Question

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying numbers of mangoes. The following was the distribution of mangoes according to the number of boxes.

No. of mangoes	50 – 52	53 – 55	56 – 58	59 – 61	62 – 64
No. of boxes	15	110	135	115	25

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Answer 1

Hint: We will first recreate the table using the data given above in the table with adding more columns or rows of data with multiplication of both f and u, one column of u. Then, we will use the formula for the assumed mean method and thus get the answer.

Complete step-by-step answer:
Let us create the new table by finding out the mid – points of the intervals, assuming a function and then their product.
Let us first find the mid – points of all the intervals.
50 – 52: the mid – point will be \[\dfrac{{50 + 52}}{2} = 51\]
53 – 55: the mid – point will be \[\dfrac{{53 + 55}}{2} = 54\]
56 – 58: the mid – point will be \[\dfrac{{56 + 58}}{2} = 57\]
59 – 61: the mid – point will be \[\dfrac{{59 + 61}}{2} = 60\]
62 – 64: the mid – point will be \[\dfrac{{62 + 64}}{2} = 63\]
Now, let us assume that ${u_i} = \dfrac{{{x_i} - 57}}{3}$
Now, let us create a new table:

Class - Interval	Mid – value $({x_i})$	Frequency $({f_i})$	${u_i} = \dfrac{{{x_i} - 57}}{3}$	${f_i}{u_i}$
50 – 52	51	15	-2	-30
53 – 55	54	110	-1	-110
56 – 58	57	135	0	0
59 – 61	60	115	1	115
62 – 64	63	25	2	50
	Total	400	0	25

Now, we also have formula of mean in assumed mean method which is given by:
$\bar x = A + h\left( {\dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}} \right)$, where A is the assumed value which is 57 here and h is the class interval which is 2 here.
Now, putting in the values as per our question, we will get:-
$ \Rightarrow \bar x = 57 + 2\left( {\dfrac{{25}}{{400}}} \right)$
Simplifying the RHS of the above expression will lead us to:-
$ \Rightarrow \bar x = 57 + \dfrac{1}{8}$
Simplifying the RHS further, we will get as follows:-
$ \Rightarrow \bar x = 57 + 0.125 = 57.125$
Hence, the mean number of mangoes kept in a packing box is 57.125

$\therefore $ The required answer is 57.125

Note: The students must know that mean refers to the average. Like if we take the example of the given question, if we pick out a random box, the average number of mangoes in it will be 57.125
We can find mean by normal methods as well in which we have to calculate the sum of product of the mid - value and the frequency and divide it by total number (frequency) but that will involve a lot of difficult calculation which eventually may lead to mistakes, therefore, we here used the Assumed mean method to calculate the same.