Question

Find the value of p, if the mean of the following distribution is 20

x:	15	17	19	20 + p	23
f:	2	3	4	5 p	6

Answer 1

Calculation of Mean

xi	fi	fi xi
15	2	30
17	3	51
19	4	76
20 + p	5p	(20+p)5p
23	6	138
Total	${\sum }_{}{f}_i$ = 15 + 5p	${\sum }_{}{f}_i{x}_i$ = 295 + (20 + p) 5p

We have:

${\sum }_{}{f}_i$ = 15 + 5p, ​${\sum }_{}{f}_i{x}_i$ = 295 + (20 + p) 5p​.

∴ Mean =$\frac{{\sum }_{}{f}_i{x}_i}{{\sum }_{}{f}_i}$

$$\begin{gathered} 20=\frac{295+(20+p)5p}{15+5p} \\ \Rightarrow 20\times (15+5p)=295+(20+p)5p\Rightarrow 300+100p=295+100p+5p \end{gathered}$$

⇒ 300 - 295 + 100p -100p = 5p²
⇒ 5 = 5p²
⇒ p² = 1
⇒ p = 1.