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Exercise 14.4

1. The following number of goals were scored by a team in a series of 10 matches:
2, 3, 4, 5, 0, 1, 3, 3, 4, 3.

Find the mean, median and mode of these scores.

Sol. Arranging the data in ascending order, we get

0 , 1 , 2, 3, 3, 3, 3, 4, 4, 5

(i) $Mean = {{0 + 1 + 2 + 3 + 3 + 3 + 3 + 4 + 4 + 5} \over {10}}$

$= {{28} \over {10}} = 2.8$

(ii) For median: n = 10.
Median is the mean of the values of observations at 5th and the 6th places.

∴   Median = ${{3 + 3} \over 2} = 3$

(iii) Mode: 3 occurs maximum number of times. Hence, mode is 3.

2. In a mathematics test given to 15 students, the following marks (out of 100) are recorded:
41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60.
Find the mean, median and mode of this data.

Sol.  Arranging the data in ascending order, we get
39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96, 98

(i) Mean = ${{39 + 40 + 40 + 41 + 42 + 46 + 48 + 52 + 52 + 52 + 60 + 62 + 96 + 98} \over {15}}$

$= {{822} \over {15}} = 54.8$

(ii) For median: n = 15, median is the value of the observation at the ${{15 + 1} \over 2}$th place, i.e., 8th place.

∴   Median = 52.

(iii) Mode: ’52’ occurs maximum number of times.

∴   Mode = 52.

3. The following observations have been arranged in ascending order If the median of the data is 63, find the value of x.
29,32, 48, 50, x, x + 2, 72, 78, 84, 95.

Sol. Given observations: 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95.

Here n = 10, Median is mean of the values of the observations at ${{10} \over 2}$th, and  $\left( {{{10} \over 2} + 1} \right)$th i.e., 5th and 6th places.

∴  Median $= {{x + (x + 2)} \over 2} \Rightarrow {{2x + 2} \over 2} = 63$

$\Rightarrow 2x = 126 - 2 = 124 \Rightarrow x = 62$

4. Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.

Sol. Arranging the data in ascending order, we have
14,     14,    14,    14,     17,     18,     18,     18,     22,     23,     25,    28
We notice 14 occurs maximum number of times.
Hence, the mode is 14.

5. Find the mean salary of 60 workers of a factory from the following table:

 Salary (in Rs.) Number of workers 3000 16 4000 12 5000 10 6000 8 70006 6 8000 4 9000 3 10000 1 Total 60

Sol.

 Salary (in Rs.)(x) Number of workers (f) xf 3000 16 48000 4000 2 48000 5000 10 50000 6000 8 48000 7000 6 42000 8000 4 32000 9000 3 27000 10000 1 10000 ∑f = 60 ∑xf = 305000

$Mean = {{\sum {xf} } \over {\sum f }} = {{305000} \over {60}} = 5083.33$

Hence, the mean salary is Rs. 5083.33

6. Give one example of a situation in which
(i) the mean is an appropriate measure of central tendency.
(ii) the mean is not an appropriate measure of central tendency but the median is an appropriate measure of central tendency.