Question

8.(i) Using step-deviation method, calculate the mean marks of the following distribution.

Class Interval	$50-55$	$55-60$	$60-65$	$65-70$	$70-75$	$75-80$	$80-85$	$85-90$
Frequency	$5$	$20$	$10$	$10$	$9$	$6$	$12$	$8$

Answer 1

Step 1: Prepare the table to find the values of $f\left(x\right)$

Class Interval	Frequency, $f$	$x$	$d=x-A=x-67.5$	$$\begin{gathered} u=\frac{d}{h} \\ h=5 \end{gathered}$$	$fu$
$50-55$	$5$	$52.5$	$-15$	$-3$	$-15$
$55-60$	$20$	$57.5$	$-10$	$-2$	$-40$
$60-65$	$10$	$62.5$	$-5$	$-1$	$-10$
$65-70$	$10$	$67.5$	$0$	$0$	$0$
$70-75$	$9$	$72.5$	$5$	$1$	$9$
$75-80$	$6$	$77.5$	$10$	$2$	$12$
$80-85$	$12$	$82.5$	$15$	$3$	$36$
$85-90$	$8$	$87.5$	$20$	$4$	$32$
Total	$\sum _{}f=80$				$\sum _{}fu=24$

Step 2: Find the value of mean marks:

Arithmetic Mean $=A+h\frac{\sum _{}fu}{\sum _{}f}$

Where,

$x$ is the midpoint of the class intervals.

$f$ is the given frequency of the intervals.

$d$ is $x-A$ (Assumed Mean $A=67.5$)

$h$ is the difference in the class intervals.

Substituting the values in the formula, we get,

Arithmetic Mean $=67.5+5\times \frac{24}{80}$

$=67.5+1.5$

$=69$

Hence, the mean for the given data is $69$.