Question

Determine the mean and standard deviation for the following distribution:

Marks	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Frequency	1	6	6	8	8	2	2	3	0	2	1	0	0	0	1

Answer 1

`x`	`f_i`	`f_ix_i`	`d_i = x_i - barx`	`f_i d_i`	`f_i d_ix^2`
2	1	2	– 4	– 4	16
3	6	18	– 3	– 18	54
4	6	24	– 2	– 12	24
5	8	40	– 1	– 8	8
6	8	48	0	0	0
7	2	14	1	2	2
8	2	16	2	4	8
9	3	27	3	9	27
10	0	0	4	0	0
11	2	2	5	10	50
12	1	12	6	6	36
13	0	0	7	0	0
14	0	0	8	0	0
15	0	0	9	0	0
16	1	16	10	10	100
	N = 40	`sumf_ix_i` = 239		`sumf_i d_i` = – 1	`sumf_i d_i^2` = 325

Mean `barx = (sumf_ix_i)/N = 239/40` = 5.9 = 6

∴ S.D. = `sigma = sqrt((sumf_i d_i^2)/N - ((sumf_i d_i)/N)^2`

= `sqrt(325/40 - ((-1)/40)^2`

= `sqrt(8.125 - 0.000625)`

= `sqrt(8.124375)`

= 2.85

Here, the required mean = 6 and M.D. = 2.85