Question

The following table shows the marks secured by $100$ students in an examination.

Marks	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
No. of students	$15$	$20$	$35$	$20$	$10$

Answer 1

Step 1: Finding the value of ${\mathit{u}}_{\mathbf{i}}$

For determining the mean, we will use the step deviation method.

Let the assumed mean be $a=25$.

Here, class size is $h=10$.

$u_{\mathrm{i}}$ can be found by substituting the corresponding values in the formula $u_i=\frac{x_i-a}{h}$.

Marks	Mid value $x_i$	Number of students $f_i$	$$\begin{gathered} u_i=\frac{x_i-a}{h} \\ {u}_i=\frac{x_i-25}{10} \end{gathered}$$	$f_i{u}_i$
$0-10$	$5$	$15$	$-2$	$-30$
$10-20$	$15$	$20$	$-1$	$-20$
$20-30$	$25$	$35$	$0$	$0$
$30-40$	$35$	$20$	$1$	$20$
$40-50$	$45$	$10$	$2$	$20$
		$\sum _{}{f}_i=100$		$\sum _{}{f}_i{u}_i=-10$

Step 2: Finding the mean

The formula used to determine the mean is $\overline{x}=a+h\left(\frac{\sum _{}{f}_i{u}_i}{\sum _{}{f}_i}\right)$.

Where, $a$ is the assumed mean,

$h$ is the class size,

$\sum _{}{f}_i$ is the total of frequencies, and

$u_i=\frac{x_i-a}{h}$

Substituting the values in the mean formula we get,

$$\begin{gathered} \Rightarrow \overline{x}=25+10\left(-\frac{10}{100}\right) \\ \Rightarrow \overline{x}=25-1 \\ \therefore \overline{x}=24 \end{gathered}$$

Hence, the mean marks secured by $100$ students is $24$ marks.