Question

In one group of students; 10% students got 0 to 20 marks; 20% students got 20 to 40 marks; 35% students got 40 to 60 marks; 20% students got 60 to 80 marks and remaining 30 students got 80 to 100 marks, then:

1. Prepare a grouped frequency distribution table.
2. Find the mode of the marks scored.

Answer 1

Let the total number of students be x

We convert the percentages into the number of students.

10% of x = `10/100 xx x = x/10`,

20% of x = `20/100 xx x = x/5`

35% of x = `35/100 xx x = (7x)/20`

Now, `x/10 + x/5 + (7x)/20 + x/5 + 30 = x`

∴ 2x + 4x + 7x + 4x + 600 = 20x    ...(Multiplying both the sides by 20)

∴ 17x + 600 = 20x

∴ 20x = 17x + 600

∴ 20x − 17x = 600

∴ 3x = 600

∴ x = `600/3`

∴ x = 200

∴ `x/10 = 200/10 = 20`

`x/5 = 200/5 = 40`

`(7x)/20 = (7 xx 200)/20 = 70`

(a) We make a tabulation:

Marks	0-20	20-40	40-60	60-80	80-100	Total
Number of students	20	40	70	40	30	200

Here, the maximum frequency, 70, is in the class 40-60.

∴ the modal class is 40-60.

(b) L = Lower class limit of the modal class = 40

h = Class interval of the modal class = 20

f₁ = Frequency of the modal class = 70

f₀ = Frequency of the class preceding the modal class = 40

f₂ = Frequency of the class succeeding the modal class = 40

Mode = `L + [(f_1 - f_0)/(2f_1 - f_0 - f_2)] xx h`   ...(Formula)

`40 + [(70 - 40)/(2(70) - 40 - 40)] xx 20`   ...(Substituting the values)

= `40 + (30/(140 - 80)) xx 20`

= `40 + 30/60 xx 20`

= 40 + 10

= 50

∴ The mode of the marks scored is 50