Question

The following table shows the marks scored by $140$ students in an examination of a certain paper:

Marks:	0-10	10-20	20-30	30-40	40-50
Number of students:	$20$	$24$	$40$	$36$	$20$

Calculate the average marks by using all the three methods: direct method, assumed mean deviation and shortcut method.

Answer 1

Direct method

size	x	f	xf
0-10	5	20	100
10-20	15	24	360
20-30	25	40	1000
30-40	35	36	1260
40-50	45	20	900

mean $=\frac{\sum fx}{\sum f}$

$=\frac{3620}{140}=25.857$

Assumed mean method

size	x	u=x-25	f	uf
0-10	5	-20	20	-400
10-20	15	-10	24	-240
20-30	25	0	40	0
30-40	35	10	36	360
40-50	45	20	20	400

mean $=A+\frac{\sum fu}{\sum f}$

$=25+\frac{120}{140}=25.875$

Step deviation method

size	x	d=x-25	u=(x-25) / 10	f	uf
0-10	5	-20	-2	20	-40
10-20	15	-10	-1	24	-24
20-30	25	0	0	40	0
30-40	35	10	1	36	36
40-50	45	20	2	20	40

mean $=A+h\times \frac{\sum fu}{\sum f}$

$=25+10\times \frac{12}{140}=25.875$