Question

The marks obtained by 80 students of class X in a mock test of Mathematics are given below in the table. Find median and the mode of 5 the data:

Marks	Number of Students
0 and above	80
10 and above	77
20 and above	72
30 and above	65
40 and above	55
50 and above	43
60 and above	28
70 and above	16
80 and above	10
90 and above	8
100 and above	0

Answer 1

Based on the cumulative distribution provided in typical versions of this problem, the marks are structured as follows:

Marks	Number of Students	Frequency	Cumulative Frequency
0 - 10	80	3	3
10 - 20	77	5	8
20 - 30	72	7	15
30 - 40	65	10	25
40 - 50	55	12	37
50 - 60	43	15	52
60 - 70	28	12	64
70 - 80	16	6	70
80 - 90	10	2	72
90 - 100	8	8	80
Total (n)		80

∴ n = 80 

`n/2 = 80/2 = 40`

It can be observed that the cumulative frequency just greater than `n/2 = 80/2 = 40` is 37, belonging to the class interval 50 - 60.

Median class = 50 - 60

Lower limit (l) of median class = 50

Cumulative frequency (cf) of class preceding median class = 37

Frequency (f₁) of median class = 15

f₀ (Frequency of class before) = 12

f₂ (Frequency of class after) = 12

Class size (h) = 10

Median = `l +((n/2 - cf)/f) xx h`

= `50 + ((40 - 37)/15) xx 10`

= `50 + ((3 xx 10)/15)`

= `50 + ((30)/15)`

= 50 + 2

= 52

Mode = `l + ((f_1 - f_0)/(2f_1 - f_0 - f_2))xxh`

= `50 + ((15 - 12)/(2(15) - 12 - 12)) xx 10`

= `50 + ((3)/(30 - 24)) xx 10`

= `50 + ((3)/(6)) xx 10`

= `50 + ((3 xx 10)/(6))`

= `50 + ((30)/(6))`

= 50 + 5

= 55