Question

Find the mean of the following distribution by step- deviation method.

Daily Expenditure (in Rs.)	100–150	150–200	200–250	250–300	300–350
No. of Households	4	5	12	2	2

Answer 1

Hint: In general there are three types of methods to find the mean of a given distribution
(i) Direct method (ii) Assumed mean method (iii) Step deviation method
As here we have been asked to solve the given problem using step deviation method, the formula used for arithmetic mean of grouped data by step deviation method is
 \[\bar{X}=A+\dfrac{\sum{fd'}}{\sum{f}}\times i\]
A = Assumed mean of given data
$\sum$ = Summation of frequencies given in the grouped data
$\sum{fd'}$= Summation of the frequencies and deviation of a given mean data
\[d’=\dfrac{(x-A)}{i}\]
i= Class interval width
\[\bar{X}\]= arithmetic mean

Complete step-by-step solution:
Problems like these would be easy to solve when computed in a tabular form of data.

Daily expenditure	Mid Point (X)	Frequency=number of households(f)	i=class interval width	A=Assumed means	\[d'=\dfrac{\left( x-A \right)}{i}\]	fd’
100-150	\[\dfrac{100+150}{2}=125\]	4	\[150-100=50\]	225	\[\dfrac{\left( 125-225 \right)}{50}=-2\]	\[4\times -2=-8\]
150-200	\[\dfrac{150+200}{2}=175\]	5	\[200-150=50\]	225	\[\dfrac{\left( 175-225 \right)}{50}=-1\]	\[5\times -1=-5\]
200-250	\[\dfrac{200+250}{2}=225\]=A	12	\[250-200=50\]	225	\[\dfrac{\left( 225-225 \right)}{50}=0\]	\[12\times 0=0\]
250-300	\[\dfrac{250+300}{2}=275\]	2	\[300-250=50\]	225	\[\dfrac{\left( 275-225 \right)}{50}=1\]	\[2\times 1=2\]
300-350	\[\dfrac{300+350}{2}=325\]	2	\[350-300=50\]	225	\[\dfrac{\left( 325-225 \right)}{50}=2\]	\[2\times 2=4\]

\[\sum{f=4+5+12+2+2=25}\]
\[\sum{fd'=(-8)+(-5)+0+2+4=-7}\]
\[\begin{align}
  & A=225 \\
 & i=50 \\
\end{align}\]
Now substituting the above values in the formula for mean from step deviation that is
\[\bar{X}=A+\dfrac{\sum{fd'}}{\sum{f}}\times i\]
\[\begin{align}
  & \bar{X}=225+\dfrac{(-7)}{25}\times 50 \\
 & \bar{X}=225+(-14)=211 \\
 & \bar{X}=211 \\
\end{align}\]
Therefore the mean for the given question solved by using step deviation method is 211.

Note: Students should read the question thoroughly and should know which method to use. While solving using the step deviation method the assumed mean should be noted down accurately and it avoids confusion. The formula should be noted down first which will make it easy to solve the problem.