Definitions
- Mean — sum divided by count. The everyday "average."
- Median — the middle value when the numbers are sorted. Robust to outliers.
- Mode — the most frequent value. Multiple modes are listed if there's a tie.
- Range — maximum minus minimum.
Mean = (x₁ + x₂ + … + xₙ) ÷ n
Working it out by hand
Take seven test scores: 72, 85, 90, 85, 64, 98, 79.
- Sum: 72 + 85 + 90 + 85 + 64 + 98 + 79 = 573.
- Mean: 573 ÷ 7 = 81.8571.
- Sort the list: 64, 72, 79, 85, 85, 90, 98.
- Median: with 7 values, the 4th one is in the middle, so the median is 85.
- Mode: 85 appears twice and every other score once, so the mode is 85.
- Range: 98 − 64 = 34.
With an even count there is no single middle value, so the median is the mean of the two middle numbers. For 3, 7, 9 and 12, the median is (7 + 9) ÷ 2 = 8.
Which average should you use?
| Situation | Best measure | Why |
|---|---|---|
| Test scores, daily temperatures, evenly spread data | Mean | Uses every value |
| Salaries, house prices, anything with a few extreme values | Median | Not pulled by outliers |
| Shoe sizes, survey answers, most popular option | Mode | Shows the most common value |
| How spread out the data is | Range | Gap between the smallest and largest value |
Outliers make a big difference. Five salaries of $42,000, $45,000, $48,000, $51,000 and $250,000 have a mean of $87,200, higher than four of the five people earn. The median, $48,000, is a much better picture of a typical salary.
Weighted averages
This calculator gives every number equal weight. If some values count more than others, such as a final exam worth half your grade, use a weighted average instead: multiply each value by its weight and add the results.
Example: homework 90 (20% weight), midterm 80 (30%) and final 70 (50%). Weighted average = 90 × 0.2 + 80 × 0.3 + 70 × 0.5 = 18 + 24 + 35 = 77. The plain mean of the three scores would be 80.