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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Similar Questions
The lengths of $40$ leaves of a plant are measured correct to the nearest millimetre,and the data obtained is represented in the following table:
Length (in mm) Number of leaves $118-126$ $3$ $127-135$ $5$ $136-144$ $9$ $145-153$ $12$ $154-162$ $5$ $163-171$ $4$ $172-180$ $2$
Find the median length of the leaves.
| Length (in mm) | Number of leaves |
| $118-126$ | $3$ |
| $127-135$ | $5$ |
| $136-144$ | $9$ |
| $145-153$ | $12$ |
| $154-162$ | $5$ |
| $163-171$ | $4$ |
| $172-180$ | $2$ |
Find the median length of the leaves.
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Find the mode of the data.
| Runs scored | Number of batsmen |
| $3000-4000$ | $4$ |
| $4000-5000$ | $18$ |
| $5000-6000$ | $9$ |
| $6000-7000$ | $7$ |
| $7000-8000$ | $6$ |
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| $10000-11000$ | $1$ |
Find the mode of the data.
Medium
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| Number of students | $2$ | $3$ | $7$ | $6$ | $6$ | $6$ |
Medium
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Life time (in hours) Number of lamps $1500-2000$ $14$ $2000-2500$ $56$ $2500-3000$ $60$ $3000-3500$ $86$ $3500-4000$ $74$ $4000-4500$ $62$ $4500-5000$ $48$
Find the median lifetime of a lamp.
| Life time (in hours) | Number of lamps |
| $1500-2000$ | $14$ |
| $2000-2500$ | $56$ |
| $2500-3000$ | $60$ |
| $3000-3500$ | $86$ |
| $3500-4000$ | $74$ |
| $4000-4500$ | $62$ |
| $4500-5000$ | $48$ |
Find the median lifetime of a lamp.
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| Number of students per teacher | Number of states/$U$.$T$. |
| $15-20$ | $3$ |
| $20-25$ | $8$ |
| $25-30$ | $9$ |
| $30-35$ | $10$ |
| $35-40$ | $3$ |
| $40-45$ | $0$ |
| $45-50$ | $0$ |
| $50-55$ | $2$ |
Medium
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