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	$65 - 85$	$85 - 105$	$105 - 125$	$125 - 145$	$145 - 165$	$165 - 185$	$185 - 205$
Number of consumers	$4$	$5$	$13$	$20$	$14$	$8$	$4$

Accepted Answer

Let's draw a table showing midpoints, frequencies and cumulative frequency.

Monthly consumption Midpoint $(x_{i})$ $d_{i} = x_{i} - 135$ No. of
consumers (frequency) $(f_{i})$ $f_{i} d_{i}$ Cumulative frequency $185 - 205$

$\begin{array}{l} 65 - 85 \end{array}$ $75$ $75 - 135 = - 60$ $4$ $4 \times - 60 = - 240$ $4$

$\begin{array}{l} 85 - 105 \end{array}$ $95$ $95 - 135 = - 40$ $5$ $5 \times - 40 = - 200$ $4 + 5 = 9$

$\begin{array}{l} 105 - 125 \end{array}$ $115$ $115 - 135 = - 20$ $13$ $13 \times - 20 = - 260$ $9 + 13 = 22$

$125 - 145$ $135$ $135 - 135 = 0$ $20$ $20 \times 0 = 0$ $22 + 20 = 42$

$145 - 165$ $155$ $155 - 135 = 20$ $14$ $14 \times 20 = 280$ $42 + 14 = 56$

$165 - 185$ $175$ $175 - 135 = 40$ $8$ $8 \times 40 = 320$ $56 + 8 = 64$

$185 - 205$ $195$ $195 - 135 = 60$ $4$ $4 \times 60 = 240$ $64 + 4 = 68$

TOTAL $Σ f_{i} = 68$ $Σ f_{i} d_{i} = 140$ $N = 68$

For median:

We have, total frequency, $N = 68$

$\frac{N}{2} = \frac{68}{2} = 34$

Observe, $c_{f} = 42$ is just greater than $34$

Thus, median class $= 125 - 145$

Median is given by

Median $= L + [\frac{\frac{N}{2} - c_{f}}{f}] \times h$

Where,

$L =$ Lower class limit of median class $= 125$

$\frac{N}{2} = 34$

$c_{f} =$ Cumulative frequency of the class preceding median class $= 22$

$f =$ Frequency of the median class $= 20$

$h =$ Class interval of the median class $= 20$

Substituting these values in the formula of median, we get,

Median $= 125 + [\frac{34 - 22}{20}] \times 20$

$\Rightarrow Median = 125 + 12$

$\Rightarrow Median = 137$

For mean:

Mean is given by

$Mean = A + \frac{\sum f_{i} d_{i}}{\sum f_{i}}$

$\Rightarrow Mean = 135 + \frac{140}{68}$

$\Rightarrow Mean = 135 + \frac{140}{68}$

$\Rightarrow Mean = 135 + 2.06$

$\Rightarrow Mean = 137.06$

For mode:

Here, the highest frequency is $20$

So, the modal class $= 125 - 145$

Mode is given by

Mode $= L + \frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}} \times h$

Where,

$L =$ Lower class limit of the modal class $= 125$

$h =$ Class interval of the modal class $= 20$

$f_{1} =$ Frequency of the modal class $= 20$

$f_{0} =$ Frequency of the class preceding the modal class $= 13$

$f_{2} =$ Frequency of the class succeeding the modal class $= 14$

Substituting values in the formula of mode,

$Mode = 125 + \frac{20 - 13}{2 \times 20 - 13 - 14} \times 20$

$\Rightarrow Mode = 125 + \frac{7}{13} \times 20$

$\Rightarrow Mode = 125 + 10.76$

$\Rightarrow Mode = 135.76$

Thus, median is $137,$ mean is $137.06$ and mode is $135.76$

Median and mean, both gives us the average value of a data, with just the difference that mean can give us quantitative measure only but median can give us both quantitative as well as qualitative measure of a data. And that is why, mean and median have come out to be very close in this question.

Mode gives the value that appears most in a given data. Thus, the maximum monthly consumption of electricity is $135.76$

Class interval	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$
Frequency	$5$	$x$	$20$	$15$	$4$	$4 \times 60 = 240$

Length (in $mm$ )	$118 - 126$	$\begin{array}{l} 127 - 135 \end{array}$	$\begin{array}{l} 136 - 144 \end{array}$	$\begin{array}{l} 145 - 153 \end{array}$	$\begin{array}{l} 154 - 162 \end{array}$	$163 - 171$	$172 - 180$
Number of leaves	$3$	$5$	$9$	$12$	$5$	$4$	$2$

Lifetime (in hours)	$\begin{array}{l} 1500 - \\ 2000 \end{array}$	$\begin{array}{l} 2000 - \\ 2500 \end{array}$	$\begin{array}{l} 2500 - \\ 3000 \end{array}$	$\begin{array}{l} 3000 - \\ 3500 \end{array}$	$\begin{array}{l} 3500 - \\ 4000 \end{array}$	$\begin{array}{l} 4000 - \\ 4500 \end{array}$	$\begin{array}{l} 4500 - \\ 5000 \end{array}$
Number of lamps	$f_{2} =$	$56$	$60$	$86$	$74$	$62$	$48$

Number of letters	$1 - 4$	$\begin{matrix} 4 - 7 \end{matrix}$	$137.06$	$\begin{matrix} 10 - 13 \end{matrix}$	$13 - 16$	$16 - 19$
Number of surnames	$6$	$30$	$40$	$16$	$4$	$4$

Weight (in $kg$ )	$\begin{array}{l} 40 - 45 \end{array}$	$\begin{array}{l} 45 - 50 \end{array}$	$\begin{array}{l} 50 - 55 \end{array}$	$\begin{array}{l} 55 - 60 \end{array}$	$\begin{array}{l} 60 - 65 \end{array}$	$65 - 70$	$70 - 75$
Number of students	$2$	$3$	$8$	$6$	$6$	$3$	$2$

Telangana Board Solutions for Chapter: Statistics, Exercise 5: Exercise

Telangana Board Mathematics Solutions for Exercise - Telangana Board Solutions for Chapter: Statistics, Exercise 5: Exercise

Questions from Telangana Board Solutions for Chapter: Statistics, Exercise 5: Exercise with Hints & Solutions

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