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 $65-85$ $4$ $85-105$ $5$ $105-125$ $13$ $125-145$ $20$ $145-165$ $14$ $165-185$ $8$ $185-205$ $4$
| Monthly consumption (in units) | Number of consumers |
| $65-85$ | $4$ |
| $85-105$ | $5$ |
| $105-125$ | $13$ |
| $125-145$ | $20$ |
| $145-165$ | $14$ |
| $165-185$ | $8$ |
| $185-205$ | $4$ |
(N/A) $1$. Mean: Using the step deviation method,$\bar{x} = a + h \left( \frac{\sum f_i u_i}{\sum f_i} \right)$. With $a = 135, h = 20, \sum f_i u_i = 7, \sum f_i = 68$,we get $\bar{x} = 135 + 20 \left( \frac{7}{68} \right) = 135 + 2.06 = 137.06$.
$2$. Mode: Modal class is $125-145$ $(f_1=20, f_0=13, f_2=14)$. Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h = 125 + \left( \frac{20 - 13}{40 - 13 - 14} \right) \times 20 = 125 + \left( \frac{7}{13} \right) \times 20 = 125 + 10.77 = 135.77$.
$3$. Median: $n=68, n/2=34$. Cumulative frequency table shows median class is $125-145$. Median $= l + \left( \frac{n/2 - cf}{f} \right) \times h = 125 + \left( \frac{34 - 22}{20} \right) \times 20 = 125 + 12 = 137$. Comparing them,Mean $\approx$ Median $\approx$ Mode.
$2$. Mode: Modal class is $125-145$ $(f_1=20, f_0=13, f_2=14)$. Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h = 125 + \left( \frac{20 - 13}{40 - 13 - 14} \right) \times 20 = 125 + \left( \frac{7}{13} \right) \times 20 = 125 + 10.77 = 135.77$.
$3$. Median: $n=68, n/2=34$. Cumulative frequency table shows median class is $125-145$. Median $= l + \left( \frac{n/2 - cf}{f} \right) \times h = 125 + \left( \frac{34 - 22}{20} \right) \times 20 = 125 + 12 = 137$. Comparing them,Mean $\approx$ Median $\approx$ Mode.
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