Smaller numbers, same mean
Subtract a fixed number from every class mark, then add it back.
Big products
In Table 13.3 the products are big: 92.5 × 6 = 555. The book asks: can we change each class mark to a smaller number to make the work easy?
Pick one class mark in the middle and call it a. Subtract a from every class mark. The new numbers are smaller, and some are negative.
The mean of the new, smaller numbers is not the mean we want. What would you do to it to get the mean of the class marks?
What this lesson covers
The idea
Assumed mean method: choose a central class mark as the assumed mean a, find deviations di = xi – a, then x̄ = a + Σfidi/Σfi; the mean obtained does not depend on the choice of a.
Big products
In Table 13.3 the products are big: 92.5 × 6 = 555. The book asks: can we change each class mark to a smaller number to make the work easy?
Pick one class mark in the middle and call it a. Subtract a from every class mark. The new numbers are smaller, and some are negative.
The mean of the new, smaller numbers is not the mean we want. What would you do to it to get the mean of the class marks?
- Add a to it
- Subtract a from it
- Multiply it by a
Choose the assumed mean
Tap a class mark to make it the assumed mean a. The table shows each deviation d = x − a and f·d. The last line adds a back. Try three different values of a and watch the final answer.
Assumed mean method
x̄ = a + Σfᵢdᵢ / Σfᵢ, where a is the assumed mean and dᵢ = xᵢ − a is the deviation of each class mark from it.
Why it works: the mean of the deviations is d̄ = Σfᵢ(xᵢ − a) / Σfᵢ = x̄ − a. So x̄ = a + d̄. We choose a to be a class mark near the centre, so the deviations are small. The mean does not depend on the choice of a.
Notes
x̄ = a + Σfᵢdᵢ / Σfᵢ, with dᵢ = xᵢ − a. The mean obtained does not depend on the choice of a.
Check yourself
In Table 13.4, a = 47.5, Σfᵢdᵢ = 435 and Σfᵢ = 30. Find the mean of the deviations, d̄ = Σfᵢdᵢ / Σfᵢ.
Answer: 14.5
435/30 = 14.5. Then x̄ = a + d̄ = 47.5 + 14.5 = 62.
NCERT Example 2: a = 50, Σfᵢdᵢ = −360 and Σfᵢ = 35. Find the mean x̄, correct to 2 decimal places.
Answer: 39.71
−360/35 = −10.2857…, and 50 − 10.2857… = 39.714…, which is 39.71 to 2 decimal places.
With a = 47.5, find the deviation d for the class mark 77.5.
Answer: 30
d = 77.5 − 47.5 = 30.
Using the same table, a friend takes a = 62.5 instead of 47.5. What mean does she get?
- 62, the same as before — correct. Yes! With a = 62.5 the mean of the deviations is −0.5, and 62.5 − 0.5 = 62. The mean does not depend on a.
- A different mean, because a is different. The deviations change too, and they cancel the change in a. Try a = 62.5 in the toy: you get 62 again.
- 62.5, because a is the mean. a is only a number we assume to make the work easy. The mean comes out the same, 62, whichever class mark we pick.