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.

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

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Solution

We may find the class marks by using the relation:

$C l a s s m a r k = \frac{u p p e r c l a s s l i m i t + l o w e r c l a s s l i m i t}{2}$

Taking 135 as assumed mean $a$ , we may find $d_{i}, u_{i}, f_{i} u_{i}$ , according to step deviation method as following:

$\begin{matrix} M o n t h l y c o n s u m p t i o n (i n u n i t s) & N u m b e r o f c o n s u m e r s (f_{i}) & x_{i} c l a s s m a r k & d_{i} = x_{i} - 135 & u_{i} = \frac{d_{i}}{20} & f_{i} u_{i} 65 - 85 & 4 & 75 & - 60 & - 3 & - 12 85 - 105 & 5 & 95 & - 40 & - 2 & - 10 105 - 125 & 13 & 115 & - 20 & - 1 & - 13 125 - 145 & 20 & 135 & 0 & 0 & 0 145 - 165 & 14 & 155 & 20 & 1 & 14 165 - 185 & 8 & 175 & 40 & 2 & 16 185 - 205 & 4 & 195 & 60 & 3 & 12 T o t a l & 68 & 7 \end{matrix}$

From the table, we may observe that:

$\sum f_{i} u_{i} = 7, \sum f_{i} = 68$ , class size $h$ = 20
$M e a n ¯ x = a + (\frac{\sum f_{i} u_{i}}{\sum f_{i}}) \times h = 135 + \frac{7}{68} \times 20 = 135 + \frac{140}{68} = 137.06$

Now, from table, it is clear that maximum class frequency is 20 belonging to class interval 125 - 145.

Modal class = 125 - 145

Lower limit $l$ of modal class = 125

Class size $h$ = 20

Frequency $(f_{1})$ of modal class = 20

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

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

$M o d e = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h = 125 + [\frac{20 - 13}{2 (20) - 13 - 14}] \times 20 = 125 + \frac{7}{13} \times 20 = 125 + \frac{140}{13} = 135.77$

We know that:

3 median = mode + 2 mean

= 135.77 + 2 (137.06)

= 135.77 + 274.12 = 409.89

Median = 409.9 / 3 = 136.63

So, median, mode, mean of given data is 136.63, 135.77, 137.06 respectively.

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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 electricity of 68 consumers of a locality.
Find the median of the data.

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

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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 consumers65−85485−1055105−12513125−14520145−16514165−1858185−2054