Find the value of p if the mean of the following distribution is 18.

Variate (xi)	13	15	17	19	20+p	23
Frequency (fi)	8	2	3	4	5p	6

Solution:

Variate (xi)	Frequency (fi)	fi xi
13	8	104
15	2	30
17	3	51
19	4	76
20+p	5p	5p2+100p
23	6	138
Total	Ʃfi = 23+5p	Ʃfi xi = 399+5p2+100p

Mean = Ʃf_i x_i / Ʃf_i

18 = (399+5p²+100p)/( 23+5p) [Given mean = 18]

18(23+5p) = 399+5p²+100p

414 + 90p = 399+5p²+100p

5p²+100p-90p+399-414 = 0

5p²+10p-15 = 0

Dividing by 5, we get

p²+2p-3 = 0

(p-1)(p+3) = 0

p-1 = 0 or p+3 = 0

p = 1 or p = -3

p cannot be negative.

So p = 1

Hence the value of p is 1.