Question

Using step deviation method find mean

X	20-40	40-60	60-80	80-100
Frequency	4	8	12	16

A.40
B.50
C.60
D.70

Answer 1

Hint: The formula for finding mean of a data using step deviation method is \[\overline X = A + \dfrac{{\sum {f{d^{'}}} }}{{\sum f }} \times i\], where \[\overline X \] is the mean of the data , \[A\] is the assumed mean, \[\sum {f{d^{'}}} \] is the summation of the frequencies and deviation of a given mean data. Here, the value of \[{d^{'}}\] will be found using formula \[{d^{'}} = \dfrac{{(x - A)}}{i}\].
We will first the class mark or the midpoint of each interval denoted by \[X\], then we will find the deviation of the data and then substitute them in the formula.

Complete step-by-step answer:
The class mark of each class will be the average of the limits of each class. For the first interval \[20 - 40\], the class mark will be \[X = \dfrac{{20 + 40}}{2} = \dfrac{{60}}{2} = 30\]. Similarly the class mark of the rest of the interval along with deviation of each class can be solved in the table below.
Also, the assumed mean for the data can be any value which we feel is closest to the actual mean. Here let us assume 50 to be the assumed mean.
Now we will solve the required values in the table below.

X	Class mark(\[X\])	Frequency(\[f\])	\[i = {\text{ }}class{\text{ }}width\]	\[A = 50\](Assumed Mean)	\[{d^{'}} = \dfrac{{(x - A)}}{i}\]	\[f{d^{'}}\]
\[20 - 40\]	30	4	20	50	-1	-4
\[40 - 60\]	\[50 = A\]	8	20	50	0	0
\[60 - 80\]	70	12	20	50	1	12
\[80 - 100\]	90	16	20	50	2	32
		\[\sum {f = 40} \]				\[\sum {f{d^{'}} = 40} \]

Now, using the formula for finding mean using step deviation method and substituting the values found we can easily find the actual mean of this data.
\[
  \overline X = A + \dfrac{{\sum {f{d^{'}}} }}{{\sum f }} \times i \\
\Rightarrow \overline X = 50 + \dfrac{{40}}{{40}} \times 20 \\
\Rightarrow \overline X = 50 + 20 \\
\Rightarrow \overline X = 70 \\
\]
Thus, the mean of this data comes out to be 70.
Hence, option (D) is correct.

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.