Page 1
ST A TISTICS 151
CHAPTER 12
STATISTICS
12.1 Graphical Representation of Data
The representation of data by tables has already been discussed. Now let us turn our
attention to another representation of data, i.e., the graphical representation. It is well
said that one picture is better than a thousand words. Usually comparisons among the
individual items are best shown by means of graphs. The representation then becomes
easier to understand than the actual data. We shall study the following graphical
representations in this section.
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(A) Bar Graphs
In earlier classes, you have already studied and constructed bar graphs. Here we
shall discuss them through a more formal approach. Recall that a bar graph is a
pictorial representation of data in which usually bars of uniform width are drawn with
equal spacing between them on one axis (say, the x-axis), depicting the variable. The
values of the variable are shown on the other axis (say, the y-axis) and the heights of
the bars depend on the values of the variable.
Example 1 : In a particular section of Class IX, 40 students were asked about the
months of their birth and the following graph was prepared for the data so obtained:
Rationalised 2023-24
Page 2
ST A TISTICS 151
CHAPTER 12
STATISTICS
12.1 Graphical Representation of Data
The representation of data by tables has already been discussed. Now let us turn our
attention to another representation of data, i.e., the graphical representation. It is well
said that one picture is better than a thousand words. Usually comparisons among the
individual items are best shown by means of graphs. The representation then becomes
easier to understand than the actual data. We shall study the following graphical
representations in this section.
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(A) Bar Graphs
In earlier classes, you have already studied and constructed bar graphs. Here we
shall discuss them through a more formal approach. Recall that a bar graph is a
pictorial representation of data in which usually bars of uniform width are drawn with
equal spacing between them on one axis (say, the x-axis), depicting the variable. The
values of the variable are shown on the other axis (say, the y-axis) and the heights of
the bars depend on the values of the variable.
Example 1 : In a particular section of Class IX, 40 students were asked about the
months of their birth and the following graph was prepared for the data so obtained:
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152 MATHEMA TICS
Fig. 12.1
Observe the bar graph given above and answer the following questions:
(i) How many students were born in the month of November?
(ii) In which month were the maximum number of students born?
Solution : Note that the variable here is the ‘month of birth’, and the value of the
variable is the ‘Number of students born’.
(i) 4 students were born in the month of November.
(ii) The Maximum number of students were born in the month of August.
Let us now recall how a bar graph is constructed by considering the following example.
Example 2 : A family with a monthly income of ` 20,000 had planned the following
expenditures per month under various heads:
Table 12.1
Heads Expenditure
(in thousand rupees)
Grocery 4
Rent 5
Education of children 5
Medicine 2
Fuel 2
Entertainment 1
Miscellaneous 1
Draw a bar graph for the data above.
Rationalised 2023-24
Page 3
ST A TISTICS 151
CHAPTER 12
STATISTICS
12.1 Graphical Representation of Data
The representation of data by tables has already been discussed. Now let us turn our
attention to another representation of data, i.e., the graphical representation. It is well
said that one picture is better than a thousand words. Usually comparisons among the
individual items are best shown by means of graphs. The representation then becomes
easier to understand than the actual data. We shall study the following graphical
representations in this section.
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(A) Bar Graphs
In earlier classes, you have already studied and constructed bar graphs. Here we
shall discuss them through a more formal approach. Recall that a bar graph is a
pictorial representation of data in which usually bars of uniform width are drawn with
equal spacing between them on one axis (say, the x-axis), depicting the variable. The
values of the variable are shown on the other axis (say, the y-axis) and the heights of
the bars depend on the values of the variable.
Example 1 : In a particular section of Class IX, 40 students were asked about the
months of their birth and the following graph was prepared for the data so obtained:
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152 MATHEMA TICS
Fig. 12.1
Observe the bar graph given above and answer the following questions:
(i) How many students were born in the month of November?
(ii) In which month were the maximum number of students born?
Solution : Note that the variable here is the ‘month of birth’, and the value of the
variable is the ‘Number of students born’.
(i) 4 students were born in the month of November.
(ii) The Maximum number of students were born in the month of August.
Let us now recall how a bar graph is constructed by considering the following example.
Example 2 : A family with a monthly income of ` 20,000 had planned the following
expenditures per month under various heads:
Table 12.1
Heads Expenditure
(in thousand rupees)
Grocery 4
Rent 5
Education of children 5
Medicine 2
Fuel 2
Entertainment 1
Miscellaneous 1
Draw a bar graph for the data above.
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ST A TISTICS 153
Solution : We draw the bar graph of this data in the following steps. Note that the unit
in the second column is thousand rupees. So, ‘4’ against ‘grocery’ means `4000.
1. We represent the Heads (variable) on the horizontal axis choosing any scale,
since the width of the bar is not important. But for clarity, we take equal widths
for all bars and maintain equal gaps in between. Let one Head be represented by
one unit.
2. We represent the expenditure (value) on the vertical axis. Since the maximum
expenditure is `5000, we can choose the scale as 1 unit = `1000.
3. To represent our first Head, i.e., grocery, we draw a rectangular bar with width
1 unit and height 4 units.
4. Similarly, other Heads are represented leaving a gap of 1 unit in between two
consecutive bars.
The bar graph is drawn in Fig. 12.2.
Fig. 12.2
Here, you can easily visualise the relative characteristics of the data at a glance, e.g.,
the expenditure on education is more than double that of medical expenses. Therefore,
in some ways it serves as a better representation of data than the tabular form.
Activity 1 : Continuing with the same four groups of Activity 1, represent the data by
suitable bar graphs.
Let us now see how a frequency distribution table for continuous class intervals
can be represented graphically.
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Page 4
ST A TISTICS 151
CHAPTER 12
STATISTICS
12.1 Graphical Representation of Data
The representation of data by tables has already been discussed. Now let us turn our
attention to another representation of data, i.e., the graphical representation. It is well
said that one picture is better than a thousand words. Usually comparisons among the
individual items are best shown by means of graphs. The representation then becomes
easier to understand than the actual data. We shall study the following graphical
representations in this section.
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(A) Bar Graphs
In earlier classes, you have already studied and constructed bar graphs. Here we
shall discuss them through a more formal approach. Recall that a bar graph is a
pictorial representation of data in which usually bars of uniform width are drawn with
equal spacing between them on one axis (say, the x-axis), depicting the variable. The
values of the variable are shown on the other axis (say, the y-axis) and the heights of
the bars depend on the values of the variable.
Example 1 : In a particular section of Class IX, 40 students were asked about the
months of their birth and the following graph was prepared for the data so obtained:
Rationalised 2023-24
152 MATHEMA TICS
Fig. 12.1
Observe the bar graph given above and answer the following questions:
(i) How many students were born in the month of November?
(ii) In which month were the maximum number of students born?
Solution : Note that the variable here is the ‘month of birth’, and the value of the
variable is the ‘Number of students born’.
(i) 4 students were born in the month of November.
(ii) The Maximum number of students were born in the month of August.
Let us now recall how a bar graph is constructed by considering the following example.
Example 2 : A family with a monthly income of ` 20,000 had planned the following
expenditures per month under various heads:
Table 12.1
Heads Expenditure
(in thousand rupees)
Grocery 4
Rent 5
Education of children 5
Medicine 2
Fuel 2
Entertainment 1
Miscellaneous 1
Draw a bar graph for the data above.
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ST A TISTICS 153
Solution : We draw the bar graph of this data in the following steps. Note that the unit
in the second column is thousand rupees. So, ‘4’ against ‘grocery’ means `4000.
1. We represent the Heads (variable) on the horizontal axis choosing any scale,
since the width of the bar is not important. But for clarity, we take equal widths
for all bars and maintain equal gaps in between. Let one Head be represented by
one unit.
2. We represent the expenditure (value) on the vertical axis. Since the maximum
expenditure is `5000, we can choose the scale as 1 unit = `1000.
3. To represent our first Head, i.e., grocery, we draw a rectangular bar with width
1 unit and height 4 units.
4. Similarly, other Heads are represented leaving a gap of 1 unit in between two
consecutive bars.
The bar graph is drawn in Fig. 12.2.
Fig. 12.2
Here, you can easily visualise the relative characteristics of the data at a glance, e.g.,
the expenditure on education is more than double that of medical expenses. Therefore,
in some ways it serves as a better representation of data than the tabular form.
Activity 1 : Continuing with the same four groups of Activity 1, represent the data by
suitable bar graphs.
Let us now see how a frequency distribution table for continuous class intervals
can be represented graphically.
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154 MATHEMA TICS
(B) Histogram
This is a form of representation like the bar graph, but it is used for continuous class
intervals. For instance, consider the frequency distribution Table 12.2, representing
the weights of 36 students of a class:
Table 12.2
Weights (in kg) Number of students
30.5 - 35.5 9
35.5 - 40.5 6
40.5 - 45.5 15
45.5 - 50.5 3
50.5 - 55.5 1
55.5 - 60.5 2
Total 36
Let us represent the data given above graphically as follows:
(i) We represent the weights on the horizontal axis on a suitable scale. We can choose
the scale as 1 cm = 5 kg. Also, since the first class interval is starting from 30.5
and not zero, we show it on the graph by marking a kink or a break on the axis.
(ii) We represent the number of students (frequency) on the vertical axis on a suitable
scale. Since the maximum frequency is 15, we need to choose the scale to
accomodate this maximum frequency.
(iii) We now draw rectangles (or rectangular bars) of width equal to the class-size
and lengths according to the frequencies of the corresponding class intervals. For
example, the rectangle for the class interval 30.5 - 35.5 will be of width 1 cm and
length 4.5 cm.
(iv) In this way, we obtain the graph as shown in Fig. 12.3:
Rationalised 2023-24
Page 5
ST A TISTICS 151
CHAPTER 12
STATISTICS
12.1 Graphical Representation of Data
The representation of data by tables has already been discussed. Now let us turn our
attention to another representation of data, i.e., the graphical representation. It is well
said that one picture is better than a thousand words. Usually comparisons among the
individual items are best shown by means of graphs. The representation then becomes
easier to understand than the actual data. We shall study the following graphical
representations in this section.
(A) Bar graphs
(B) Histograms of uniform width, and of varying widths
(C) Frequency polygons
(A) Bar Graphs
In earlier classes, you have already studied and constructed bar graphs. Here we
shall discuss them through a more formal approach. Recall that a bar graph is a
pictorial representation of data in which usually bars of uniform width are drawn with
equal spacing between them on one axis (say, the x-axis), depicting the variable. The
values of the variable are shown on the other axis (say, the y-axis) and the heights of
the bars depend on the values of the variable.
Example 1 : In a particular section of Class IX, 40 students were asked about the
months of their birth and the following graph was prepared for the data so obtained:
Rationalised 2023-24
152 MATHEMA TICS
Fig. 12.1
Observe the bar graph given above and answer the following questions:
(i) How many students were born in the month of November?
(ii) In which month were the maximum number of students born?
Solution : Note that the variable here is the ‘month of birth’, and the value of the
variable is the ‘Number of students born’.
(i) 4 students were born in the month of November.
(ii) The Maximum number of students were born in the month of August.
Let us now recall how a bar graph is constructed by considering the following example.
Example 2 : A family with a monthly income of ` 20,000 had planned the following
expenditures per month under various heads:
Table 12.1
Heads Expenditure
(in thousand rupees)
Grocery 4
Rent 5
Education of children 5
Medicine 2
Fuel 2
Entertainment 1
Miscellaneous 1
Draw a bar graph for the data above.
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ST A TISTICS 153
Solution : We draw the bar graph of this data in the following steps. Note that the unit
in the second column is thousand rupees. So, ‘4’ against ‘grocery’ means `4000.
1. We represent the Heads (variable) on the horizontal axis choosing any scale,
since the width of the bar is not important. But for clarity, we take equal widths
for all bars and maintain equal gaps in between. Let one Head be represented by
one unit.
2. We represent the expenditure (value) on the vertical axis. Since the maximum
expenditure is `5000, we can choose the scale as 1 unit = `1000.
3. To represent our first Head, i.e., grocery, we draw a rectangular bar with width
1 unit and height 4 units.
4. Similarly, other Heads are represented leaving a gap of 1 unit in between two
consecutive bars.
The bar graph is drawn in Fig. 12.2.
Fig. 12.2
Here, you can easily visualise the relative characteristics of the data at a glance, e.g.,
the expenditure on education is more than double that of medical expenses. Therefore,
in some ways it serves as a better representation of data than the tabular form.
Activity 1 : Continuing with the same four groups of Activity 1, represent the data by
suitable bar graphs.
Let us now see how a frequency distribution table for continuous class intervals
can be represented graphically.
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154 MATHEMA TICS
(B) Histogram
This is a form of representation like the bar graph, but it is used for continuous class
intervals. For instance, consider the frequency distribution Table 12.2, representing
the weights of 36 students of a class:
Table 12.2
Weights (in kg) Number of students
30.5 - 35.5 9
35.5 - 40.5 6
40.5 - 45.5 15
45.5 - 50.5 3
50.5 - 55.5 1
55.5 - 60.5 2
Total 36
Let us represent the data given above graphically as follows:
(i) We represent the weights on the horizontal axis on a suitable scale. We can choose
the scale as 1 cm = 5 kg. Also, since the first class interval is starting from 30.5
and not zero, we show it on the graph by marking a kink or a break on the axis.
(ii) We represent the number of students (frequency) on the vertical axis on a suitable
scale. Since the maximum frequency is 15, we need to choose the scale to
accomodate this maximum frequency.
(iii) We now draw rectangles (or rectangular bars) of width equal to the class-size
and lengths according to the frequencies of the corresponding class intervals. For
example, the rectangle for the class interval 30.5 - 35.5 will be of width 1 cm and
length 4.5 cm.
(iv) In this way, we obtain the graph as shown in Fig. 12.3:
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ST A TISTICS 155
Fig. 12.3
Observe that since there are no gaps in between consecutive rectangles, the resultant
graph appears like a solid figure. This is called a histogram, which is a graphical
representation of a grouped frequency distribution with continuous classes. Also, unlike
a bar graph, the width of the bar plays a significant role in its construction.
Here, in fact, areas of the rectangles erected are proportional to the corresponding
frequencies. However, since the widths of the rectangles are all equal, the lengths of
the rectangles are proportional to the frequencies. That is why, we draw the lengths
according to (iii) above.
Now, consider a situation different from the one above.
Example 3 : A teacher wanted to analyse the performance of two sections of students
in a mathematics test of 100 marks. Looking at their performances, she found that a
few students got under 20 marks and a few got 70 marks or above. So she decided to
group them into intervals of varying sizes as follows: 0 - 20, 20 - 30, . . ., 60 - 70,
70 - 100. Then she formed the following table:
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