Step by step
If all deviations are multiples of one number, divide by it. The deviations then become smaller.
- Direct gives the mean.
- Assumed mean gives the same mean.
- Step deviation gives it too.
The third route
Take 30 made-up readings in five classes of width 10. The class marks are 5, 15, 25, 35 and 45. Their frequencies are 8, 6, 9, 4 and 3. Use the assumed mean a = 25 and the deviation d = x − a. Class marks are 10 apart, and a is one of them. So every deviation d is a multiple of 10. Divide each d by 10, the class width h. The results are the step deviations, u = d ÷ h.
| Mark x | d = x − 25 | u = d ÷ 10 | Frequency f | f × u |
|---|---|---|---|---|
| 5 | −20 | −2 | 8 | −16 |
| 15 | −10 | −1 | 6 | −6 |
| 25 | 0 | 0 | 9 | 0 |
| 35 | 10 | 1 | 4 | 4 |
| 45 | 20 | 2 | 3 | 6 |
The f × u column adds up to −12. Here n = 30. Sharing is dividing by n: −12 ÷ 30 = −0.4. Now multiply by h to undo the division: 10 × (−0.4) = −4. Then 25 + (−4) = 25 − 4 = 21. In symbols, mean = a + h × Σ f u ÷ Σ f.
Choose h to divide every d, so u stays whole. Divide each d by h to get u. Weight each u by its frequency. Share the total over n. Multiply by h, then add a.
Try h = 3. Then −20 ÷ 3 is not a whole number. The u column turns into fractions. The mean is still 21, but the sums are harder. A good h divides every d.
Watch out!
Do not round −20 ÷ 3 to −6.67. Rounding changes the mean. Keep it exact, or pick an h that divides every d.
Slide h. Watch the u column and the mean.
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