Question

Find mean, variance and S.D. of the following data:

Classes	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70	70  –  80	80 – 90	90 – 100
Frequency	7	14	6	13	9	15	11	10	15

Answer 1

We prepare the following table to compute mean, variance, and S.D.:

C.I.	Mid value (xi)	fi	ui = `(x_"i" - "A")/"h"` A = 55 h = 10	fiui	fiui2 = fiui x ui
10 – 20	15	7	– 4	– 28	112
20 – 30	25	14	– 3	– 42	126
30 – 40	35	6	– 2	– 12	24
40 – 50	45	13	– 1	– 13	13
50 – 60	55	9	0	0	0
60 – 70	65	15	1	15	15
70 – 80	75	11	2	22	44
80 – 90	85	10	3	30	90
90 – 100	95	15	4	60	240
		`sumf_"i"` = 100		`sumf_"i"u_"i"` = 32	`sumf_"i"u_"i"^2` = 664

From the table, `sumf_"i"u_"i" = 32, sumf_"i" = 100, sumf_"i"u_"i"^2 = 644`

∴ `bar(u) = (sumf_"i"u_"i")/(sumf_"i")`

`bar(u) = 32/100`

`bar(u)` = 0.32

∴ `bar(x) = "A" + "h"bar(u)`

= 55 + 10(0.32)

= 58.2

`sigma_u^2 = (sumf_"i"u_"i"^2)/(sumf_"i") - (baru)^2`

`= 644/100 - (0.32)^2`

= 6.64 – 0.1024

= 6.5376

Var(x) = `sigma_x^2`

`= "h"^2sigma_u^2`

= 100 × 6.5376

= 653.76

S.D. = `sigma_x`

`= sqrt(653.76)`

= 25.56

Hence, mean = 58.2, variance 653.76, S.D. = 25.56