The monthly expenditure on fruits in 200 families of a Housing Society is given below. Find the value of x and also find the mode and mean…

Class Interval	Frequency \(\displaystyle \left(f_{i}\right)\)	\(\displaystyle x_{i}\)	\(\displaystyle f_{i} x_{i}\)
$\displaystyle 1000$-$\displaystyle 1500$	$\displaystyle 24$	$\displaystyle 1250$	$\displaystyle 30000$
$\displaystyle 1500$-$\displaystyle 2000$	$\displaystyle 40$	$\displaystyle 1750$	$\displaystyle 70000$
$\displaystyle 2000$-$\displaystyle 2500$	$\displaystyle 33$	$\displaystyle 2250$	$\displaystyle 74250$
$\displaystyle 2500$-$\displaystyle 3000$	$\displaystyle 28$	$\displaystyle 2750$	$\displaystyle 77000$
$\displaystyle 3000$-$\displaystyle 3500$	\(\displaystyle x\)	$\displaystyle 3250$	\(\displaystyle 3250 x\)
$\displaystyle 3500$-$\displaystyle 4000$	$\displaystyle 22$	$\displaystyle 3750$	$\displaystyle 82500$
$\displaystyle 4000$-$\displaystyle 4500$	$\displaystyle 16$	$\displaystyle 4250$	$\displaystyle 68000$
$\displaystyle 4500$-$\displaystyle 5000$	$\displaystyle 7$	$\displaystyle 4750$	$\displaystyle 33250$
Total	\(\displaystyle 170+x\)		$\displaystyle 435000$ + $\displaystyle 3250$ x

\(\displaystyle 170+x=200 \Rightarrow x=30\) $\displaystyle 1500$ - $\displaystyle 2000$ is the modal class \(\displaystyle \Rightarrow\) Mode \(\displaystyle =1500+\left(\frac{40-24}{2 \times 40-24-33}\right) \times 500\) \[\text { = } 1500 \text { + } 347.82 \] Hence the modal monthly expenditure is ₹ $\displaystyle 1847.82$ \(\displaystyle \therefore \sum f_{i} x_{i}=435000+3250 \times 30=532500\) Mean \(\displaystyle =\frac{532500}{200}=2662.50\) Hence the mean monthly expenditure is ₹ $\displaystyle 2662.50$ ALTERNATE SOLUTION: \(\displaystyle \sum f_{i}=170+x=200\) \(\displaystyle \Rightarrow x=30\)

Class Interval	Frequency \(\displaystyle \left(f_{i}\right)\)	\(\displaystyle x_{i}\)	\(\displaystyle u_{i}\)	\(\displaystyle f_{i} u_{i}\)
$\displaystyle 1000$-$\displaystyle 1500$	$\displaystyle 24$	$\displaystyle 1250$	-$\displaystyle 3$	-$\displaystyle 72$
$\displaystyle 1500$-$\displaystyle 2000$	$\displaystyle 40$	$\displaystyle 1750$	-$\displaystyle 2$	-$\displaystyle 80$
$\displaystyle 2000$-$\displaystyle 2500$	$\displaystyle 33$	$\displaystyle 2250$	-$\displaystyle 1$	-$\displaystyle 33$
$\displaystyle 2500$-$\displaystyle 3000$	$\displaystyle 28$	$\displaystyle 2750$	$\displaystyle 0$	$\displaystyle 0$
$\displaystyle 3000$-$\displaystyle 3500$	$\displaystyle 30$	$\displaystyle 3250$	$\displaystyle 1$	$\displaystyle 30$
$\displaystyle 3500$-$\displaystyle 4000$	$\displaystyle 22$	$\displaystyle 3750$	$\displaystyle 2$	$\displaystyle 44$
$\displaystyle 4000$-$\displaystyle 4500$	$\displaystyle 16$	$\displaystyle 4250$	$\displaystyle 3$	$\displaystyle 48$
$\displaystyle 4500$-$\displaystyle 5000$	$\displaystyle 7$	$\displaystyle 4750$	$\displaystyle 4$	$\displaystyle 28$
Total	$\displaystyle 200$			-$\displaystyle 35$

$\displaystyle 1500$ - $\displaystyle 2000$ is the modal class \[\begin{aligned} \Rightarrow \text { Mode } & =1500+\left(\frac{40-24}{2 \times 40-24-33}\right) \times 500 \\ & =1500+347.82 \end{aligned} \] \[=1847.82 \] Hence the modal monthly expenditure is ₹ $\displaystyle 1847.82$ \[\begin{aligned} \text { Mean } & =2750+\frac{(-35)}{200} \times 500 \\ & =2662.50 \end{aligned} \] Hence the mean monthly expenditure is ₹ $\displaystyle 2662.50$