8 min readPathway sheetA 3-mark question; often asked with a missing frequency

Mean by the step-deviation method

When every dᵢ shares the class size h, divide it out: uᵢ = (xᵢ − a) ÷ h. Then x̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ).

Formulax̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ)

Step deviationuᵢ = (xᵢ − a) ÷ h

hThe class size, the common factor of all the dᵢ

Best forLarge class marks with equal class sizes

Making the numbers even smaller

In the assumed mean method, the dᵢ are often −40, −20, 0, 20, 40. They are all multiples of the class size h = 20. Divide each by h to get uᵢ: −2, −1, 0, 1, 2. Multiplying by such small numbers is quick and safe. At the end, multiply by h to undo the division.

Three methods, one mean

Method	Extra columns	Formula	Use it when
Direct	fᵢxᵢ	x̄ = Σfᵢxᵢ ÷ Σfᵢ	Class marks are small
Assumed mean	dᵢ = xᵢ − a, fᵢdᵢ	x̄ = a + Σfᵢdᵢ ÷ Σfᵢ	Class marks are large
Step deviation	uᵢ = (xᵢ − a) ÷ h, fᵢuᵢ	x̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ)	Class marks are large and all dᵢ share a factor h

Worked example 1: homework time

Problem

The time (in minutes) 40 students spend on homework daily is: 10-30: 5, 30-50: 8, 50-70: 12, 70-90: 9, 90-110: 6. Find the mean time by the step-deviation method.

Working

Formula: x̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ), where uᵢ = (xᵢ − a) ÷ h.
Class marks xᵢ: 20, 40, 60, 80, 100. Take a = 60. Class size h = 20.
uᵢ = (xᵢ − 60) ÷ 20: −2, −1, 0, 1, 2.
fᵢuᵢ: −10, −8, 0, 9, 12.
Σfᵢ = 40; Σfᵢuᵢ = −18 + 21 = 3.
x̄ = 60 + 20 × (3 ÷ 40) = 60 + 1.5 = 61.5.

Answer

Mean homework time = 61.5 minutes.

Worked example 2: a missing frequency

Problem

The mean of this distribution is 51. Find k. Classes: 0-20: 4, 20-40: 10, 40-60: k, 60-80: 12, 80-100: 4.

Working

Formula: x̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ).
Class marks: 10, 30, 50, 70, 90. Take a = 50 and h = 20.
uᵢ: −2, −1, 0, 1, 2.
fᵢuᵢ: −8, −10, 0, 12, 8. So Σfᵢuᵢ = 2.
Σfᵢ = 4 + 10 + k + 12 + 4 = 30 + k.
51 = 50 + 20 × 2 ÷ (30 + k), so 1 = 40 ÷ (30 + k).
30 + k = 40, so k = 10.
Check by the direct method: Σfᵢxᵢ = 40 + 300 + 500 + 840 + 360 = 2040, and 2040 ÷ 40 = 51.

Answer

k = 10.

dᵢ or uᵢ?

dᵢ (deviation)dᵢ = xᵢ − a. Used in the assumed mean method. Values like −20, −10, 0, 10, 20.

uᵢ (step deviation)uᵢ = (xᵢ − a) ÷ h. Used in the step-deviation method. Values like −2, −1, 0, 1, 2. Remember to multiply by h at the end.

The step-deviation method works neatly only when all the classes have the same size.

In real life

A school counsellor can find the average daily study time of a class from a grouped survey. A sports teacher can find the average time for a 400 m race in the same way.

Exam angle

2 marks: “In the step-deviation method, a = 50, h = 10, Σfᵢuᵢ = −12 and Σfᵢ = 60. Find the mean.” x̄ = 50 + 10 × (−12 ÷ 60) = 50 − 2 = 48.

3 marks: “The monthly electricity used (in units) by 40 homes is: 0-50: 4, 50-100: 8, 100-150: 12, 150-200: 10, 200-250: 6. Find the mean by the step-deviation method.” Take a = 125 and h = 50, so uᵢ = −2, −1, 0, 1, 2. Σfᵢuᵢ = −8 − 8 + 0 + 10 + 12 = 6 and Σfᵢ = 40. So x̄ = 125 + 50 × 6 ÷ 40 = 132.5 units.

Remember it

Step deviation: subtract a, divide by h, and at the end multiply back by h.