Divide the deviations too

The idea

Step-deviation method: with assumed mean a and class size h, take ui = (xi – a)/h; then x̄ = a + h(Σfiui/Σfi). It suits data whose di share a common factor and holds for any non-zero a, h.

A common factor

With a = 47.5 the deviations are −30, −15, 0, 15, 30, 45. Every one of them is a multiple of 15, the class size.

Dividing every deviation by 15 would give small whole numbers.

What do you expect will happen to the numbers we multiply by f?

• They become smaller whole numbers
• They stay the same
• They become decimals

Choose the class size h

Here a = 47.5. Tap a value of h and look at the column u = d / h. Some choices give whole numbers and some do not. Find the largest h for which every u is a whole number.

Step-deviation method

x̄ = a + h (Σfᵢuᵢ / Σfᵢ), where uᵢ = (xᵢ − a) / h, a is the assumed mean and h is the class size.

Now the numbers are the smallest: the u values are −2, −1, 0, 1, 2, 3. Why it works: the mean of the u values is ū = (x̄ − a) / h, so x̄ = a + h·ū.

It is convenient when all the deviations dᵢ have a common factor. The formula holds for any non-zero a and h. With h = 10 the mean is still 62, only the u values are not whole numbers.

Notes

x̄ = a + h (Σfᵢuᵢ / Σfᵢ), with uᵢ = (xᵢ − a) / h. It holds for any non-zero a and h, and is convenient when the dᵢ share a common factor.

Check yourself

Table 13.5 has a = 47.5, h = 15, Σfᵢuᵢ = 29 and Σfᵢ = 30. Find x̄ = a + h (Σfᵢuᵢ / Σfᵢ).

Answer: 62

15 × 29/30 = 14.5, and 47.5 + 14.5 = 62.

NCERT Example 2: a = 50, h = 10, Σfᵢuᵢ = −36 and Σfᵢ = 35. Find x̄, correct to 2 decimal places.

Answer: 39.71

10 × (−36/35) = −10.2857…, and 50 − 10.2857… = 39.714…, which is 39.71.

The deviations are 30, 60, 90, 120. Which h turns them into 1, 2, 3, 4?

With a = 47.5 and h = 15, find u for the class mark 92.5.

Answer: 3

u = (92.5 − 47.5)/15 = 45/15 = 3.

• h = 30 — correct. Yes! 30/30 = 1, 60/30 = 2, 90/30 = 3, 120/30 = 4. 30 is the largest number that divides all four.
• h = 10. With h = 10 you get 3, 6, 9, 12. They work, but they are bigger than they need to be.
• h = 60. With h = 60 you get 0.5, 1, 1.5, 2. Not all whole numbers.