The assumed mean
Show on a big screenBig class marks make big products. An assumed mean makes them small.
- The marks, with a = 25 picked.
- Subtract 25 — small deviations remain.
- Their share, −4, added back: 25 − 4 = 21.
Subtract first, add back later
Here are 30 made-up readings, in minutes played.
| Minutes played | Frequency |
|---|---|
| 0-10 | 8 |
| 10-20 | 6 |
| 20-30 | 9 |
| 30-40 | 4 |
| 40-50 | 3 |
Pick any class mark and call it the assumed mean, a. Here a = 25. The deviation of a mark is d = x − a. It shows how far the mark is from a. A negative d means the mark is below a.
| Mark x | d = x − a | Frequency f | f × d |
|---|---|---|---|
| 5 | −20 | 8 | −160 |
| 15 | −10 | 6 | −60 |
| 25 | 0 | 9 | 0 |
| 35 | 10 | 4 | 40 |
| 45 | 20 | 3 | 60 |
The f × d column adds up to −120. Here n = 30, the number of readings. To share is to divide by n: −120 ÷ 30 = −4. Each d was measured from a. So the average d shows how far the mean is from a. The mean is 25 − 4 = 21. In symbols, mean = a + Σ f d ÷ Σ f.
Pick an assumed mean a. Subtract it from every class mark to get d. Weight each d by its frequency. Share the total over n. Add that share back to a.
Tap different values of a and compare.
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Try the lab to finish this lesson.