NCERT Textbook: Statistics

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ST A TISTICS 171
13
13.1 Introduction
In Class IX, you have studied the classification of given data into ungrouped as well as
grouped frequency distributions. You have also learnt to represent the data pictorially
in the form of various graphs such as bar graphs, histograms (including those of varying
widths) and frequency polygons. In fact, you went a step further by studying certain
numerical representatives of the ungrouped data, also called measures of central
tendency, namely, mean, median and mode. In this chapter, we shall extend the study
of these three measures, i.e., mean, median and mode from ungrouped data to that of
grouped data. We shall also discuss the concept of cumulative frequency, the
cumulative frequency distribution and how to draw cumulative frequency curves, called
ogives.
13.2 Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the
observations divided by the total number of observations. From Class IX, recall that if
x
1
, x
2
,. . ., x
n
are observations with respective frequencies f
1
, f
2
, . . ., f
n
, then this
means observation x
1
occurs f
1
times, x
2
occurs f
2
times, and so on.
Now, the sum of the values of all the observations = f
1
x
1
+ f
2
x
2
+ . . . + f
n
x
n
,  and
the number of observations = f
1
+ f
2
+ . . . + f
n
.
So, the mean x of the data is given by
x
=
11 2 2
12
nn
n
fx fx fx
ff f
?? ?
?? ?
?
?
Recall that we can write this in short form by using the Greek letter ? (capital
sigma) which means summation. That is,
ST A TISTICS
Rationalised 2023-24
Page 2

ST A TISTICS 171
13
13.1 Introduction
In Class IX, you have studied the classification of given data into ungrouped as well as
grouped frequency distributions. You have also learnt to represent the data pictorially
in the form of various graphs such as bar graphs, histograms (including those of varying
widths) and frequency polygons. In fact, you went a step further by studying certain
numerical representatives of the ungrouped data, also called measures of central
tendency, namely, mean, median and mode. In this chapter, we shall extend the study
of these three measures, i.e., mean, median and mode from ungrouped data to that of
grouped data. We shall also discuss the concept of cumulative frequency, the
cumulative frequency distribution and how to draw cumulative frequency curves, called
ogives.
13.2 Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the
observations divided by the total number of observations. From Class IX, recall that if
x
1
, x
2
,. . ., x
n
are observations with respective frequencies f
1
, f
2
, . . ., f
n
, then this
means observation x
1
occurs f
1
times, x
2
occurs f
2
times, and so on.
Now, the sum of the values of all the observations = f
1
x
1
+ f
2
x
2
+ . . . + f
n
x
n
,  and
the number of observations = f
1
+ f
2
+ . . . + f
n
.
So, the mean x of the data is given by
x
=
11 2 2
12
nn
n
fx fx fx
ff f
?? ?
?? ?
?
?
Recall that we can write this in short form by using the Greek letter ? (capital
sigma) which means summation. That is,
ST A TISTICS
Rationalised 2023-24
172 MATHEMA TICS
x
=
1
1
n
ii
i
n
i
i
fx
f
?
?
?
?
which, more briefly, is written as x =
?
?
ii
i
fx
f
, if it is understood that i varies from
1 to n.
Let us apply this formula to find the mean in the following example.
Example 1 : The marks obtained by 30 students of Class X of a certain school in a
Mathematics paper consisting of 100 marks are presented in table below. Find the
mean of the marks obtained by the students.
Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95
(x
i
)
Number of 113 4324 4112 3 1
students ( f
i
)
Solution: Recall that to find the mean marks, we require the product of each x
i
with
the corresponding frequency f
i
. So, let us put them in a column as shown in Table 13.1.
Table 13.1
Marks obtained (x
i
) Number of students ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
Total ?f
i
= 30 ?f
i
x
i
= 1779
Rationalised 2023-24
Page 3

ST A TISTICS 171
13
13.1 Introduction
In Class IX, you have studied the classification of given data into ungrouped as well as
grouped frequency distributions. You have also learnt to represent the data pictorially
in the form of various graphs such as bar graphs, histograms (including those of varying
widths) and frequency polygons. In fact, you went a step further by studying certain
numerical representatives of the ungrouped data, also called measures of central
tendency, namely, mean, median and mode. In this chapter, we shall extend the study
of these three measures, i.e., mean, median and mode from ungrouped data to that of
grouped data. We shall also discuss the concept of cumulative frequency, the
cumulative frequency distribution and how to draw cumulative frequency curves, called
ogives.
13.2 Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the
observations divided by the total number of observations. From Class IX, recall that if
x
1
, x
2
,. . ., x
n
are observations with respective frequencies f
1
, f
2
, . . ., f
n
, then this
means observation x
1
occurs f
1
times, x
2
occurs f
2
times, and so on.
Now, the sum of the values of all the observations = f
1
x
1
+ f
2
x
2
+ . . . + f
n
x
n
,  and
the number of observations = f
1
+ f
2
+ . . . + f
n
.
So, the mean x of the data is given by
x
=
11 2 2
12
nn
n
fx fx fx
ff f
?? ?
?? ?
?
?
Recall that we can write this in short form by using the Greek letter ? (capital
sigma) which means summation. That is,
ST A TISTICS
Rationalised 2023-24
172 MATHEMA TICS
x
=
1
1
n
ii
i
n
i
i
fx
f
?
?
?
?
which, more briefly, is written as x =
?
?
ii
i
fx
f
, if it is understood that i varies from
1 to n.
Let us apply this formula to find the mean in the following example.
Example 1 : The marks obtained by 30 students of Class X of a certain school in a
Mathematics paper consisting of 100 marks are presented in table below. Find the
mean of the marks obtained by the students.
Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95
(x
i
)
Number of 113 4324 4112 3 1
students ( f
i
)
Solution: Recall that to find the mean marks, we require the product of each x
i
with
the corresponding frequency f
i
. So, let us put them in a column as shown in Table 13.1.
Table 13.1
Marks obtained (x
i
) Number of students ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
Total ?f
i
= 30 ?f
i
x
i
= 1779
Rationalised 2023-24
ST A TISTICS 173
Now,
?
?
?
i i
i
f x
x
f
=
1779
30
= 59.3
Therefore, the mean marks obtained is 59.3.
In most of our real life situations, data is usually so large that to make a meaningful
study it needs to be condensed as grouped data. So, we need to convert given ungrouped
data into grouped data and devise some method to find its mean.
Let us convert the ungrouped data of Example 1 into grouped data by forming
class-intervals of width, say 15. Remember that, while allocating frequencies to each
class-interval, students falling in any upper class-limit would be considered in the next
class, e.g., 4 students who have obtained 40 marks would be considered in the class-
interval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped
frequency distribution table (see Table 13.2).
Table 13.2
Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100
Number of students 2 3 7 6 6 6
Now, for each class-interval, we require a point which would serve as the
representative of the whole class. It is assumed that the frequency of each class-
interval is centred around its mid-point. So the mid-point (or class mark) of each
class can be chosen to represent the observations falling in the class. Recall that we
find the mid-point of a class (or its class mark) by finding the average of its upper and
lower limits. That is,
Class mark =
Upper class limit + Lower class limit
2
With reference to Table 13.2, for the class 10-25, the class mark is
10 25
2
+
, i.e.,
17.5. Similarly, we can find the class marks of the remaining class intervals. We put
them in Table 13.3. These class marks serve as our x
i
’s. Now, in general, for the ith
class interval, we have the frequency f
i
corresponding to the class mark x
i
. We can
now proceed to compute the mean in the same manner as in Example 1.
Rationalised 2023-24
Page 4

ST A TISTICS 171
13
13.1 Introduction
In Class IX, you have studied the classification of given data into ungrouped as well as
grouped frequency distributions. You have also learnt to represent the data pictorially
in the form of various graphs such as bar graphs, histograms (including those of varying
widths) and frequency polygons. In fact, you went a step further by studying certain
numerical representatives of the ungrouped data, also called measures of central
tendency, namely, mean, median and mode. In this chapter, we shall extend the study
of these three measures, i.e., mean, median and mode from ungrouped data to that of
grouped data. We shall also discuss the concept of cumulative frequency, the
cumulative frequency distribution and how to draw cumulative frequency curves, called
ogives.
13.2 Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the
observations divided by the total number of observations. From Class IX, recall that if
x
1
, x
2
,. . ., x
n
are observations with respective frequencies f
1
, f
2
, . . ., f
n
, then this
means observation x
1
occurs f
1
times, x
2
occurs f
2
times, and so on.
Now, the sum of the values of all the observations = f
1
x
1
+ f
2
x
2
+ . . . + f
n
x
n
,  and
the number of observations = f
1
+ f
2
+ . . . + f
n
.
So, the mean x of the data is given by
x
=
11 2 2
12
nn
n
fx fx fx
ff f
?? ?
?? ?
?
?
Recall that we can write this in short form by using the Greek letter ? (capital
sigma) which means summation. That is,
ST A TISTICS
Rationalised 2023-24
172 MATHEMA TICS
x
=
1
1
n
ii
i
n
i
i
fx
f
?
?
?
?
which, more briefly, is written as x =
?
?
ii
i
fx
f
, if it is understood that i varies from
1 to n.
Let us apply this formula to find the mean in the following example.
Example 1 : The marks obtained by 30 students of Class X of a certain school in a
Mathematics paper consisting of 100 marks are presented in table below. Find the
mean of the marks obtained by the students.
Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95
(x
i
)
Number of 113 4324 4112 3 1
students ( f
i
)
Solution: Recall that to find the mean marks, we require the product of each x
i
with
the corresponding frequency f
i
. So, let us put them in a column as shown in Table 13.1.
Table 13.1
Marks obtained (x
i
) Number of students ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
Total ?f
i
= 30 ?f
i
x
i
= 1779
Rationalised 2023-24
ST A TISTICS 173
Now,
?
?
?
i i
i
f x
x
f
=
1779
30
= 59.3
Therefore, the mean marks obtained is 59.3.
In most of our real life situations, data is usually so large that to make a meaningful
study it needs to be condensed as grouped data. So, we need to convert given ungrouped
data into grouped data and devise some method to find its mean.
Let us convert the ungrouped data of Example 1 into grouped data by forming
class-intervals of width, say 15. Remember that, while allocating frequencies to each
class-interval, students falling in any upper class-limit would be considered in the next
class, e.g., 4 students who have obtained 40 marks would be considered in the class-
interval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped
frequency distribution table (see Table 13.2).
Table 13.2
Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100
Number of students 2 3 7 6 6 6
Now, for each class-interval, we require a point which would serve as the
representative of the whole class. It is assumed that the frequency of each class-
interval is centred around its mid-point. So the mid-point (or class mark) of each
class can be chosen to represent the observations falling in the class. Recall that we
find the mid-point of a class (or its class mark) by finding the average of its upper and
lower limits. That is,
Class mark =
Upper class limit + Lower class limit
2
With reference to Table 13.2, for the class 10-25, the class mark is
10 25
2
+
, i.e.,
17.5. Similarly, we can find the class marks of the remaining class intervals. We put
them in Table 13.3. These class marks serve as our x
i
’s. Now, in general, for the ith
class interval, we have the frequency f
i
corresponding to the class mark x
i
. We can
now proceed to compute the mean in the same manner as in Example 1.
Rationalised 2023-24
174 MATHEMA TICS
Table 13.3
Class interval Number of students ( f
i
) Class mark (x
i
) f
i
x
i
10 - 25 2 17.5 35.0
25 - 40 3 32.5 97.5
40 - 55 7 47.5 332.5
55 - 70 6 62.5 375.0
70 - 85 6 77.5 465.0
85 - 100 6 92.5 555.0
Total S f
i
= 30 S f
i
x
i
= 1860.0
The sum of the values in the last column gives us S f
i
x
i
. So, the mean x of the
given data is given by
x
=
1860.0
62
30
i i
i
f x
f
S
= =
S
This new method of finding the mean is known as the Direct Method.
We observe that Tables 13.1 and 13.3 are using the same data and employing the
same formula for the calculation of the mean but the results obtained are different.
Can you think why this is so, and which one is more accurate? The difference in the
two values is because of the mid-point assumption in Table 13.3, 59.3 being the exact
mean, while 62 an approximate mean.
Sometimes when the numerical values of x
i
and f
i
are large, finding the product
of x
i
and f
i
becomes tedious and time consuming. So, for such situations, let us think of
a method of reducing these calculations.
We can do nothing with the f
i
’s, but we can change each x
i
to a smaller number
so that our calculations become easy. How do we do this? What about subtracting a
fixed number from each of these x
i
’s? Let us try this method.
The first step is to choose one among the x
i
’s as the assumed mean, and denote
it by ‘a’. Also, to further reduce our calculation work, we may take ‘a’ to be that x
i
which lies in the centre of x
1
, x
2
, . . ., x
n
. So, we can choose a = 47.5 or a = 62.5. Let
us choose a = 47.5.
The next step is to find the difference d
i
between a and each of the x
i
’s, that is,
the deviation of ‘a’ from each of the x
i
’s.
i.e., d
i
= x
i
– a = x
i
– 47.5
Rationalised 2023-24
Page 5

ST A TISTICS 171
13
13.1 Introduction
In Class IX, you have studied the classification of given data into ungrouped as well as
grouped frequency distributions. You have also learnt to represent the data pictorially
in the form of various graphs such as bar graphs, histograms (including those of varying
widths) and frequency polygons. In fact, you went a step further by studying certain
numerical representatives of the ungrouped data, also called measures of central
tendency, namely, mean, median and mode. In this chapter, we shall extend the study
of these three measures, i.e., mean, median and mode from ungrouped data to that of
grouped data. We shall also discuss the concept of cumulative frequency, the
cumulative frequency distribution and how to draw cumulative frequency curves, called
ogives.
13.2 Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the
observations divided by the total number of observations. From Class IX, recall that if
x
1
, x
2
,. . ., x
n
are observations with respective frequencies f
1
, f
2
, . . ., f
n
, then this
means observation x
1
occurs f
1
times, x
2
occurs f
2
times, and so on.
Now, the sum of the values of all the observations = f
1
x
1
+ f
2
x
2
+ . . . + f
n
x
n
,  and
the number of observations = f
1
+ f
2
+ . . . + f
n
.
So, the mean x of the data is given by
x
=
11 2 2
12
nn
n
fx fx fx
ff f
?? ?
?? ?
?
?
Recall that we can write this in short form by using the Greek letter ? (capital
sigma) which means summation. That is,
ST A TISTICS
Rationalised 2023-24
172 MATHEMA TICS
x
=
1
1
n
ii
i
n
i
i
fx
f
?
?
?
?
which, more briefly, is written as x =
?
?
ii
i
fx
f
, if it is understood that i varies from
1 to n.
Let us apply this formula to find the mean in the following example.
Example 1 : The marks obtained by 30 students of Class X of a certain school in a
Mathematics paper consisting of 100 marks are presented in table below. Find the
mean of the marks obtained by the students.
Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95
(x
i
)
Number of 113 4324 4112 3 1
students ( f
i
)
Solution: Recall that to find the mean marks, we require the product of each x
i
with
the corresponding frequency f
i
. So, let us put them in a column as shown in Table 13.1.
Table 13.1
Marks obtained (x
i
) Number of students ( f
i
) f
i
x
i
10 1 10
20 1 20
. 36 3 108
40 4 160
50 3 150
56 2 112
60 4 240
70 4 280
72 1 72
80 1 80
88 2 176
92 3 276
95 1 95
Total ?f
i
= 30 ?f
i
x
i
= 1779
Rationalised 2023-24
ST A TISTICS 173
Now,
?
?
?
i i
i
f x
x
f
=
1779
30
= 59.3
Therefore, the mean marks obtained is 59.3.
In most of our real life situations, data is usually so large that to make a meaningful
study it needs to be condensed as grouped data. So, we need to convert given ungrouped
data into grouped data and devise some method to find its mean.
Let us convert the ungrouped data of Example 1 into grouped data by forming
class-intervals of width, say 15. Remember that, while allocating frequencies to each
class-interval, students falling in any upper class-limit would be considered in the next
class, e.g., 4 students who have obtained 40 marks would be considered in the class-
interval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped
frequency distribution table (see Table 13.2).
Table 13.2
Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100
Number of students 2 3 7 6 6 6
Now, for each class-interval, we require a point which would serve as the
representative of the whole class. It is assumed that the frequency of each class-
interval is centred around its mid-point. So the mid-point (or class mark) of each
class can be chosen to represent the observations falling in the class. Recall that we
find the mid-point of a class (or its class mark) by finding the average of its upper and
lower limits. That is,
Class mark =
Upper class limit + Lower class limit
2
With reference to Table 13.2, for the class 10-25, the class mark is
10 25
2
+
, i.e.,
17.5. Similarly, we can find the class marks of the remaining class intervals. We put
them in Table 13.3. These class marks serve as our x
i
’s. Now, in general, for the ith
class interval, we have the frequency f
i
corresponding to the class mark x
i
. We can
now proceed to compute the mean in the same manner as in Example 1.
Rationalised 2023-24
174 MATHEMA TICS
Table 13.3
Class interval Number of students ( f
i
) Class mark (x
i
) f
i
x
i
10 - 25 2 17.5 35.0
25 - 40 3 32.5 97.5
40 - 55 7 47.5 332.5
55 - 70 6 62.5 375.0
70 - 85 6 77.5 465.0
85 - 100 6 92.5 555.0
Total S f
i
= 30 S f
i
x
i
= 1860.0
The sum of the values in the last column gives us S f
i
x
i
. So, the mean x of the
given data is given by
x
=
1860.0
62
30
i i
i
f x
f
S
= =
S
This new method of finding the mean is known as the Direct Method.
We observe that Tables 13.1 and 13.3 are using the same data and employing the
same formula for the calculation of the mean but the results obtained are different.
Can you think why this is so, and which one is more accurate? The difference in the
two values is because of the mid-point assumption in Table 13.3, 59.3 being the exact
mean, while 62 an approximate mean.
Sometimes when the numerical values of x
i
and f
i
are large, finding the product
of x
i
and f
i
becomes tedious and time consuming. So, for such situations, let us think of
a method of reducing these calculations.
We can do nothing with the f
i
’s, but we can change each x
i
to a smaller number
so that our calculations become easy. How do we do this? What about subtracting a
fixed number from each of these x
i
’s? Let us try this method.
The first step is to choose one among the x
i
’s as the assumed mean, and denote
it by ‘a’. Also, to further reduce our calculation work, we may take ‘a’ to be that x
i
which lies in the centre of x
1
, x
2
, . . ., x
n
. So, we can choose a = 47.5 or a = 62.5. Let
us choose a = 47.5.
The next step is to find the difference d
i
between a and each of the x
i
’s, that is,
the deviation of ‘a’ from each of the x
i
’s.
i.e., d
i
= x
i
– a = x
i
– 47.5
Rationalised 2023-24
ST A TISTICS 175
The third step is to find the product of d
i
with the corresponding f
i
, and take the sum
of all the f
i
d
i
’s. The calculations are shown in Table 13.4.
Table 13.4
Class interval Number of Class mark d
i
= x
i
– 47.5 f
i
d
i
students (f
i
) (x
i
)
10 - 25 2 17.5 –30 –60
25 - 40 3 32.5 –15 –45
40 - 55 7 47.5 0 0
55 - 70 6 62.5 15 90
70 - 85 6 77.5 30 180
85 - 100 6 92.5 45 270
Total Sf
i
= 30 Sf
i
d
i
= 435
So, from Table 13.4, the mean of the deviations, d =
i i
i
f d
f
S
S
.
Now, let us find the relation between d and x .
Since in obtaining d
i
, we subtracted ‘a’ from each x
i
, so, in order to get the mean
x , we need to add ‘a’ to d . This can be explained mathematically as:
Mean of deviations, d =
i i
i
f d
f
S
S
So,
d
=
( )
i i
i
f x a
f
S -
S
=
i i i
i i
f x f a
f f
S S
-
S S
=
i
i
f
x a
f
S
-
S
= x a -
So,
x
= a + d
i.e.,
x
=
i i
i
f d
a
f
S
+
S
Rationalised 2023-24
```

## Mathematics (Maths) Class 10

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## FAQs on NCERT Textbook: Statistics - Mathematics (Maths) Class 10

 1. What is the importance of statistics in our daily lives?
Ans. Statistics plays a crucial role in our daily lives as it helps in organizing, analyzing, and interpreting data. It enables us to make informed decisions, evaluate the outcomes of various scenarios, and understand the patterns and trends in different fields such as economics, medicine, and social sciences. By using statistical techniques, we can draw conclusions, identify relationships, and solve real-world problems efficiently.
 2. How can statistics be applied in business management?
Ans. Statistics find wide applications in business management. It helps in analyzing market trends, understanding customer preferences, and making data-driven decisions. Statistical techniques like regression analysis, hypothesis testing, and forecasting assist in assessing the impact of various factors on business performance, identifying areas of improvement, and optimizing resources. By using statistics, businesses can identify patterns, estimate demand, and implement effective strategies for growth and profitability.
 3. What are the different types of data used in statistics?
Ans. In statistics, data can be broadly classified into two types: qualitative and quantitative. Qualitative data represents attributes or characteristics and is non-numerical in nature. Examples include gender, color, or occupation. On the other hand, quantitative data is numerical and can be measured. It further consists of two subtypes: discrete data (countable) such as the number of students in a class, and continuous data (measurable) such as weight or height. Understanding the type of data is essential for selecting appropriate statistical techniques for analysis.
 4. How is probability related to statistics?
Ans. Probability is a branch of mathematics that deals with the likelihood of events occurring. It forms the foundation of statistical reasoning. In statistics, probability is used to measure uncertainty and make predictions based on data. By applying probability concepts, statisticians can estimate the likelihood of certain outcomes, construct sampling distributions, and determine confidence intervals. Probability theory provides a mathematical framework to analyze and interpret data, allowing us to make informed decisions and draw reliable conclusions.
 5. What are the key steps involved in the statistical analysis of data?
Ans. The statistical analysis of data typically involves the following key steps: 1. Data Collection: Gathering relevant data through surveys, experiments, or observations. 2. Data Cleaning: Checking for errors, missing values, and outliers in the collected data and addressing them appropriately. 3. Data Description: Summarizing the data using measures such as mean, median, mode, and standard deviation to gain insights into the data set. 4. Data Analysis: Applying statistical techniques such as hypothesis testing, regression analysis, or ANOVA to draw conclusions and identify patterns in the data. 5. Interpretation and Conclusion: Interpreting the results obtained from the analysis and drawing meaningful conclusions based on statistical evidence. These steps help researchers and analysts in understanding the data, making valid inferences, and presenting the findings accurately.

## Mathematics (Maths) Class 10

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