RS AGGARWAL CLASS 9 CHAPTER 16 PRESENTAION OF DATA IN TABULAR FORM EXERCISE 16

  EXERCISE 16

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Question 1:

Define statistics as a subject.

Answer 1:

Statistics is the science which deals with the collection, presentation, analysis and interpretation of numerical data.

Question 2:

Define some fundamental characteristics of statistics.

Answer 2:

The fundamental characteristics of data (statistics) are as follows:
(i) Numerical facts alone constitute data.
(ii) Qualitative characteristics like intelligence and poverty, which cannot be measured numerically, do not form data.
(iii) Data are aggregate of facts. A single observation does not form data.
(iv) Data collected for a definite purpose may not be suited for another purpose.
(v) Data in different experiments are comparable.

Question 3:

What are primary data and secondary data? Which of the two is more reliable and why?

Answer 3:

Primary data: The data collected by the investigator himself with a definite plan in mind are known as primary data.
Secondary data: The data collected by someone other than the investigator are known as secondary data.

Primary data are highly reliable and relevant because they are collected by the investigator himself with a definite plan in mind,  whereas secondary data are collected with a purpose different from that of the investigator and may not be fully relevant to the investigation.

Question 4:

Explain the meaning of each of the following terms:
(i) Variate
(ii) Class interval
(iii) Class size
(iv) Class mark
(v) Class limit
(vi) True class limits
(vii) Frequency of a class
(viii) Cumulative frequency of a class

Answer 4:

(i) Variate : Any character which is capable of taking several different values is called a variant or a variable.
(ii) Class interval : Each group into which the raw data is condensed is called class interval .
(iii) Class size: The difference between the true upper limit and the true lower limit of a class is called its class size.
(iv) Class mark of a class: The class mark is given by $\left(\frac{\mathrm{Upper}\mathrm{limit}+\mathrm{Lower}\mathrm{limit}}{2}\right)$$\left(\frac{\mathrm{Upper} \mathrm{limit}+\mathrm{Lower} \mathrm{limit}}{2}\right)$.
(v) Class limit: Each class is bounded by two figures, which are called class limits.
(vi) True class limits: In the exclusive form, the upper and lower limits of a class are respectively known as true upper limit and true lower limit.
In the inclusive form of frequency distribution, the true lower limit of a class is obtained by subtracting 0.5 from the lower limit and the true upper limit of the class is obtained by adding 0.5 to the upper limit.
(vii) Frequency of a class: Frequency of a class is the number of times an observation occurs in that class.
(viii) Cumulative frequency of a class: Cummulative frequency of a class is the sum total of all the frequencies up to and including that class.

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Question 5:

The blood groups of 30 students of a class are recorded as under:
A, B, O, O, AB, O, A, O, A, B, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O.
(i) Represent this data in the form of a frequency distribution table.
(ii) Find out which is the most common and which is the rarest blood group among these students.

Answer 5:

(i) 

Blood group	tally marks	Number of students
A		9
B		6
O		12
AB		3

(ii) AB is rarest and O is most common. 

Question 6:

Three coins are tossed 30 times. Each time the number of heads occurring was noted down as follows:
0, 1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 2, 2, 0, 1, 2, 1, 0, 3, 0, 2, 1, 1, 3, 2, 0, 2.
Prepare a frequency distribution table.

Answer 6:

Number of heads	tally marks	Frequency
0		6
1		10
2		9
3		5

Question 7:

Following data gives the number of children in 40 families:
1,2,6,5,1,5,1,3,2,6,2,3,4,2,0,4,4,3,2,2,0,0,1,2,2,4,3,2,1,0,5,1,2,4,3,4,1,6,2,2.
Represent it in the form of a frequency distribution, taking classes 0−2, 1−4, etc.

Answer 7:

The minimum observation is 0 and the maximum observation is 8.
Therefore, classes of  the same size covering the given data are 0-2, 2-4, 4-6 and 6-8.          .

Frequency distribution table:

Class	Tally mark	Frequency
0-2		11
2-4		17
4-6		9
6-8		3

Question 8:

Thirty children were asked about the number of hours they watched TV programmes in the previous week. The results were found as under:
8, 4, 8, 5, 1, 6, 2, 5, 3, 12, 3, 10, 4, 12, 2, 8, 15, 1, 6, 17, 5, 8, 2, 3, 9, 6, 7, 8, 14, 12.
(i) Make a grouped frequency distribution table for this data, taking class width 5 and one of the class interval as 5 –10.
(ii) How many children watched television for 15 or more hours a week?

Answer 8:

(i) 

Class interval	tally marks	Frequency
0-5		10
5-10		13
10-15		5
15-20		2

(ii) As we can see from the table, there are 2 children who watched tv for 15 hours or more. 

Question 9:

The marks obtained by 40 students of a class in an examination are given below.
3,20,13,1,21,13,3,23,16,13,18,12,5,12,5,24,9,2,7,18,20,3,10,12,7,18,2,5,7,10,16,8,16,17,8,23,24,6,23, 15.
Present the data in the form of a frequency distribution using equal class size, one such class being 10−15 (15 not included).

Answer 9:

The minimum observation is 0 and the maximum observation is 25.
Therefore, classes of the same size covering the given data are 0-5, 5-10, 10-15, 15-20 and 20-25.
Frequency distribution table:

Class	Tally mark	Frequency
0-5		6
5-10		10
10-15		8
15-20		8
20-25		8

Question 10:

Construct a frequency table for the following ages (in years) of 30 students using equal class intervals, one of them being 9−12, where 12 is not included.
18,12,7,6,11,15,21,9,8,13,15,17,22,19,14,21,23,8,12,17,15,6,18,23,22,16,9,21,11,16.

Answer 10:

The minimum observation is 6 and the maximum observation is 24.
Therefore, classes of the same size covering the given data are 6-9, 9-12, 12-15, 15-18, 18-21 and 21-24.
Frequency distribution table:

Class	Tally mark	Frequency
6-9		5
9-12		4
12-15		4
15-18		7
18-21		3
21-24		7

Question 11:

Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 labourers working in a factory, taking one of the class intervals as 210−230 (230 not included).
220,268,258,242,210,268,272,242,311,290,300,320,319,304,302,318,306,292,254,278,210,240,280,316,306,215,256,236.

Answer 11:

The minimum observation is 210 and the maximum observation is 330.
Therefore, classes of the same size covering the given data are 210-230, 230-250,250-270,270-290,290-310 and 310-330.
Frequency distribution table:
 

Class	Tally mark	Frequency
210-230		4
230-250		4
250-270		5
270-290		3
290-310		7
310-330		5

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Question 12:

The weights (in grams) of 40 oranges picked at random from a basket are as follows:
40,50,60,65,45,55,30,90,75,85,70,85,75,80,100,110,70,55,30,35,45,70,80,85,95,70,60,70,75,100,65,60,40,100,75,110,30,45,84.
Construct a frequency table as well as a cumulative frequency table.

Answer 12:

The minimum observation is 30 and the maximum observation is 120.
                                                
Frequency distribution table:

Class	Tally mark	Frequency
30-40		4
40-50		6
50-60		3
60-70		5
70-80		9
80-90		6
90-100		2
100-110		3
110-120		2

                                     
Cumulative frequency table:

Class	Tally mark	Frequency	Cumulative frequency
30-40		4	4
40-50		6	10
50-60		3	13
60-70		5	18
70-80		9	27
80-90		6	33
90-100		2	35
100-110		3	38
110-120		2	40

Question 13:

The heights (inAns cm) of 30 students of a class are given below:
161, 155, 159, 153, 150, 158, 154, 158, 160, 148, 149, 162, 163, 159, 148,
153, 157, 151, 154, 157, 153, 156, 152, 156, 160, 152, 147, 155, 155, 157.
Prepare a frequency table as well as a cumulative frequency table with 160 – 165  (165 not included) as one of the class intervals.

Answer 13:

Class	tally marks	Frequency	Cumulative frequency
145-150		4	4
150-155		9	4 + 9 = 13
155-160		12	13 + 12 = 25
160-165		5	25 + 5 = 30

Question 14:

Following are the ages (in years) of 360 patients, getting medical treatment in a hospital:

Ages (in years)	10−20	20−30	30−40	40−50	50−60	60−70
Number of patients	90	50	60	80	50	30

Construct the cumulative frequency table for the above data.

Answer 14:

The cumulative frequency table can be presented as given below:
 

Age (in years )	No. of patients	Cumulative frequency
10-20	90	90
20-30	50	140
30-40	60	200
40-50	80	280
50-60	50	330
60-70	30	360

Question 15:

Present the following as an ordinary grouped frequency table:

Marks (below)	10	20	30	40	50	60
Number of students	5	12	32	40	45	48

Answer 15:

The grouped frequency table can be presented as given below:
 

Marks	No. of students
0-10	5
10-20	7
20-30	20
30-40	8
40-50	5
50-60	3

Question 16:

Given below is a cumulative frequency table:

Marks	Number of students
Below 10	17
Below 20	22
Below 30	29
Below 40	37
Below 50	50
Below 60	60

Extract a frequency table from the above.

Answer 16:

The frequency table can be presented as given below:
 

Marks	Number of students
0-10	17
10-20	5
20-30	7
30-40	8
40-50	13
50-60	10

Question 17:

Make a frequency table from the following:

Marks obtained	Number of students
More than 60	0
More than 50	16
More than 40	40
More than 30	75
More than 20	87
More than 10	92
More than 0	100

Answer 17:

The frequency table can be presented as below:
 

Class	Frequency
0-10	8
10-20	5
20-30	12
30-40	35
40-50	24
50-60	16

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Question 18:

The marks obtained by 17 students in a mathematics test (out of 100) are given below:
90, 79, 76, 82, 65, 96, 100, 91, 82, 100, 49, 46, 64, 48, 72, 66, 68.
Find the range of the above data.

Answer 18:

Range = Maximum value $-$$-$ minimu value
= 100 $-$$-$ 46 = 54
Thus, the range is 54.

Question 19:

(i) Find the class mark of the class 90 – 120.
(ii) In a frequency distribution, the mid-value of the class is 10 and width of the class is 6. Find the lower limit of the class.
(iii) The width of each of five continuous classes in a frequency distribution is 5 and lower class limit of the lowest class is 10. What is the upper class limit of the highest class?
(iv) The class marks of a frequency distribution are 15, 20, 25, ... . Find the class corresponding to the class mark 20.
(v) In the class intervals 10 – 20, 20 – 30, find the class in which 20 is included.

Answer 19:

(i) $\mathrm{class}\mathrm{mark}=\frac{\mathrm{upper}\mathrm{limit}+\mathrm{lower}\mathrm{limit}}{2}=\frac{120+90}{2}=\frac{210}{2}=105$$\mathrm{class} \mathrm{mark}=\frac{\mathrm{upper} \mathrm{limit}+\mathrm{lower} \mathrm{limit}}{2}=\frac{120+90}{2}=\frac{210}{2}=105$
(ii) mid-value = 10
width = 6
Let the lower limit of the class be x. 
upper limit = x + 6
$\mathrm{class}\mathrm{mark}/\mathrm{mid}-\mathrm{value}=\frac{\mathrm{upper}\mathrm{limit}+\mathrm{lower}\mathrm{limit}}{2}$$\mathrm{class} \mathrm{mark}/\mathrm{mid}-\mathrm{value}=\frac{\mathrm{upper} \mathrm{limit}+\mathrm{lower} \mathrm{limit}}{2}$
$\frac{x+\left(x+6\right)}{2}=10\Rightarrow x=7$$$\begin{gathered} \frac{x+\left(x+6\right)}{2}=10 \\ \Rightarrow x=7 \end{gathered}$$
(iii) width = 5
lower class limit of lowest class = 10
The classes will be 10-15, 15-20, 20-25, 25-30, 30-35.
Upper class limit of the highest class = 35.
(iv) Class marks = 15, 20, 25, ...
class size = 20 $-$$-$ 15 = 5
Let lower limit of class be x.
$\frac{x+\left(x+5\right)}{2}=20\Rightarrow x=17.5$$$\begin{gathered} \frac{x+\left(x+5\right)}{2}=20 \\ \Rightarrow x=17.5 \end{gathered}$$
Thus, the class is 17.5-22.5. 
(v) 20 will be included in the class interval 20-30. 

Question 20:

Find the values of a, b, c, d, e, f, g from the following frequency distribution of the heights of 50 students in a class:
 

Height (in cm)	Frequency	Cumulative frequency
160 – 165	15	a
165 – 170	b	35
170 – 175	12	c
175 – 180	d	50
180 – 185	e	55
185 –  190	5	f
	g

Answer 20:

The  complete table will be

Height (in cm)	Frequency	Cumulative frequency
160 – 165	15	a = 15
165 – 170	b = 35 – 15 = 20	35
170 – 175	12	c = 35 + 12 = 47
175 – 180	d = 50 – 47 = 3	50
180 – 185	e = 55 – 50 = 5	55
185 –  190	5	f = 55 + 5 = 60
	g = 15 + 20 + 12 + 3 + 5 + 5 = 60