Question

In a college, there are 500 students in junior college, 5% score less than 25 marks, 68 score from 26 to 30 marks, 30% score from 31 to 35 marks, 70 score from 36 to 40 marks, 20% score from 41 to 45 marks and the rest score 46 and above marks. What is the median marks?

Answer 1

Given data can be written in tabulated form as follows:

Marks	No. of students
Less than 25	5% of 500 = `5/100 xx 500` = 25
26 – 30	68
31 – 35	30% of 500 = `30/100 xx 500` = 150
36 – 40	70
41 – 45	20% of 500 = `20/100 xx 500` = 100
46 and above	500 – (25 + 68 + 150 + 70 + 100) = 87

Since, the given data is not continuous, we have to convert it in the continuous form by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of every class interval.
∴ the class intervals will be
Less than 25.5, 25.5 – 30.5 etc.
We construct the less than cumulative frequency table as given below:

Marks	No. of students (f)	Less than cumulative frequency (c.f.)
Less than 25.5	25	25
25.5 – 30.5	68	93
30.5 – 35.5	150	243
35.5 – 40.5	70	313 ← Q2
40.5 – 45.5	100	413
45.5 and above	87	500
Total	500

Here, N = 500

Q₂ class = class containing `((2"N")/4)^"th"` observation

∴ `(2"N")/4 =(2xx500)/4` = 250

Cumulative frequency which is just greater than (or equal to) 250 is 313.
∴ Q₂ lies in the class 35.5 – 40.5.
∴ L = 35.5, h = 5, f = 70, c.f. = 243

∴ Median = Q₂ = `"L"+"h"/"f"((2"N")/4-"c.f.")`

= `35.5 + 5/70 (250 - 243)`

= `35.5 + 1/14(7)`

= 35.5 + 0.5
= 36