What Is Mean Median and Standard Deviation?
The mean, median and standard deviation answer two questions about a data set: where is the centre, and how spread out is it? The mean and median give two different centres, and the standard deviation measures spread around the mean.
Enter at least three values to get all three statistics, the gap between mean and median and the sample and population SD. A table then drops the largest value and shows how far each statistic moves, so you can see which summary is robust to outliers.
The sections below give the formulas, a worked example with an outlier and answers on when to prefer the median.
How to Use the Mean Median and Standard Deviation Calculator
- Enter at least three values.
- Read the mean, median and SD.
- Compare the mean and median.
- Check the outlier-effect table.
Mean Median and Standard Deviation Formula
Mean
Sample SD
Mean Median and Standard Deviation Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Mean | |
| Median | Middle sorted value | |
| s | Sample standard deviation | |
| n | Number of values |
Mean Median and Standard Deviation Worked Example
For 12, 14, 15, 15, 16, 18, 19, 21, 22, 64 the mean is 21.6 and the median is 17. The value 64 pulls the mean up. Without it the mean is 16.9 while the median is 16, so the median barely moves.
With the example values loaded in the calculator above, the steps are:
- Mean = 216 ÷ 10 = 21.6.
- Median = middle of the sorted values = 17.
- Sample SD = √(2086.4 ÷ 9) = 15.225709.
- Dropping the largest value (64) moves the mean by -4.71111 but the median by only -1.
Result: Mean = 21.6; Median = 17; Standard deviation (sample) = 15.225709.
How to Read Mean, Median and Standard Deviation Together
When the mean and median differ a lot, the data are skewed or have an outlier. For 12, 14, 15, 15, 16, 18, 19, 21, 22, 64 the mean is 21.6 but the median is 17. The value 64 pulls the mean up.
Dropping the largest value moves the mean from 21.6 to 16.9, while the median moves from 17 to 16. The median barely moves, so it is the more robust summary here. The standard deviation also falls sharply once 64 is gone.
Where Mean Median and Standard Deviation Is Used
Choosing a summary
Decide between the mean and the median from how far apart they are.
Outlier checks
See how much one extreme value shifts each statistic.
Reports
Quote the centre and spread together.
Homework
Check all three statistics in one step.
Common Mean Median and Standard Deviation Mistakes to Avoid
- A large gap between the mean and median points to skew or an outlier.
- Use the median and IQR when the data are skewed.
- Report which standard deviation you used, sample or population.
- Check the outlier before you drop it. It may be a real value.
Mean Median and Standard Deviation Terms and Definitions
- Robust
- Little affected by extreme values.
Mean Median and Standard Deviation FAQs
How do I find the mean, median and standard deviation?
Add the values and divide by n for the mean, find the middle value for the median, and take the square root of the average squared distance from the mean for the standard deviation, dividing by n − 1 for a sample.
Why are the mean and median different?
The mean uses every value, so extreme values pull it toward them. The median is the middle value and ignores how large the extremes are.
Which is more robust?
The median. It changes little when an outlier is added, while the mean and standard deviation change a lot.
Sample or population SD?
Use the sample version when the data come from a larger group, which is most cases. Use the population version only for the whole group.