8 min readConcept sheetA 3-mark question; the class mark alone is a 1-mark MCQ

Mean of grouped data: the direct method

For grouped data, take each class's class mark as its value. Then mean = Σfᵢxᵢ ÷ Σfᵢ: multiply, add, divide.

Class markxᵢ = (lower limit + upper limit) ÷ 2

Direct methodx̄ = Σfᵢxᵢ ÷ Σfᵢ

Class sizeh = upper limit − lower limit, e.g. 20 − 10 = 10

AssumptionAll values in a class are taken to sit at its class mark

Words you need

Grouped data is data arranged in classes such as 10-20, 20-30. The frequency fᵢ is how many values fall in a class. In 10-20, 10 is the lower limit and 20 is the upper limit. A value of exactly 20 goes in the next class, 20-30. The class mark xᵢ is the middle of a class: (10 + 20) ÷ 2 = 15. The symbol Σ (sigma) means 'add them all'.

How to set up the table

1
Make four columns: Class, fᵢ, xᵢ, fᵢxᵢ.
2
Find each class mark xᵢ.
3
Multiply fᵢ × xᵢ in each row.
4
Add the fᵢ column to get Σfᵢ and the fᵢxᵢ column to get Σfᵢxᵢ.
5
Divide: x̄ = Σfᵢxᵢ ÷ Σfᵢ.

Worked example 1: marks in a class test

Problem

The marks (out of 50) of 40 students are: 0-10: 3 students, 10-20: 7, 20-30: 12, 30-40: 10, 40-50: 8. Find the mean mark by the direct method.

Working

Formula: x̄ = Σfᵢxᵢ ÷ Σfᵢ.
Class marks xᵢ: 5, 15, 25, 35, 45.
fᵢxᵢ: 3 × 5 = 15, 7 × 15 = 105, 12 × 25 = 300, 10 × 35 = 350, 8 × 45 = 360.
Σfᵢ = 3 + 7 + 12 + 10 + 8 = 40.
Σfᵢxᵢ = 15 + 105 + 300 + 350 + 360 = 1130.
x̄ = 1130 ÷ 40 = 28.25.

Answer

Mean mark = 28.25.

The working table for example 1

Marks	fᵢ	xᵢ	fᵢxᵢ
0-10	3	5	15
10-20	7	15	105
20-30	12	25	300
30-40	10	35	350
40-50	8	45	360
Total	Σfᵢ = 40		Σfᵢxᵢ = 1130

Worked example 2: a missing frequency

Problem

The table shows the daily travel time to school of some students in Pune: 0-10 min: 5, 10-20: 8, 20-30: p, 30-40: 12, 40-50: 5. The mean time is 26 minutes. Find p.

Working

Formula: x̄ = Σfᵢxᵢ ÷ Σfᵢ.
Class marks xᵢ: 5, 15, 25, 35, 45.
Σfᵢ = 5 + 8 + p + 12 + 5 = 30 + p.
Σfᵢxᵢ = 25 + 120 + 25p + 420 + 225 = 790 + 25p.
So (790 + 25p) ÷ (30 + p) = 26.
790 + 25p = 780 + 26p, so p = 10.
Check: Σfᵢ = 40, Σfᵢxᵢ = 1040, and 1040 ÷ 40 = 26.

Answer

p = 10.

In real life

A school uses grouped marks to find a class's average score. A ration shop can find the average amount of rice taken per family in the same way.

Exam angle

1 marks: “Find the class mark of the class 25-35.” (25 + 35) ÷ 2 = 30.

3 marks: “Find the mean of this data by the direct method: 0-20: 4, 20-40: 6, 40-60: 10, 60-80: 7, 80-100: 3.” Class marks are 10, 30, 50, 70, 90. Σfᵢ = 30 and Σfᵢxᵢ = 40 + 180 + 500 + 490 + 270 = 1480. So x̄ = 1480 ÷ 30 ≈ 49.33.

3 marks: “The mean of the distribution 10-20: 4, 20-30: k, 30-40: 10, 40-50: 6 is 30. Find k.” Σfᵢ = 20 + k and Σfᵢxᵢ = 680 + 25k. Then 680 + 25k = 30(20 + k) = 600 + 30k, so 5k = 80 and k = 16.

Remember it

Middle, multiply, add, divide: find the class mark, multiply by f, add both columns, divide.