Question

Calculate the average daily income (in rupees) of the following data about men working in a company:

Daily Income (₹)	<100	<200	<300	<400	<500
Number of men	12	28	34	41	50

Answer 1

Hint:
Here, we will calculate the mean that is average value by using the formula \[\overline X = \left( {a + \dfrac{{{\sum _i}{f_i}{u_i}}}{{{\sum _i}{f_i}}} \times h} \right)\] and we will calculate mid value i.e \[A = 250\], \[\sum {f_i},{x_i},{u_i},{f_i}{u_i}\] and \[\sum {f_i}{u_i}\]
Formula Used: \[\overline X = \left( {a + \dfrac{{{\sum _i}{f_i}{u_i}}}{{{\sum _i}{f_i}}} \times h} \right)\] and also calculate \[{u_i} = \dfrac{{{x_i} - A}}{h}\]
Here, \[\overline X = \] means.
\[A\] and \[h\] are arbitrary constants and \[{f_i}\] is the frequency.

Complete step by step solution:
From the given data, we can calculate \[h\]by using the above formula i.e \[h = {x_n} - {x_{n - 1}}\] which comes out to be \[h = 100 - 0 = 100\] So, \[h = 100\] and here A is the mid value which is calculated from \[{x_i}\] i.e calculated from the given intervals which is mid value of \[0{\rm{ }}and{\rm{ }}100 = 50\]. Similarly, calculated all the values of \[{x_i}\].Now, next step is to calculate \[{u_i}\]from the above formula i.e \[{u_i} = \dfrac{{{x_i} - A}}{h}\]. Put h as 100 , A as 250 and \[{x_i}\] has different values for different intervals. Then, calculate \[{f_i}{u_i}\]. At the end calculate the sum of \[{f_i}\] , \[{f_i}{u_i}\].

Class Interval (rupees)	Number of men \[\left( {{f_i}} \right)\]	\[{x_i}\]	\[{u_i} = \dfrac{{{x_i} - A}}{h}\]	\[{f_i}{u_i}\]
\[0 - 100\]	\[12\]	\[50\]	\[ - 2\]	\[ - 24\]
\[100 - 200\]	\[16\]	\[150\]	\[ - 1\]	\[ - 16\]
\[200 - 300\]	\[6\]	\[250 \to A\]	\[0\]	\[0\]
\[300 - 400\]	\[7\]	\[350\]	\[1\]	\[7\]
\[400 - 500\]	\[9\]	\[450\]	\[2\]	\[18\]
	\[50\]			\[ - 15\]

As, we will use the formula of step deviation of mean that is \[\overline X = \left( {a + \dfrac{{{\sum _i}{f_i}{u_i}}}{{{\sum _i}{f_i}}} \times h} \right)\] to calculate average daily income (in rupees)
After substituting all the calculated values we get,
\[\overline X = \left( {250 + \dfrac{{\left( { - 15} \right)}}{{50}} \times 100} \right)\]
On simplifying we get,
 \[\overline X = \left( {250 - 30} \right)\]
\[\overline X = 220\]

Hence, the average daily income of the men is 220 rupees.

Note:
Note: To solve these types of questions you can use the alternative method by just calculating \[{f_i}{x_i}\] and using the formula of mean that is \[\overline X = \dfrac{{\sum {x_i}{f_i}}}{{\sum {f_i}}}\] as shown below.

Class Interval (rupees)	Number of men \[\left( {{f_i}} \right)\]	\[{x_i}\]	\[{f_i}{x_i}\]
\[0 - 100\]	\[12\]	\[50\]	\[600\]
\[100 - 200\]	\[16\]	\[150\]	\[2400\]
\[200 - 300\]	\[6\]	\[250\]	\[1500\]
\[300 - 400\]	\[7\]	\[350\]	\[2450\]
\[400 - 500\]	\[9\]	\[450\]	\[4050\]
	\[50\]		\[11,000\]

Now, we will calculate Mean \[\overline X = \dfrac{{\sum {x_i}{f_i}}}{{\sum {f_i}}}\] .
By substituting all the calculated values we get,
\[\overline X = \dfrac{{11000}}{{50}}\]
After dividing we get,
\[\overline X = 220\]
Hence, average income comes out to be 220 rupees.