Question

The table below gives the percentage distribution of female teachers in the primary school of rural areas of various States and Union Territories (UT) of India. Find the median, mean and mode of the data.

% of female teachers	$15-25$	$25-35$	$35-45$	$45-55$	$55-65$	$65-75$	$75-85$
No. of states/UT	$6$	$11$	$7$	$4$	$4$	$2$	$1$

Answer 1

Step 1: Completing the table and finding the mean

Prepare a cumulative frequency and $x_i{f}_i$ in the table as below:

% of female teachers	$\left(x_i\right)$	No. of states/UT $\left(f_i\right)$	$x_i{f}_i$	Cumulative Frequency ($cf$)
$15-25$	$20$	$6$	$120$	$6$
$25-35$	$30$	$11$	$330$	$17$
$35-45$	$40$	$7$	$280$	$24$
$45-55$	$50$	$4$	$200$	$28$
$55-65$	$60$	$4$	$240$	$32$
$65-75$	$70$	$2$	$140$	$34$
$75-85$	$80$	$1$	$80$	$35$
Total		$\sum _{}{f}_i=35$	$\sum _{}{f}_i{x}_i=1390$

The formula of the mean is $Mean=\frac{\sum _{}{f}_i{x}_i}{\sum _{}{f}_i}$,

where $\sum _{}{f}_i$ is the total frequency and $x_i$ is the mid-value of the class-interval.

$$\begin{gathered} \Rightarrow Mean=\frac{1390}{35} \\ \therefore Mean=39.71 \end{gathered}$$

Thus, the mean is $39.71\%$.

Step 2: Calculating the median

The formula for the median is,

$Median=l+\left(\frac{{\displaystyle \frac{\mathrm{n}}{2}-\mathrm{c}}}{\mathrm{f}}\right)\times h$.

Here,

$l$ is the lower limit of the median class,

$n$ is the number of observations,

$c$ is the cumulative frequency of the class preceding the median class,

$f$ is the frequency of the median class, and

$h$ is the class size

Here $n=35$

$\Rightarrow \frac{35}{2}=17.5$

The observation $17.5$ lies in the class $35-45$.

So, we have

$l=35,c=17,f=7,h=10$

$$\begin{gathered} \Rightarrow Median=35+\left(\frac{17.5-17}{7}\right)\times 10 \\ \Rightarrow Median=35+0.7 \\ \therefore Median=35.7 \end{gathered}$$

Thus, the median is $35.7\%$.

Step 3: Calculating the mode

The formula for the mode is,

$Mode=l+\left(\frac{{\displaystyle f_1-f_0}}{2f_1-f_0-f_2}\right)\times h$.

Here,

$l$ is the lower limit of the modal class,

$f_1$ is the frequency of the modal class,

$f_0$ is the frequency of the class preceding the modal class,

$f_2$ is the frequency of the class succeeding the modal class, and

$h$ is the class size

The class $25-35$ has a maximum frequency, so the modal class is $25-35$.

So, we have

$l=25,f_1=11,f_0=6,f_2=7,h=10$

$$\begin{gathered} \Rightarrow Mode=25+\left(\frac{11-6}{22-6-7}\right)\times 10 \\ \Rightarrow Mode=25+5.5 \\ \therefore Mode=30.5 \end{gathered}$$

Thus, the mode is $30.5\%$.

Hence, the given distribution the value of mean is $39.71\%$, median is $35.7\%$ and mode is $30.5\%$.