Question

Find the value of the missing variable $x$ from the following distribution whose mean is $31.87$.

$x$	$12$	$20$	$27$	$33$	$x$	$54$
$f$	$8$	$16$	$48$	$90$	$30$	$8$

Answer 1

Step 1: Calculate the value of $\sum _{i=1}^n{x}_i{f}_i$and $\sum _{i=1}^n{f}_i$:

As per the given table, values of $x_i{f}_i$ are:

$x_i$	$f_i$	$x_i{f}_i$
$12$	$8$	$96$
$20$	$16$	$320$
$27$	$48$	$1296$
$33$	$90$	$2970$
$x$	$30$	$30x$
$54$	$8$	$432$
	$\sum _{i=1}^n{f}_i=200$	$\sum _{i=1}^n{x}_i{f}_i=5114+30x$

Therefore,

$\sum _{i=1}^n{f}_i=8+16+48+90+30+8$

$=200$

$\sum _{i=1}^n{x}_i{f}_i=96+320+1296+2970+30x+432$

$=5114+30x$

Step 2: Calculate the value of $x$:

$\overline{x}=\frac{{\sum }_{i=1}^n{x}_i{f}_i}{{\sum }_{i=1}^n{f}_i}$

Where,

$\overline{x}=$mean of the given distribution

$\sum _{i=1}^n{f}_i=$sum of the values of frequency

$\sum _{i=1}^n{x}_i{f}_i=$summation of all the values of $x_i{f}_i$

Put the values in the formula:

$\overline{x}=\frac{{\sum }_{i=1}^n{x}_i{f}_i}{{\sum }_{i=1}^n{f}_i}$

$\Rightarrow 31.87=\frac{5114+30x}{200}$

$\Rightarrow 31.87\times 200=5114+30x$

$\Rightarrow 6374=5114+30x$

$\Rightarrow 6374-5114=30x$

$\Rightarrow 1260=30x$

$\Rightarrow x=\frac{1260}{30}$

$\Rightarrow x=42$

Hence, the value of $x$ in the given distribution is $42$.