Question

On the occasion of New year’s day a sweet stall prepared sweet packets. Number of sweet packets and the cost of each packet are given as follows.

Cost of packet (in Rs)	Rs.25	Rs.50	Rs.75	Rs.100	Rs.125	Rs.150
No. of packets	20	26	32	29	22	11

Find the mean, median and mode of the data.

Answer 1

Hint: To solve this problem we have to know about the mean, median and mode. Mean, median and mode are statistical values in statistics and probability theory. Mean is the average of the given set of data which can be expressed as the ratio of the sum of all the observations to the total number of observations. Median is the middle value of the observation in a given set of data. Mode is the value of the variable of which the frequency is maximum.

Complete step-by-step solution:
We can find the mean and the mode from the given data, but to solve the median we need to find the cumulative frequency of the given distributed data.

Cost of packet (in Rs) $({x_i})$	Number of packets $({f_i})$	Cumulative frequency $(cf)$
25	20	20
50	26	46
75	32	78
100	29	107
125	22	129
150	11	140

Mean is mathematically expressed by the following expression below:
$ \Rightarrow \dfrac{{\sum\limits_i {{f_i}{x_i}} }}{{\sum\limits_i {{f_i}} }}$
Where $\sum\limits_i {{f_i}} = 140$, is the total no. of observations.
$\sum\limits_i {{f_i}{x_i}} $ is the sum of total no. of observations.
Hence the mean is given by:
$ \Rightarrow \dfrac{{20(25) + 26(50) + 32(75) + 29(100) + 22(125) + 11(150)}}{{140}}$
$ \Rightarrow \dfrac{{11500}}{{140}} = 82.1428$
$\therefore $Mean is Rs.82.1428.
As mode is the value which occurs for a maximum number of times, or mostly recurring data value in the distribution.
Here out of all the Rs.75 packets have occurred for 32 times, out of all packets.
Hence the mode is 75.
$\therefore $Mode is Rs.75.
Now we have to find the median of the data.
As there are of total 140 packets, which is even, so the middle ones would be 70th packet and 71st packet.
Thus the median would be the cost of the packet corresponding to the cumulative frequency of 70 and 71, which is Rs.75
$\therefore $Median is Rs.75.

Mean is Rs.82.1428, Median is Rs.75, Mode is Rs.75.

Note: While finding the median of any given data it is always important to observe the total number of observations correctly, else everything could go wrong. As already discussed, if the total number of observations are odd, then the mean is the middle value of the observation else the median would be the average of the two middle values in the data. While mode would be the mostly recurring element.