Question

Calculate the mean and median from the grouped data.

Exam Score	Number of Applicants
61-65	20
66-70	13
71-75	47
76-80	56
81-85	33
86-90	27
91-95	41
96-100	34

Answer 1

Solution:

Given that,

Class	Frequency	Mid -point	d	fd	Cumulative Frequency
61-65	20	63	-4	- 80	20
66-70	13	68	-3	- 39	33
71-75	47	73	-2	- 94	80
76-80	56	78	-1	- 56	136
81-85	33	83	0	0	169
86-90	27	88	1	27	196
91-95	41	93	2	82	237
96-100	34	98	3	102	271
Total	n = 271			fd = - 58

a ) The sample mean is

Mean   = (A + (fd / n) * h)

= ( 83+( - 58 / 271 ) * 5

= 83 + - 214.5 * 5

= 83 + - 1.0701

= 81.9299

Mean   =  81.9299

b ) Median

Median = Value of ( n / 2 )^th Observation

  = Value of ( 271 / 2 )^th Observation

  = Value of 135^th Observation

From the column of cumulative frequency cf, we find that the 135th observation lies in the class 76-80.

∴ The median class is 75.5-80.5.

Now,
∴L=lower boundary point of median class =75.5

∴n=Total frequency =271

∴cf=Cumulative frequency of the class preceding the median class =80

∴f=Frequency of the median class =56

∴c=class length of median class =5

M = L + (( n / 2 ) - c .f / f ) * c

= 75.5 + ( (135 - 80 )/ 56 ) * 5

= 75.5 + (55 / 56 ) * 5

= 75.5 + 4.9107

= 80.4554

Median = 80.4554