Question

Find the mean marks of students from the following cumulative frequency distribution:

Marks	Number of students
o 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

Here we have, the cumulative frequency distribution. So, first we convert it into an ordinary frequency distribution. we observe that are 80 students getting marks greater than or equal to 0 and 77 students have secured 10 and more marks. Therefore, the number of students getting marks between 0 and 10 is 80-77= 3.
Similarly, the number of students getting marks between 10 and 20 is 77-72= 5 and so on. Thus, we obtain the following frequency distribution.

Marks	Number of students
0-10	3
10-20	5
20-30	7
30-40	10
40-50	12
50-60	15
60-70	12
70-80	6
80-90	2
90-100	8

Now, we compute mean arithmetic mean by taking 55 as the assumed mean.
Computative of Mean

Marks $\left(x_i\right)$	Mid-value $\left(f_i\right)$	Frequency	$u_i\frac{x_i-55}{10}$	$f_i{u}_i$
0-10	5	3	-5	-15
10-20	15	5	-4	-20
20-30	25	7	-3	-21
30-40	35	10	-2	-20
40-50	45	12	-1	-20
50-60	55	15	0	0
60-70	65	12	1	12
70-80	75	6	2	12
80-90	85	2	3	6
90-100	95	8	4	32
Total	$\sum f_i=80$			$\sum f_i{u}_i$= -26

We have,
N= $sum{f}_i=80,\sum f_i{u}_i=-26$, A= 55 and h= 10
$\therefore \overline{X}=A+h[\frac{1}{N}\sum f_i{u}_i]$
$\Rightarrow \overline{X}=A+h[\frac{1}{N}\sum f_i{u}_i]$
$\Rightarrow \overline{X}=55+10\times \frac{-26}{80}=55-3.25=51.75$Marks.