How the averages are calculated
Mean = sum ÷ count; weighted mean = sum(value × weight) ÷ sum(weights)
- Add the values and divide by their count to find the mean.
- Sort the list to find the median. For an even count, average the two middle values.
- Find the most frequent values for the modes. Unique values give no mode.
- Subtract the minimum from the maximum for the range.
- For weighted averages, divide the weighted sum by the total weight.
Worked example: For 2, 4, 4, 4, 5, 5, 7 and 9, the sum is 40 and the mean is 5. The median is 4.5 and the mode is 4. Population standard deviation is 2; sample standard deviation is about 2.1381.
Example averages
| Values | Mean | Median | Special average |
|---|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 4.5 | Not applicable |
| 10, 20, 30 | 20 | 20 | Not applicable |
| 1, 2, 3, 4, 100 | 22 | 3 | Not applicable |
| 40, 60 | 50 | 50 | Harmonic: 48 |
| Grades 80:20, 90:30, 70:50 | Not applicable | Not applicable | Weighted: 78 |
Swipe the table sideways to see the rest.
Things to consider
- Extreme values can pull the mean away from the median.
- Weights may be counts, proportions or percentages. They need not total 100.
- Use population deviation for the whole group and sample deviation for a sample.
- Harmonic speed averages assume equal distances, not equal travel times.
Frequently asked questions
What is the difference between mean, median and mode?
The mean is the sum divided by the count. The median is the middle of a sorted list. The mode is the most common value; tied values are all listed, and a list of unique values has no mode.
When should I use the median?
The median is often useful for skewed data such as incomes or house prices, where a few extreme values can pull the mean up or down. It describes the middle observation, not the total spread.
How do I calculate a weighted average?
Multiply each value by its weight, add the products and divide by the sum of weights. For grades 80:20, 90:30 and 70:50, the result is 78. Portfolio weights work the same way. The weights need not total 100. The Test Grade Calculator handles single scores.
What is population vs sample standard deviation?
Both describe spread around the mean. Population deviation divides the sum of squared differences by n. Sample deviation divides by n − 1 to estimate the spread of a larger population from a sample. Sample deviation is undefined for one value.
When is geometric mean useful for growth rates?
Geometric mean combines positive multiplicative factors. Convert growth rates to factors first: 10% becomes 1.10 and −10% becomes 0.90. Their geometric mean minus 1 gives the average rate per period. All factors must be greater than 0.
When should I use harmonic mean for average speed?
Use it for speeds over equal distances. Travelling equal distances at 40 and 60 gives a harmonic average speed of 48. For equal travel times, use the arithmetic mean instead. This calculator requires non-zero values with the same sign.
Results are estimates for general information only, not financial, medical or professional advice.