8 min readPathway sheetA 3-mark question; the method may be named in the question

Mean by the assumed mean method

Pick a class mark near the middle as the assumed mean a. Find the deviations dᵢ = xᵢ − a, then x̄ = a + Σfᵢdᵢ ÷ Σfᵢ.

Formulax̄ = a + (Σfᵢdᵢ ÷ Σfᵢ)

Deviationdᵢ = xᵢ − a

Choosing aA class mark near the middle of the table, often the one with the highest frequency

Why use itSmaller numbers to multiply when class marks are large

Why assume a mean?

When class marks are big numbers like 145 or 155, fᵢxᵢ gets large and messy. So we guess a value a, called the assumed mean. We then work with small deviations dᵢ = xᵢ − a. Some dᵢ are negative and some are positive. The correction Σfᵢdᵢ ÷ Σfᵢ fixes our guess. Any choice of a gives the same mean.

Pathway: the assumed mean method

1
Find the class marks xᵢ.
2
Choose a: a class mark near the middle.
3
Find dᵢ = xᵢ − a for each class. The row of a has dᵢ = 0.
4
Find fᵢdᵢ for each class, keeping the minus signs.
5
Add: Σfᵢ and Σfᵢdᵢ.
6
x̄ = a + Σfᵢdᵢ ÷ Σfᵢ.

Worked example 1: milk at a dairy society

Problem

The milk (in litres) given by 50 farmers to a dairy society on one day is: 20-30: 6, 30-40: 11, 40-50: 15, 50-60: 10, 60-70: 8. Find the mean by the assumed mean method.

Working

Formula: x̄ = a + Σfᵢdᵢ ÷ Σfᵢ, where dᵢ = xᵢ − a.
Class marks xᵢ: 25, 35, 45, 55, 65. Take a = 45.
dᵢ = xᵢ − 45: −20, −10, 0, 10, 20.
fᵢdᵢ: 6 × (−20) = −120, 11 × (−10) = −110, 0, 10 × 10 = 100, 8 × 20 = 160.
Σfᵢ = 50; Σfᵢdᵢ = −120 − 110 + 0 + 100 + 160 = 30.
x̄ = 45 + 30 ÷ 50 = 45 + 0.6 = 45.6.

Answer

Mean milk given = 45.6 litres per farmer.

Worked example 2: a different a, the same mean

Problem

Find the mean of the same dairy data, this time taking a = 55.

Working

dᵢ = xᵢ − 55: −30, −20, −10, 0, 10.
fᵢdᵢ: 6 × (−30) = −180, 11 × (−20) = −220, 15 × (−10) = −150, 0, 8 × 10 = 80.
Σfᵢdᵢ = −180 − 220 − 150 + 0 + 80 = −470.
x̄ = 55 + (−470) ÷ 50 = 55 − 9.4 = 45.6.

Answer

Mean = 45.6 litres again. The choice of a does not change the mean.

Direct method vs assumed mean method

Direct methodUses fᵢxᵢ. Easy when class marks are small, like 5, 15, 25.

Assumed mean methodUses fᵢdᵢ, with dᵢ = xᵢ − a. Easier when class marks are large. It gives the same answer.

If Σfᵢdᵢ is negative, the mean is less than a. If it is positive, the mean is more than a.

In real life

A milk cooperative like those around Anand, Gujarat, can find the average milk per farmer quickly with this method. It is the same idea as guessing a typical value and then correcting the guess.

Exam angle

2 marks: “In the assumed mean method, a = 45, Σfᵢdᵢ = −60 and Σfᵢ = 40. Find the mean.” x̄ = 45 + (−60 ÷ 40) = 45 − 1.5 = 43.5.

3 marks: “Find the mean by the assumed mean method: 100-120: 10, 120-140: 20, 140-160: 30, 160-180: 15, 180-200: 5.” Take a = 150, so dᵢ = −40, −20, 0, 20, 40. Σfᵢdᵢ = −400 − 400 + 0 + 300 + 200 = −300 and Σfᵢ = 80. So x̄ = 150 − 300 ÷ 80 = 146.25.

Remember it

Guess, gap, fix: guess a, find the gaps dᵢ, then fix the guess with Σfᵢdᵢ ÷ Σfᵢ.