Question

Find the missing frequency (p) for the following distribution whose mean is 7.68

x:	3	5	7	9	11	13
f:	6	8	15	p	8	4

Answer 1

Calculation of Mean

xi	fi	fi xi
3	6	18
5	8	40
7	15	105
9	p	9p
11	8	88
13	4	52
Total	${\sum }_{}{f}_i$ = 41 + p	${\sum }_{}{f}_i{x}_i$ = 303 + 9p

We have:

${\sum }_{}{f}_i$ = 41+ p, ​ ${\sum }_{}{f}_i{x}_i$ = 303 +9p

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

$$\begin{gathered} 7.68=\frac{303+9p}{41+p} \\ \Rightarrow 7.68\times (41+p)=303+9p\Rightarrow 314.88+7.68p=303+9p \\ \Rightarrow 314.88-303=9p-7.68p \\ \Rightarrow 11.88=1.32p \\ \Rightarrow p=\frac{11.88}{1.32} \\ \Rightarrow p=9. \end{gathered}$$