Question

Marks	0 - 5	5 -10	10 - 15	15 - 20	20 - 25	25 - 30	30 - 35	35 - 40	40 - 45	45 - 50
Frequency	3	5	7	8	10	11	14	19	15	13

For the following distribution, find the mean using the step deviation method. (Round your answer to the nearest whole number)
(A) 29
(B) 31
(C) 35
(D) 37

Answer 1

In the given question, we have to find the mean. Thus, we will use the step deviation method to get the answer. To find the mean, we will use the formula, $a+\dfrac{\Sigma {{f}_{i}}{{d}_{i}}}{\Sigma f}\times h$, where a is the assumed mean, ${{d}_{i}}=\dfrac{\left( {{x}_{i}}-a \right)}{h}$, h is the height and f is the frequency. Thus, we will find the mid-value of the given marks and then substitute all these values in the given formula to get the required solution for the problem.

Complete step by step answer:
According to the problem, we have to find the mean using the step-deviation method. So, we will first find the mid-value of the given interval, which is equal to, $\dfrac{\text{upper limit + lower limit}}{2}$, thus we get,

Marks	Frequency $\left( {{f}_{i}} \right)$	Mid-value $\left( {{x}_{i}} \right)$
0 - 5	3	2.5
5 - 10	5	7.5
10 - 15	7	12.5
15 - 20	8	17.5
20 - 25	10	22.5
25 - 30	11	27.5
30 - 35	14	32.5
35 - 40	19	37.5
40 - 45	15	42.5
45 - 50	13	47.5

So, let us assume that the mid-value $=a=22.5$. Therefore, we will now calculate the value of ${{d}_{i}}$, which is equal to $\dfrac{{{x}_{i}}-a}{h}$, where h is the height.
Thus, h = 5, therefore, we get,

Marks	Frequency $\left( {{f}_{i}} \right)$	Mid-value $\left( {{x}_{i}} \right)$	${{d}_{i}}=\dfrac{{{x}_{i}}-22.5}{5}$
0 - 5	3	2.5	-4
5 - 10	5	7.5	-3
10 - 15	7	12.5	-2
15 - 20	8	17.5	-1
20 - 25	10	22.5	0
25 - 30	11	27.5	1
30 - 35	14	32.5	2
35 - 40	19	37.5	3
40 - 45	15	42.5	4
45 - 50	13	47.5	5

Now we will find the summation of frequencies and deviation of the given mean data, thus we get,

Marks	Frequency $\left( {{f}_{i}} \right)$	Mid-value $\left( {{x}_{i}} \right)$	${{d}_{i}}=\dfrac{{{x}_{i}}-22.5}{5}$	${{f}_{i}}{{d}_{i}}$
0 - 5	3	2.5	-4	-12
5 - 10	5	7.5	-3	-15
10 - 15	7	12.5	-2	-14
15 - 20	8	17.5	-1	-8
20 - 25	10	22.5	0	0
25 - 30	11	27.5	1	11
30 - 35	14	32.5	2	28
35 - 40	19	37.5	3	57
40 - 45	15	42.5	4	60
45 - 50	13	47.5	5	65
105			172

Now we have,
$\begin{aligned} & \Sigma {{f}_{i}}{{d}_{i}}=172\ldots \ldots \ldots \left( 1 \right) \\\ & \Sigma {{f}_{i}}=105\ldots \ldots \ldots \left( 2 \right) \\\ & h=5\ldots \ldots \ldots \left( 3 \right) \\\ & a=22.5\ldots \ldots \ldots \left( 4 \right) \\\ \end{aligned}$
Now, mean for step=deviation formula is equal to, mean = $a+\dfrac{\Sigma {{f}_{i}}{{d}_{i}}}{\Sigma f}\times h$. So, we will put equation (1), (2), (3) and (4) in the formula and we get mean as,
$\begin{aligned} & =22.5+\dfrac{172}{105}\times 5 \\\ & \Rightarrow 22.5+\dfrac{172}{21} \\\ \end{aligned}$
On further simplification, we get,
Mean $=22.5+8.19$
$\Rightarrow$ Mean $=30.69\approx 31$
Therefore, for the given distribution, the mean using step-deviation method is equal to 31 approximately.
So, the correct answer is “Option B”.

Note: While solving this problem, do mention all the steps properly to avoid error and confusion. Do mention the formulas accurately. Do not forget that ${{d}_{i}}$ is calculated by subtracting ${{x}_{i}}$ and assumed mean, divided by the width of the class interval.