Question

Find the mean, mode and median of the following frequency distribution:
$\begin{array}{cccccccc}\text{Class}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70\\ \text{Frequency}& 5& 10& 18& 30& 20& 12& 5\end{array}$

Answer 1

Let assumed mean be $35,h=10,$ now we have

$\begin{array}{cccccc}\text{Class}& \text{Frequency}\left(f_i\right)& \text{Mid value}\left(x_i\right)& u_i=\frac{x_i-A}{h}& C.F& f_i{u}_i\\ 0-10& 5& 5& -3& 5& -15\\ 10-20& 10& 15& -2& 15& -20\\ 20-30& 18& 25& -1& 33& -18\\ 30-40& 30& 35=A& 0& 63& 0\\ 40-50& 20& 45& 1& 83& 20\\ 50-60& 12& 55& 2& 95& 24\\ 60-70& 5& 65& 3& 100& 15\\ & N=100& & & & \sum f_i{u}_i=6\end{array}$

(i) Mean, $\overline{x}$ $=A+h\left(\frac{\sum f_i{u}_i}{N}\right)$

$=35+10\times \left(\frac{6}{100}\right)$
$=35+0.6=35.6$

(ii) $N=100,\frac{N}{2}=50$

Cumulative frequency just after $50$ is $63.$

Median class is $30-40$

$I=30,h=10,N=100,c=33,f=30$

Therefore,
Median $=I+h\times \left(\frac{\frac{N}{2}-c)}{f}\right)$

$=30+10\left(\frac{50-33}{30}\right)$

$=30+10\left(\frac{17}{30}\right)$

$=30+5.67=35.67$

(iii) Mode $=3\times$median $-2\times$mean

$=3\times 35.67-2\times 35.6=107.01-71.2$

$=35.81$

Thus, Mean $=35.6,$ Median $=35.67$ and Mode $=35.81$