The direct mean
One number can describe a whole table.
- Four marks, same size.
- Sized by frequency: 8, 5, 4 and 3.
- Weighted together, they give 16.
The direct method
Here are 20 made-up readings, grouped into four classes of width 10. Call each class mark x. Call the frequency of its class f.
| Minutes played | Mark x | Frequency f | f × x |
|---|---|---|---|
| 0-10 | 5 | 8 | 40 |
| 10-20 | 15 | 5 | 75 |
| 20-30 | 25 | 4 | 100 |
| 30-40 | 35 | 3 | 105 |
| Total | 20 | 320 |
The products add up to 320. The frequencies add up to 20. So the mean is 320 ÷ 20 = 16. The sign Σ means "add them all up". In symbols, mean = Σ f x ÷ Σ f. Here Σ f x is the total of the f × x column. Σ f is the total of the f column.
Multiply each class mark by its frequency. Add the products. Divide the sum by the total frequency. That is the mean of grouped data.
Before grouping, the 20 readings add up to 307. And 307 ÷ 20 = 15.35. The table gave 16, a little higher. Grouping hides detail, so a grouped mean is an estimate. An estimate is a close answer, not a perfect one.
A pivot is the point a seesaw turns on. Slide it until the bars balance.
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