Question

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures

No of Students per teacher	Number of states / U.T
$15-20$	$3$
$20-25$	$8$
$25-30$	$9$
$30-35$	$10$
$35-40$	$3$
$40-45$	$0$
$45-50$	$0$
$50-55$	$2$

Answer 1

Step 1: Find the mode of the given data.

The modal class is $30-35$; $l=30,$ and the frequencies are$f_m=10,f_1=9,f_2=3andh=5$

The formula to find the mode is

$Mode=l+\left[\frac{(f_m-f_1)}{(2f_m-f_1-f_2)}\right]\times h$

$$\begin{gathered} =30+\left(\frac{(10-9)}{(20-9-3)}\right)\times 5 \\ =30+\left(\frac{5}{8}\right) \\ =30+0.625 \\ =30.625 \end{gathered}$$

Therefore, the mode of the given data is $30.625$

Step 2: Find the mean of the given data.

Find the class mark using the formula, $x_i=\frac{(upperlimit+lowerlimit)}{2}$

Class Interval	Frequency ($f_i$)	Class mark ($x_i$)	$f_i{x}_i$
$15-20$	$3$	$17.5$	$52.5$
$20-25$	$8$	$22.5$	$180.0$
$25-30$	$9$	$27.5$	$247.5$
$30-35$	$10$	$32.5$	$325.0$
$35-40$	$3$	$37.5$	$112.5$
$40-45$	$0$	$42.5$	$0$
$45-50$	$0$	$47.5$	$0$
$50-55$	$2$	$52.5$	$105.5$
Total	$\sum f_i=35$		$\sum f_i{x}_i=1022.5$

The formula to calculate the mean,

Mean $=\overline{x}̄=\frac{\sum f_i{x}_i}{\sum f_i}$

$$\begin{gathered} =\frac{1022.5}{35} \\ =29.2 \end{gathered}$$

Hence, the mean of the given data is $29.2$