Rs Aggarwal 2021 2022 for Class 9 Maths Chapter 16 - Presentation Of Data In Tabular Form

Answer:

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

Answer:

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.

Answer:

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.

Answer:

(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 $\left(\frac{\mathrm{Upper} \mathrm{limit}+\mathrm{Lower} \mathrm{limit}}{2}\right)$.
(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.

Answer:

(i) 

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

Answer:

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

Answer:

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.          .

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

Answer:

(i) 

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

The cumulative frequency table can be presented as given below:
 

Answer:

The grouped frequency table can be presented as given below:
 

Answer:

The frequency table can be presented as given below:
 

Answer:

The frequency table can be presented as below:
 

Answer:

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

Answer:

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

Answer:

The  complete table will be

Answer:

(d) 26

We have:
Maximum value = 32
Minimum value = 6
We know:
Range = Maximum value $-$ Minimum value 
          =32 $-$ 6
          =26

Answer:

(b) 110

Class mark = $\frac{\mathrm{Upper} \mathrm{limit}+\mathrm{Lower} \mathrm{limit}}{2}=\frac{120+100}{2}=110$

Answer:

(b) 20−30
This is the continuous form of frequency distribution. Here, the upper limit of each class is excluded, while the lower limit is included. So, the number 20 is included in the class interval 20−30.

Answer:

(b) 17.5$-$22.5

We are given frequency distribution 15, 20, 25, 30,...
Class size = 20 $-$ 15 = 5
Class marks = 20
Now,
$$\begin{gathered} \mathrm{Lower} \mathrm{limit}=\left(20-\frac{5}{2}\right)=\frac{35}{2}=17.5 \\ \mathrm{Upper} \mathrm{limit} =\left(20+\frac{5}{2}\right)=\frac{45}{2}=22.5 \end{gathered}$$

Thus, the required class is 17.5$-$22.5.

Answer:

(b) 7

Given:
Mid value of the class = 10
Width of each class = 6
Now,
Let the lower limit be x.
We know:
Upper limit = Lower limit + Class size
                  = x + 6
Also,
$$\begin{gathered} \mathrm{Mid} \mathrm{value}=\frac{x+x+6}{2}=\frac{2x+6}{2}=x+3 \\ \Rightarrow x+3=10 \\ \Rightarrow x=7 \end{gathered}$$

Thus, the lower limit is 7.

Answer:

(a) 37–47

Let the lower limit be x.
Here,
Class size = 10
∴ Upper limit = Class size + Lower limit
Upper limit = (x + 10)
Mid value of the class interval = 42

$$\begin{gathered} \therefore \frac{x+x+10}{2}=42 \\ \Rightarrow \frac{2x+10}{2}=42 \\ \Rightarrow 2x+10=84 \\ \Rightarrow 2x=74 \\ \Rightarrow x=37 \\ \mathrm{Thus}, \mathrm{we} \mathrm{have}: \\ \mathrm{Lower} \mathrm{limit}=37 \\ \mathrm{Upper} \mathrm{limit}=37+10=47 \end{gathered}$$

Answer:

(a) 2m $-$ u

Given:
Mid value = m
Upper limit = u

We know:
$$\begin{gathered} \frac{\mathrm{Lower} \mathrm{limit}+\mathrm{Upper} \mathrm{limit}}{2}=\mathrm{Mid} \mathrm{value} \\ \Rightarrow \frac{\mathrm{Lower} \mathrm{limit} +u}{2}=m \\ \Rightarrow \mathrm{Lower} \mathrm{limit}+u=2m \\ \Rightarrow \mathrm{Lower} \mathrm{limit}=2m\mathit{-}u \end{gathered}$$

Answer:

(c) 35

We have:
Class width = 5
Lower class limit of the lowest class = 10
Now,
Upper class limit of the highest class = 10 + 5 $\times$ 5 = 35

Answer:

(c) 2m$-$L

$$\begin{gathered} \mathrm{Mid} \mathrm{value}=\frac{\mathrm{Lower} \mathrm{limit}+\mathrm{Upper} \mathrm{limit}}{2} \\ \Rightarrow m=\frac{L+U}{2} \\ \Rightarrow U=2m-L \\ \therefore \mathrm{Upper} \mathrm{class} \mathrm{boundary} \mathrm{of} \mathrm{the} \mathrm{class}=2m-L \end{gathered}$$