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.
$\begin{matrix} Monthly consumption & Number of consumers (in units) 65 - 85 & 4 85 - 105 & 5 105 - 125 & 13 125 - 145 & 20 145 - 165 & 14 165 - 185 & 8 185 - 205 & 4 \end{matrix}$

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Solution

$\begin{matrix} Class Interval & Frequency(f_{i}) & Cumulative frequency (c f) 65 - 85 & 4 & 4 85 - 105 & 5 & 9 105 - 125 & 13 & 22 125 - 145 & 20 & 42 145 - 165 & 14 & 56 165 - 185 & 8 & 64 185 - 205 & 4 & 68 \end{matrix}$

$∙$ Here, $n = 68$ and $\frac{n}{2} = 34$ . Therefore, median class is $125 - 145$

$∙$ Here, $l = 125, f = 20, c f = 22$ and $h = 20$

$M e d i a n = l + ⎛ ⎜ ⎜ ⎝ \frac{\frac{n}{2} - c f}{f} ⎞ ⎟ ⎟ ⎠ \times h$

$= 125 + (\frac{34 - 22}{20}) \times 20 = 125 + 12 = 137$

$∙$ Therefore, the median of the given data is $137$ units.

$\begin{matrix} Class Interval & Frequency (f_{i}) & Class mark (x_{i}) & u_{i} = (\frac{x_{i} - 135}{h}) & (f_{i} u_{i}) 65 - 85 & 4 & 75 & - 3 & - 12 85 - 105 & 5 & 95 & - 2 & - 10 105 - 125 & 13 & 115 & - 1 & - 13 125 - 145 & 20 & 135 & 0 & 0 145 - 165 & 14 & 155 & 1 & 14 165 - 185 & 8 & 175 & 2 & 16 185 - 205 & 4 & 195 & 3 & 12 \sum f_{i} = 68 & \sum f_{i} u_{i} = 7 \end{matrix}$

$∙$ Here, $a = 135$ and $h = 20$

$M e a n = a + h ⎛ ⎝ \frac{\sum f_{i} u_{i}}{\sum f_{i}} ⎞ ⎠$

$= 135 + 20 (\frac{7}{68}) = 135 + 2.05 = 137.05$

$∙$ Therefore, the mean of the monthly consumption of electricity is $137.05$ units.

$∙$ Highest frequency is $20$ , therefore the modal class is $125 - 145$

$∙$ Here, $l = 125, f_{1} = 20, f_{0} = 13, f_{2} = 14$ and $h = 20$

$M o d e = l + [\frac{(f_{1} - f_{0})}{(2 f_{1} - f_{0} - f_{2})}] \times h$

$= 125 + [\frac{20 - 13}{40 - 13 - 14}] \times 20$

$= 125 + (\frac{7}{13}) \times 20 = 125 + 10.77 = 135.77$

$∙$ Therefore, the mode of the monthly consumption of electricity is $135.77$ units.

$∙$ The median of the monthly consumption of electricity is $137$ units

$∙$ The mean of the monthly consumption of electricity is $137.05$ units

$∙$ The mode of the monthly consumption of electricity is $135.77$ units.

The three measures are approximately the same in this case.

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Similar questions

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.

$\begin{matrix} M o n t h l y c o n s u m p t i o n (i n u n i t s) & N u m b e r o f c o n s u m e r s 65 - 85 & 4 85 - 105 & 5 105 - 125 & 13 125 - 145 & 20 145 - 165 & 14 165 - 185 & 8 185 - 205 & 4 \end{matrix}$

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

Monthly consumption(in units)	No. of customers
$65 - 85$	$4$
$85 - 105$	$5$
$105 - 125$	$13$
$125 - 145$	$20$
$145 - 165$	$14$
$165 - 185$	$8$
$185 - 205$	$4$

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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 consumptionNumber of consumers(in units)65−85485−1055105−12513125−14520145−16514165−1858185−2054