The following frequency distribution gives the monthly consumption of electricity of 68 con sumers of a locality.
Question:

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

Solution:

(i) 

$\mathrm{n}=68$ gives $\frac{\mathbf{n}}{\mathbf{2}}=34$

So, we have the median class $(125-145)$

$\ell=125, \mathrm{n}=68, \mathrm{f}=20, \mathrm{cf}=22, \mathrm{~h}=20$

$\operatorname{Median}=\ell+\left\{\frac{\frac{\mathbf{n}}{\mathbf{z}}-\mathbf{c f}}{\mathbf{f}}\right\} \times \mathrm{h}$

$=125+\left\{\frac{\mathbf{3 4}-\mathbf{2 2}}{\mathbf{2 0}}\right\} \times 20=137$ units.

(ii) Modal class is $(125-145)$ having maximum frequency $\mathrm{f}_{\mathrm{m}}=20, \mathrm{f}_{1}=13, \mathrm{f}_{2}=14, \ell=$ 125 and $\mathrm{h}=20$

Mode $=\ell+\left\{\frac{\mathbf{f}_{\mathbf{m}}-\mathbf{f}}{\mathbf{2 f}_{\mathbf{m}}-\mathbf{f}-\mathbf{f}_{\mathbf{Z}}}\right\} \times \mathbf{h}$

$=125+\left\{\frac{\mathbf{2 0}-\mathbf{1 3}}{\mathbf{4 0}-\mathbf{1 3}-\mathbf{1 4}}\right\} \times 20=125+\frac{\mathbf{7} \times \mathbf{2 0}}{\mathbf{1 3}}$

$=125+\frac{\mathbf{1 4 0}}{\mathbf{1 3}}=125+10.76=135.76$ units

(iii) $n=68, a=135, h=20$ and $\Sigma f_{i} u_{i}=7$

$\mathrm{n}=68, \mathrm{a}=135, \mathrm{~h}=20$ and $\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{u}_{\mathrm{i}}=7$

By step-deviation method.

Mean $=\mathrm{a}+\mathrm{h} \times \frac{\mathbf{1}}{\mathbf{n}} \times \Sigma \mathrm{f}_{\mathrm{i}} \mathrm{u}_{\mathrm{i}}=135+20 \times \frac{\mathbf{1}}{\mathbf{6 8}} \times 7$

$=135+\frac{\mathbf{3 5}}{\mathbf{1 7}}=135+2.05=137.05$ units

 

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