entire group.
What is the midrange?
It is the average of the smallest and largest values, (min + max) ÷ 2. It is quick to compute but depends only on the extremes.