18. The runs scored by two teams A and B on the first 60 balls in a cricket match are given below
Represent the data of both the teams on the same graph by frequency polygons.
Solution:
The given class intervals are not continuous. Therefore, we first modify the distribution as continuous.
Now, the required frequency polygons are as shown below:
19. A random survey of the number of children of various age groups playing in a park was found as follows:
Draw a histogram to represent the data above.
Solution:
Here, the class sizes are different. So, we calculate the adjusted frequencies corresponding to each rectangle i.e., length of the rectangle.
Adjusted frequency or length of the rectangle
\(= \left[\frac{\text{Minimum class size}}{\text{Class size}}\right] \times \text{Frequency}\)
Here, the minimum class size = 2 – 1 = 1
∴ We have the following table for adjusted frequencies or length of rectangles:
Now, the required histogram is shown below:
20. 100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows
(i) Draw a histogram to depict the given information.
(ii) Write the class interval in which the maximum number of surnames lie.
Solution:
(i) Since, class intervals of the given frequency distribution are unequal, and the minimum class size = 6 – 4 = 2.
Therefore, we have the following table for length of rectangles.
The required histogram is shown below:
(ii) The maximum frequency is 44, which is corresponding to the class interval 6 – 8.
∴ Maximum number of surnames lie in the class interval 6 – 8.
21. 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.
Solution:
To find the mean:
Here, n = 10
Thus, mean = 2.8
To find median:
Now arranging the given data in ascending order,
we have 0,1, 2, 3, 3, 3, 3, 4, 4, 5
∵ n = 10, an even number
Thus, median = 3
To find mode:
In the given data, the observation 3 occurs 4 times,
i.e., maximum number of times.
Thus, mode = 3
22. 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.
Solution:
To find the mean:
Here, n = 15
Thus, mean = 54.8
To find median:
Arranging the given data in ascending order,
we have
39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96,98
∵ n = 15, an odd number
Thus, median = 52
To find mode:
In the given data, the observation 52 occurs 3 times,
i.e., the maximum number of times.
Thus, mode = 52
23. 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
Solution:
Here, the given observations are in ascending order.
Since, n = 10 (an even number)
Since, median = 63 [Given]
∵ x + 1 = 63
⇒ x = 63 – 1 = 62
Thus, the required value of x is 62.
24. Find the mode of 14, 25,14, 28,18,17,18,14, 23, 22,14 and 18.
Solution:
Arranging the given data in ascending order, we have 14, 14, 14, 14, 17, 18, 18, 18, 22, 23 25, 28.
Since the observation 14 is occuring the maximum number of times (i.e. 4 times)
∴ Mode of the given data = 14
25. Find the mean salary of 60 workers of a factory from the following table
Solution:
Thus, the required mean salary = Rs. 5083.33
26. 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.
Solution:
(i) Mean height of the students of a class.
(ii) Median weight of a pen, a book, a rubber band, a match box and a chair.