Question

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Monthly consumption (in units)	Number of consumers
$65 - 85$	$4$
$85 - 105$	$5$
$105 - 125$	$13$
$125 - 145$	$20$
$145 - 165$	$14$
$165 - 185$	$8$
$185 - 205$	$4$

Answer 1

According to given in the question we have to determine the mean for the given frequency distribution. So, first of all to find the mean we have to understand mean as explained below:
Mean: The mean is basically a way to find the average of the given data but before that we have to arrange the all data or given number into ascending order means we have to arrange all the given data from the smaller number to the largest number and after that we have to add up all the given data or number and same as we have to determine the total number of given data or terms.

Formula used:
$\overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}........................(a)$
Obtained by But first of all we have to determine the class which can be obtained by dividing the sum of upper limit and lower limit by 2.
Now, we have to calculate ${f_i}{x_i}$ with the help of the table obtained, and after applying the formula (a) as given above we can obtain the required mean.
Now, we have to find the median but before that we have to determine the value of N which is equal to the sum of all the frequencies $\sum {{f_i}}$and also the class interval for the obtain mean so that we can determine upper and lower limits. Formula to find median is given as below:
Median$= l + \dfrac{{\dfrac{N}{2} - c.f}}{f} \times h...................(b)$
Where, c.f is the cumulative frequency and h is the height of the interval.
Now, we have to obtain the height for the obtained class interval, frequency for the obtained interval and the cumulative frequency which is explained below:
Cumulative frequency: The cumulative frequency can be calculated or obtained by adding each given frequency from a frequency distribution table to the sum of its predecessors and its last value will be always equal to the total for all observations, since all the frequencies will already have been added to the previous total.
Now, we have to obtain the mode but first of all we have to understand about mode which is explained below:
Mode: The mode of a given data set is the number or term that occurs most frequently in that set and to find the mode we have to put the given numbers or terms in ascending order or in the order from least to the greatest number or term after that we have to count that which number is occurred the most hence, that the number that occurs the most is the mode.
Mode$= l + \dfrac{{f_1^{} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}....................(c)$

Complete step-by-step solution:
Step 1: First of all we have to determine the class which can ${x_i}$ be obtained by the sum of upper limit and lower limit for that class interval divided by 2. Hence,

Monthly consumption (in units)	Number of consumers ${f_i}$	${x_i}$
$65 - 85$	$4$	$\dfrac{{65 + 85}}{2} = 75$
$85 - 105$	$5$	$\dfrac{{85 + 105}}{2} = 95$
$105 - 125$	$13$	$\dfrac{{105 + 125}}{2} = 115$
$125 - 145$	$20$	$\dfrac{{125 + 145}}{2} = 135$
$145 - 165$	$14$	$\dfrac{{145 + 165}}{2} = 155$
$165 - 185$	$8$	$\dfrac{{165 + 185}}{2} = 175$
$185 - 205$	$4$	$\dfrac{{185 + 205}}{2} = 195$

Step 2:
Now, we have to find the value of ${f_i}{x_i}$ which can be obtained by the multiplication of ${f_i}$ and $({x_i})$ as mentioned in the solution hint. Hence,

Monthly consumption (in units)	Number of consumers ${f_i}$	${x_i}$	${f_i}{x_i}$
$65 - 85$	$4$	$\dfrac{{65 + 85}}{2} = 75$	$4 \times 75 = 300$
$85 - 105$	$5$	$\dfrac{{85 + 105}}{2} = 95$	$5 \times 95 = 475$
$105 - 125$	$13$	$\dfrac{{105 + 125}}{2} = 115$	$13 \times 115 = 1495$
$125 - 145$	$20$	$\dfrac{{125 + 145}}{2} = 135$	$20 \times 135 = 2700$
$145 - 165$	$14$	$\dfrac{{145 + 165}}{2} = 155$	$14 \times 155 = 2170$
$165 - 185$	$8$	$\dfrac{{165 + 185}}{2} = 175$	$8 \times 175 = 1400$
$185 - 205$	$4$	$\dfrac{{185 + 205}}{2} = 195$	$4 \times 195 = 780$

Step 3: Now, we have to find $\sum {{f_i}{x_i}}$ which can be obtained by adding all the obtained ${f_i}{x_i}$
Hence,
$\Rightarrow \sum {{f_i}{x_i}} = 300 + 475 + 1495 + 2700 + 2170 + 1400 + 780 \\\ \Rightarrow \sum {{f_i}{x_i}} = 9320$
Now, same as we have to determine the value of $\sum {{f_i}}$ hence,
$\Rightarrow \sum {{f_i}} = 4 + 5 + 13 + 20 + 14 + 8 + 4 \\\ \Rightarrow \sum {{f_i}} = 68$
Step 4: On substituting all the values of $\sum {{f_i}{x_i}}$ and $\sum {{f_i}}$ in the formula (a) as mentioned in the solution hint. Hence,
$\Rightarrow (\overline x ) = \dfrac{{9820}}{{68}} \\\ \Rightarrow (\overline x ) = 137.05$
Step 5: Now, as we know that the mean is 137.05 hence the median class interval for the mean is 125-145. So the lower limit l = 125 and cumulative frequency is 22 and frequency f = 20 and height of the obtained class interval is 20 hence, on substituting all the values in the formula (b) as mentioned in the solution hint.
$= 125 + \dfrac{{\dfrac{{68}}{2} - 22}}{{20}} \times 20 \\\ = 125 + \dfrac{{38 - 22}}{{20}} \times 20 \\\ = 125 + 12 \\\ = 137$
Median = 137
Step 6: Now, we have to obtain the mode but before that we have to obtain the modal class from the obtained table which is 125-145. Hence, lower limit l = 125, ${f_0} = 13,{f_1} = 20,{f_2} = 14$ and h = 20 so, on substituting all the values in formula (c) as mentioned in the solution hint.
Mode:
$= 125 + \dfrac{{20 - 13}}{{2 \times 20 - 13 - 14}} \times 20 \\\ = 125 + 10.77 \\\ = 135.77$

Hence, with the help of formula (a) and formula (b) we have obtain the mean = 137.05, median = 137, and mode = 135.77

Note: Mean can be determined by arranging the given data in ascending order and them we have to divide the sum of all the given data with the total number of given data and mean of the absolute values of the numerical differences between the numbers of a set such as, a static data and their mean and median.
The cumulative frequency of a set of a data or class interval of a frequency table is the sum of frequencies of the data up to a required level and it can also be used to determine the number of items that have values below a particular level.
To find the mode, modal value is the best way to arrange the given number into ascending order means from the smallest number to the largest number then we have to check which number is occurring most of the time hence, that number or interval is the mode for the given data.