Question

Frequency distribution of daily commission received by $100$ salesmen is given.

Daily commission (in Rs.)	No. of salesmen
$100 - 120$	$20$
$120 - 140$	$45$
$140 - 160$	$22$
$160 - 180$	$09$
$180 - 200$	$04$

Find mean daily commission received by salesmen, by assumed mean method.

Answer 1

Hint: In this question, we are given frequency distribution of daily commission of $100$ salesmen. We have been asked to find mean daily commission by assuming a mean method. First step will be to find the mid-points. Then we assume one of the mid-points as mean. Using this assumed mean, we will find the deviations. Then we will multiply these deviations with the frequency. It will use ${f_i}{d_i}$. Now we have all the required values. Put them in the formula to find the mean.

Formula used: $\bar X = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}$

Complete step-by-step solution:
We are given a frequency distribution of daily commission of $100$ salesmen. We have been asked to find mean daily commission by assuming a mean method.
In this method, we have to first find the mid-points of the given class intervals. Then we assume one mid-point as mean. Next, we find the deviations from the assumed mean. Deviations can be calculated by - ${d_i} = {x_i} - A$. After finding the deviations, multiply the deviations with frequency. It will give us ${f_i}{d_i}$. Sum up all the ${f_i}{d_i}$.
See the below table:

Daily commission (in Rs.)	No. of salesmen$({f_i})$	Mid-points$({x_i})$	${d_i} = {x_i} - A$$A = 150$	${f_i}{d_i}$
$100 - 120$	$20$	$110$	$ - 40$	$ - 800$
$120 - 140$	$45$	$130$	$ - 20$	$ - 900$
$140 - 160$	$22$	$150$$ = A$	$0$	$0$
$160 - 180$	$09$	$170$	$20$	$180$
$180 - 200$	$04$	$190$	$40$	$160$
	$\sum {{f_i} = 100} $			$\sum {{f_i}{d_i} = - 1360} $

Now, we have all the required values. Let us put them in the formula –
$ \Rightarrow \bar X = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}$
Putting the values in the formula –
$ \Rightarrow \bar X = 150 + \left( {\dfrac{{ - 1360}}{{100}}} \right)$
On simplifying we get,
$ \Rightarrow \bar X = 150 - 13.60$
$ \Rightarrow \bar X = 136.4$

The mean daily commission is 136.4.

Note: 1) Mean can be founded by many ways- direct method, shortcut method. But since it is mentioned that we have to use the assumed mean method, we will use it only otherwise you will not be awarded any marks even if the entire solution and steps are correct.
2) Take the middle most number as ‘A’ in case of odd numbers and in case of even numbers, take any one out of the two middle numbers. Taking the middle number helps in making the calculations easier.