Question

Calculate the mean of the following distribution using the step deviation method.

Marks	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
Number of students	10	9	25	30	16	10

Answer 1

Hint: In the step deviation method, we take an assumed mean from the frequencies, and by using the formula of the step deviation method, we can find the arithmetic mean. The values which are given here as y are the frequencies. We also need to find class marks while finding the mean in the assumed mean method. It will give the desired answer for the problem.

Complete step by step solution:
We know that the formula used to find the arithmetic mean for a data using step deviation is,
$\bar x = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}$
Where A is assumed mean,
and ${d_i}$ is $\left( {{y_i} - A} \right)$
and $\sum {{f_i}} $ is the sum of all the frequencies.

Classes	Class mark (${y_i}$)	${d_i} = \left( {{y_i} - A} \right)$	${f_i}$	${f_i}{d_i}$
0 – 10	5	-20	10	-200
10 – 20	15	-10	9	-90
20 – 30	25 = A	0	25	0
30 – 40	35	10	30	300
40 – 50	45	20	16	320
50 – 60	55	30	10	300
Total			$\sum {{f_i}} = 100$	$\sum {{f_i}{d_i} = 630} $

By substituting the above information in the mean formula that is mentioned above we get,
$ \Rightarrow \bar x = 25 + \dfrac{{630}}{{100}}$
Divide the numerator by denominator,
$ \Rightarrow \bar x = 25 + 6.3$
Add the terms,
$ \Rightarrow \bar x = 31.3$

Hence, the mean by using the step deviation method is 31.3.

Note: While solving such questions where we have to find many values then add them and then substitute the summed value, it is best to draw a table as it makes it much easier to solve. Also it doesn’t matter what the assumed meaning is. It is simply used for calculation, though it is best to use one of the class marks as assumed to decrease the calculations.