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.

Number of mangoes	\[50 -52\]	\[53 - 55\]	\[56 - 58\]	\[59 - 61\]	\[62 - 64\]
Number 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 have to find the mean number of mangoes kept in a packing box and which method we choose for finding the mean.
The mean (or average) of observations, is the sum of the values of all the observations divided by the total number of observations.
If \[{x_1},{x_2},{x_3},......,{x_n}\] are observations with respective frequencies \[{f_1},{f_2},{f_3},........,{f_n}\] then this means observation \[{x_1}\] occurs \[{f_1}\] times, \[{x_2}\] occurs \[{f_2}\] times, and so on.
Now, the sum of the values of all the observations =\[{f_1}{x_1} + {f_2}{x_2} + ...... + {f_n}{x_n}\], and sum of the number of observations = \[{f_1} + {f_2} + {f_3} + ........ + {f_n}\]

Formula used: So, the mean x of the data is given by
\[x = \dfrac{{{f_1}{x_1} + {f_2}{x_2} + ...... + {f_n}{x_n}}}{{{f_1} + {f_2} + {f_3} + ........ + {f_n}}}\]
Or
\[x = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]

Complete step-by-step answer:
It is given that, 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.

Number of mangoes	Number of boxes
\[50 - 52\]	\[15\]
\[53 - 55\]	\[110\]
\[56 - 58\]	\[135\]
\[59 - 61\]	\[115\]
\[62 - 64\]	\[25\]

We need to find out the mean number of mangoes kept in a packing box.
The observation \[{x_i}\] is given by \[\dfrac{{{\text{(upper class limit + lower class limit)}}}}{{\text{2}}}\]

Number of mangoes	Number of boxes (\[{f_i}\])	Observation \[{x_i}\]	\[{f_i}{x_i}\]
\[50 - 52\]	\[15\]	\[\dfrac{{50 + 52}}{2} = \dfrac{{102}}{2} = 51\]	\[51 \times 15 = 765\]
\[53 - 55\]	\[110\]	\[\dfrac{{53 + 55}}{2} = \dfrac{{108}}{2} = 54\]	\[54 \times 110 = 5940\]
\[56 - 58\]	\[135\]	\[\dfrac{{56 + 58}}{2} = \dfrac{{114}}{2} = 57\]	\[57 \times 135 = 7695\]
\[59 - 61\]	\[115\]	\[\dfrac{{59 + 61}}{2} = \dfrac{{120}}{2} = 60\]	\[60 \times 115 = 6900\]
\[62 - 64\]	\[25\]	\[\dfrac{{62 + 64}}{2} = \dfrac{{126}}{2} = 63\]	\[63 \times 25 = 1575\]

\[\sum\limits_{i = 1}^n {{f_i}} = 15 + 110 + 135 + 115 + 25 = 400\]
\[\sum\limits_{i = 1}^n {{f_i}{x_i}} = 765 + 5940 + 7695 + 6900 + 1575 = 22875\]
Mean number of mangoes kept in a packing box = \[\dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }} = \dfrac{{22875}}{{400}} = 57.1875 = 57.19\].

Note: Mean
There are several kinds of means in mathematics, especially in statistics. For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.
\[{\text{m = }}\dfrac{{{\text{Sum of the terms}}}}{{{\text{Number of terms}}}}\].