260

MATHEMATICS

STATISTICS

14

There are lies, damned lies and statistics.
— by Disraeli

14.1 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.

14.2 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
x1, x2,. . ., xn are observations with respective frequencies f1, f2, . . ., fn, then this
means observation x1 occurs f1 times, x2 occurs f2 times, and so on.
Now, the sum of the values of all the observations = f1x1 + f2x2 + . . . + fnxn, and
the number of observations = f1 + f2 + . . . + fn.
So, the mean x of the data is given by
f x + f 2 x2 + L + f n xn
x = 1 1
f1 + f 2 + L + f n
Recall that we can write this in short form by using the Greek letter Σ (capital
sigma) which means summation. That is,

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261
n

∑fx
x =

i =1
n

i i

∑f
i =1

i

Σ fi xi
which, more briefly, is written as x =
, if it is understood that i varies from
Σ fi
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
(x i )
Number of
student ( fi)

1

20

36

40

50

56

60

70

72

80

88

92 95

1

3

4

3

2

4

4

1

1

2

3

1

Solution: Recall that to find the mean marks, we require the product of each xi with
the corresponding frequency fi. So, let us put them in a column as shown in Table 14.1.
Table 14.1

.

Marks obtained (xi )

Number of students ( fi )

10
20
36
40
50
56
60
70
72
80
88
92
95

1
1
3
4
3
2
4
4
1
1
2
3
1

Total

Σfi = 30

f ix i
10
20
108
160
150
112
240
280
72
80
176
276
95
Σfixi = 1779

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MATHEMATICS

x=

Now,

1779
Σ fi xi
=
= 59.3
30
Σ fi

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 classinterval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped
frequency distribution table (see Table 14.2).
Table 14.2
Class interval
Number of students

10 - 25

25 - 40

40 - 55

55 - 70

70 - 85

85 - 100

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 classinterval 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

10 25
, i.e.,
2
17.5. Similarly, we can find the class marks of the remaining class intervals. We put
them in Table 14.3. These class marks serve as our xi’s. Now, in general, for the ith
class interval, we have the frequency fi corresponding to the class mark xi. We can
now proceed to compute the mean in the same manner as in Example 1.
With reference to Table 14.2, for the class 10-25, the class mark is

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263

Table 14.3
Class interval

Number of students ( fi )

Class mark (xi )

fi xi

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

Σ fi = 30

Σ fi xi = 1860.0

The sum of the values in the last column gives us Σ fi xi. So, the mean x of the
given data is given by
Σfi xi 1860.0
x =
=
= 62
30
Σ fi
This new method of finding the mean is known as the Direct Method.
We observe that Tables 14.1 and 14.3 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 14.3, 59.3 being the exact
mean, while 62 an approximate mean.
Sometimes when the numerical values of xi and fi are large, finding the product
of xi and fi 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 fi’s, but we can change each xi 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 xi’s? Let us try this method.
The first step is to choose one among the xi’s as the assumed mean, and denote
it by ‘a’. Also, to further reduce our calculation work, we may take ‘a’ to be that xi
which lies in the centre of x1, x2, . . ., xn. 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 di between a and each of the xi’s, that is,
the deviation of ‘a’ from each of the xi’s.
i.e.,

di = xi – a = xi – 47.5

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MATHEMATICS

The third step is to find the product of di with the corresponding fi, and take the sum
of all the fi di’s. The calculations are shown in Table 14.4.
Table 14.4
Class interval
10 - 25
25 - 40
40 - 55
55 - 70
70 - 85
85 - 100
Total

Number of
students ( fi )

Class mark
(xi )

di = xi – 47.5

fid i

2
3
7
6
6
6
Σfi = 30

17.5
32.5
47.5
62.5
77.5
92.5

–30
–15
0
15
30
45

–60
–45
0
90
180
270
Σfidi = 435

So, from Table 14.4, the mean of the deviations, d =

Σfi di
.
Σfi

Now, let us find the relation between d and x .
Since in obtaining di, we subtracted ‘a’ from each xi, 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 =

So,

d =

Σfi di
Σfi

Σfi ( xi − a )
Σfi
Σfi xi Σf i a
−
=
Σf i
Σfi
= x −a

Σfi
Σfi

= x−a
So,

x = a+ d

i.e.,

x = a+

Σfi di
Σfi

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265

Substituting the values of a, Σfidi and Σfi from Table 14.4, we get
x = 47.5 +

435
= 47.5 + 14.5 = 62 .
30

Therefore, the mean of the marks obtained by the students is 62.
The method discussed above is called the Assumed Mean Method.
Activity 1 : From the Table 14.3 find the mean by taking each of xi (i.e., 17.5, 32.5,
and so on) as ‘a’. What do you observe? You will find that the mean determined in
each case is the same, i.e., 62. (Why ?)
So, we can say that the value of the mean obtained does not depend on the
choice of ‘a’.
Observe that in Table 14.4, the values in Column 4 are all multiples of 15. So, if
we divide the values in the entire Column 4 by 15, we would get smaller numbers to
multiply with fi. (Here, 15 is the class size of each class interval.)
So, let ui =

xi − a
, where a is the assumed mean and h is the class size.
h

Now, we calculate ui in this way and continue as before (i.e., find fi ui and
then Σ fi ui). Taking h = 15, let us form Table 14.5.
Table 14.5
Class interval
10 - 25
25 - 40
40 - 55
55 - 70
70 - 85
85 - 100
Total

Let

fi

xi

2
3
7
6
6
6

17.5
32.5
47.5
62.5
77.5
92.5

di = xi – a
–30
–15
0
15
30
45

ui =

xi – a
h
–2
–1
0
1
2
3

f iu i
–4
–3
0
6
12
18
Σfiui = 29

Σfi = 30
u =

Σfi ui
Σf i

Here, again let us find the relation between u and x .

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MATHEMATICS

We have,

Therefore,

ui =

u =

xi − a
h
Σf i

( xi − a )
1 ⎡ Σ f x − a Σf i ⎤
h
= ⎢ i i
⎥
h⎣
Σfi
Σf i
⎦

=

Σf ⎤
1 ⎡ Σfi xi
−a i⎥
⎢
h ⎣ Σf i
Σfi ⎦

=

1
[ x − a]
h

hu = x − a

So,

x = a + hu

i.e.,

⎛ Σf u ⎞
x = a + h⎜ i i ⎟
⎝ Σfi ⎠
Now, substituting the values of a, h, Σfiui and Σfi from Table 14.5, we get
So,

⎛ 29 ⎞
x = 47.5 + 15 × ⎜ ⎟
⎝ 30 ⎠
= 47.5 + 14.5 = 62
So, the mean marks obtained by a student is 62.
The method discussed above is called the Step-deviation method.
We note that :
z

the step-deviation method will be convenient to apply if all the di’s have a
common factor.

z

The mean obtained by all the three methods is the same.

z

The assumed mean method and step-deviation method are just simplified
forms of the direct method.
The formula x = a + h u still holds if a and h are not as given above, but are

z

xi − a
.
h
Let us apply these methods in another example.
any non-zero numbers such that ui =

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267

Example 2 : The table below gives the percentage distribution of female teachers in
the primary schools of rural areas of various states and union territories (U.T.) of
India. Find the mean percentage of female teachers by all the three methods discussed
in this section.
Percentage of
15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85
female teachers
Number of
States/U.T.

6

11

7

4

4

2

1

Source : Seventh All India School Education Survey conducted by NCERT
Solution : Let us find the class marks, xi, of each class, and put them in a column
(see Table 14.6):
Table 14.6
Percentage of female

Number of

teachers

States /U.T. ( f i )

15 - 25

6

20

25 - 35

11

30

35 - 45

7

40

45 - 55

4

50

55 - 65

4

60

65 - 75

2

70

75 - 85

1

80

Here we take a = 50, h = 10, then di = xi – 50 and ui
We now find di and ui and put them in Table 14.7.

xi

xi 50
.
10

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MATHEMATICS

Table 14.7
Percentage of
female
teachers

15 - 25
25 - 35
35 - 45
45 - 55
55 - 65
65 - 75
75 - 85
Total

Number of
states/U.T.
( fi )

6
11
7
4
4
2
1

xi

di = xi – 50

20
30
40
50
60
70
80

–30
–20
–10
0
10
20
30

ui =

xi −50
10

fixi

–3
–2
–1
0
1
2
3

120 –180 –18
330 –220 –22
280 –70 –7
200
0
0
240
40
4
140
40
4
80
30
3

35

fidi

fiui

1390 –360 –36

From the table above, we obtain Σfi = 35, Σfixi = 1390,
Σfi di = – 360,

Σfiui = –36.

Σf i xi 1390
=
= 39.71
Σfi
35
Using the assumed mean method,

Using the direct method, x =

Σfi di
( −360)
= 39.71
= 50 +
Σfi
35
Using the step-deviation method,
x = a+

⎛ Σf u ⎞
⎛ – 36 ⎞
x = a + ⎜ i i ⎟ × h = 50 + ⎜
⎟ × 10 = 39.71
⎝ 35 ⎠
⎝ Σfi ⎠
Therefore, the mean percentage of female teachers in the primary schools of
rural areas is 39.71.
Remark : The result obtained by all the three methods is the same. So the choice of
method to be used depends on the numerical values of xi and fi. If xi and fi are
sufficiently small, then the direct method is an appropriate choice. If xi and fi are
numerically large numbers, then we can go for the assumed mean method or
step-deviation method. If the class sizes are unequal, and xi are large numerically, we
can still apply the step-deviation method by taking h to be a suitable divisor of all the di’s.

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269

Example 3 : The distribution below shows the number of wickets taken by bowlers in
one-day cricket matches. Find the mean number of wickets by choosing a suitable
method. What does the mean signify?
Number of
wickets

20 - 60

Number of
bowlers

7

60 - 100 100 - 150 150 - 250 250 - 350 350 - 450
5

16

12

2

3

Solution : Here, the class size varies, and the xi,s are large. Let us still apply the stepdeviation method with a = 200 and h = 20. Then, we obtain the data as in Table 14.8.
Table 14.8
ui =

di
20

Number of
wickets
taken

Number of
bowlers
( fi )

xi

di = xi – 200

20 - 60

7

40

–160

–8

–56

60 - 100

5

80

–120

–6

–30

100 - 150

16

125

–75

–3.75

–60

150 - 250

12

200

0

0

0

250 - 350

2

300

100

5

10

350 - 450

3

400

200

10

30

Total

45

So, u =

u i fi

–106

⎛ −106 ⎞
−106
⋅ Therefore, x = 200 + 20 ⎜
⎟ = 200 – 47.11 = 152.89.
⎝ 45 ⎠
45

This tells us that, on an average, the number of wickets taken by these 45 bowlers
in one-day cricket is 152.89.
Now, let us see how well you can apply the concepts discussed in this section!

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MATHEMATICS

Activity 2 :
Divide the students of your class into three groups and ask each group to do one of the
following activities.
1. Collect the marks obtained by all the students of your class in Mathematics in the
latest examination conducted by your school. Form a grouped frequency distribution
of the data obtained.
2. Collect the daily maximum temperatures recorded for a period of 30 days in your
city. Present this data as a grouped frequency table.
3. Measure the heights of all the students of your class (in cm) and form a grouped
frequency distribution table of this data.
After all the groups have collected the data and formed grouped frequency
distribution tables, the groups should find the mean in each case by the method which
they find appropriate.

EXERCISE 14.1
1. A survey was conducted by a group of students as a part of their environment awareness
programme, in which they collected the following data regarding the number of plants in
20 houses in a locality. Find the mean number of plants per house.
Number of plants

0-2

2-4

4-6

6-8

8 - 10

10 - 12

12 - 14

Number of houses

1

2

1

5

6

2

3

Which method did you use for finding the mean, and why?
2. Consider the following distribution of daily wages of 50 workers of a factory.
Daily wages (in Rs)

100 - 120

120 - 140

140 - 160

160 - 180

180 - 200

Number of workers

12

14

8

6

10

Find the mean daily wages of the workers of the factory by using an appropriate method.
3. The following distribution shows the daily pocket allowance of children of a locality.
The mean pocket allowance is Rs 18. Find the missing frequency f.
Daily pocket
allowance (in Rs)
Number of children

11 - 13

13 - 15

15 - 17

17 - 19

19 - 21

21 - 23

23 - 25

7

6

9

13

f

5

4

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271

4. Thirty women were examined in a hospital by a doctor and the number of heart beats per
minute were recorded and summarised as follows. Find the mean heart beats per minute
for these women, choosing a suitable method.
Number of heart beats 65 - 68
per minute
Number of women

2

68 - 71

71 - 74

74 - 77

77 - 80

80 - 83

83 - 86

4

3

8

7

4

2

5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These
boxes contained varying number of mangoes. The following was the distribution of
mangoes according to the number of boxes.
Number of mangoes

50 - 52

53 - 55

56 - 58

59 - 61

62 - 64

15

110

135

115

25

Number of boxes

Find the mean number of mangoes kept in a packing box. Which method of finding
the mean did you choose?
6. The table below shows the daily expenditure on food of 25 households in a locality.
Daily expenditure
(in Rs)
Number of
households

100 - 150

150 - 200

200 - 250

250 - 300

300 - 350

4

5

12

2

2

Find the mean daily expenditure on food by a suitable method.
7. To find out the concentration of SO2 in the air (in parts per million, i.e., ppm), the data
was collected for 30 localities in a certain city and is presented below:
Concentration of SO2 (in ppm)

Frequency

0.00 - 0.04

4

0.04 - 0.08

9

0.08 - 0.12

9

0.12 - 0.16

2

0.16 - 0.20

4

0.20 - 0.24

2

Find the mean concentration of SO2 in the air.

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MATHEMATICS

8. A class teacher has the following absentee record of 40 students of a class for the whole
term. Find the mean number of days a student was absent.
Number of
days

0-6

Number of
students

11

6 - 10 10 - 14 14 - 20 20 - 28 28 - 38 38 - 40
10

7

4

4

3

1

9. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean
literacy rate.
Literacy rate (in %)

45 - 55

55 - 65

65 - 75

75 - 85

85 - 95

3

10

11

8

3

Number of cities

14.3 Mode of Grouped Data
Recall from Class IX, a mode is that value among the observations which occurs most
often, that is, the value of the observation having the maximum frequency. Further, we
discussed finding the mode of ungrouped data. Here, we shall discuss ways of obtaining
a mode of grouped data. It is possible that more than one value may have the same
maximum frequency. In such situations, the data is said to be multimodal. Though
grouped data can also be multimodal, we shall restrict ourselves to problems having a
single mode only.
Let us first recall how we found the mode for ungrouped data through the following
example.
Example 4 : The wickets taken by a bowler in 10 cricket matches are as follows:
2

6

4

5

0

2

1

3

2

3

Find the mode of the data.
Solution : Let us form the frequency distribution table of the given data as follows:
Number of
wickets

0

1

2

3

4

5

6

Number of
matches

1

1

3

2

1

1

1

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273

Clearly, 2 is the number of wickets taken by the bowler in the maximum number
(i.e., 3) of matches. So, the mode of this data is 2.
In a grouped frequency distribution, it is not possible to determine the mode by
looking at the frequencies. Here, we can only locate a class with the maximum
frequency, called the modal class. The mode is a value inside the modal class, and is
given by the formula:

⎛
f1 − f0 ⎞
Mode = l + ⎜
⎟×h
⎝ 2 f1 − f 0 − f 2 ⎠
where l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f1 = frequency of the modal class,
f0 = frequency of the class preceding the modal class,
f2 = frequency of the class succeeding the modal class.
Let us consider the following examples to illustrate the use of this formula.
Example 5 : A survey conducted on 20 households in a locality by a group of students
resulted in the following frequency table for the number of family members in a
household:
Family size

1-3

3-5

5-7

7-9

9 - 11

Number of
families

7

8

2

2

1

Find the mode of this data.
Solution : Here the maximum class frequency is 8, and the class corresponding to this
frequency is 3 – 5. So, the modal class is 3 – 5.
Now
modal class = 3 – 5, lower limit (l ) of modal class = 3, class size (h) = 2
frequency ( f1 ) of the modal class = 8,
frequency ( f0 ) of class preceding the modal class = 7,
frequency ( f2 ) of class succeeding the modal class = 2.
Now, let us substitute these values in the formula :

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MATHEMATICS

⎛
f1 − f0 ⎞
Mode = l + ⎜
⎟×h
⎝ 2 f1 − f 0 − f 2 ⎠
⎛
8−7
= 3+⎜
⎝2×8− 7 −

⎞
2
⎟ × 2 = 3 + = 3.286
2⎠
7

Therefore, the mode of the data above is 3.286.
Example 6 : The marks distribution of 30 students in a mathematics examination are
given in Table 14.3 of Example 1. Find the mode of this data. Also compare and
interpret the mode and the mean.
Solution : Refer to Table 14.3 of Example 1. Since the maximum number of students
(i.e., 7) have got marks in the interval 40 - 55, the modal class is 40 - 55. Therefore,
the lower limit ( l ) of the modal class = 40,
the class size ( h) = 15,
the frequency ( f1 ) of modal class = 7,
the frequency ( f0 ) of the class preceding the modal class = 3,
the frequency ( f2 ) of the class succeeding the modal class = 6.
Now, using the formula:

⎛
f1 − f0 ⎞
Mode = l + ⎜
⎟ × h,
⎝ 2 f1 − f 0 − f 2 ⎠
we get

⎛ 7−3 ⎞
Mode = 40 + ⎜
⎟ × 15 = 52
⎝ 14 − 6 − 3 ⎠

So, the mode marks is 52.
Now, from Example 1, you know that the mean marks is 62.
So, the maximum number of students obtained 52 marks, while on an average a
student obtained 62 marks.
Remarks :
1. In Example 6, the mode is less than the mean. But for some other problems it may
be equal or more than the mean also.
2. It depends upon the demand of the situation whether we are interested in finding the
average marks obtained by the students or the average of the marks obtained by most

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275

of the students. In the first situation, the mean is required and in the second situation,
the mode is required.
Activity 3 : Continuing with the same groups as formed in Activity 2 and the situations
assigned to the groups. Ask each group to find the mode of the data. They should also
compare this with the mean, and interpret the meaning of both.
Remark : The mode can also be calculated for grouped data with unequal class sizes.
However, we shall not be discussing it.

EXERCISE 14.2
1. The following table shows the ages of the patients admitted in a hospital during a year:
Age (in years)

5 - 15

15 - 25

25 - 35

35 - 45

45 - 55

55 - 65

6

11

21

23

14

5

Number of patients

Find the mode and the mean of the data given above. Compare and interpret the two
measures of central tendency.
2. The following data gives the information on the observed lifetimes (in hours) of 225
electrical components :
Lifetimes (in hours)

0 - 20

20 - 40

40 - 60

60 - 80

80 - 100

100 - 120

10

35

52

61

38

29

Frequency

Determine the modal lifetimes of the components.
3. The following data gives the distribution of total monthly household expenditure of 200
families of a village. Find the modal monthly expenditure of the families. Also, find the
mean monthly expenditure :
Expenditure (in Rs)

Number of families

1000 - 1500
1500 - 2000
2000 - 2500
2500 - 3000
3000 - 3500
3500 - 4000
4000 - 4500
4500 - 5000

24
40
33
28
30
22
16
7

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MATHEMATICS

4. The following distribution gives the state-wise teacher-student ratio in higher
secondary schools of India. Find the mode and mean of this data. Interpret the two
measures.
Number of students per teacher

Number of states / U .T.

15 - 20

3

20 - 25

8

25 - 30

9

30 - 35

10

35 - 40

3

40 - 45

0

45 - 50

0

50 - 55

2

5. The given distribution shows the number of runs scored by some top batsmen of the
world in one-day international cricket matches.
Runs scored

Number of batsmen

3000 - 4000

4

4000 - 5000

18

5000 - 6000

9

6000 - 7000

7

7000 - 8000

6

8000 - 9000

3

9000 - 10000

1

10000 - 11000

1

Find the mode of the data.
6. A student noted the number of cars passing through a spot on a road for 100
periods each of 3 minutes and summarised it in the table given below. Find the mode
of the data :
Number of cars
Frequency

0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80
7

14

13

12

20

11

15

8

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277

14.4 Median of Grouped Data
As you have studied in Class IX, the median is a measure of central tendency which
gives the value of the middle-most observation in the data. Recall that for finding the
median of ungrouped data, we first arrange the data values of the observations in

⎛ n + 1⎞
ascending order. Then, if n is odd, the median is the ⎜
⎟ th observation. And, if n
⎝ 2 ⎠
is even, then the median will be the average of the

n
⎛n
⎞
th and the ⎜ + 1⎟ th observations.
2
2
⎝
⎠

Suppose, we have to find the median of the following data, which gives the
marks, out of 50, obtained by 100 students in a test :
Marks obtained

20

29

28

33

42

38

43

25

Number of students

6

28

24

15

2

4

1

20

First, we arrange the marks in ascending order and prepare a frequency table as
follows :
Table 14.9
Marks obtained

Number of students
(Frequency)

20

6

25

20

28

24

29

28

33

15

38

4

42

2

43

1

Total

100

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MATHEMATICS

Here n = 100, which is even. The median will be the average of the

n th and the
2

⎞
⎛n
⎜ + 1 ⎟ th observations, i.e., the 50th and 51st observations. To find these
⎝2
⎠
observations, we proceed as follows:
Table 14.10
Marks obtained

Number of students

20

6

upto 25

6 + 20 = 26

upto 28

26 + 24 = 50

upto 29

50 + 28 = 78

upto 33

78 + 15 = 93

upto 38

93 + 4 = 97

upto 42

97 + 2 = 99

upto 43

99 + 1 = 100

Now we add another column depicting this information to the frequency table
above and name it as cumulative frequency column.
Table 14.11
Marks obtained

Number of students

Cumulative frequency

20

6

6

25

20

26

28

24

50

29

28

78

33

15

93

38

4

97

42

2

99

43

1

100

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279

From the table above, we see that:
50th observaton is 28

(Why?)

51st observation is 29
So,

Median =

28 + 29
= 28.5
2

Remark : The part of Table 14.11 consisting Column 1 and Column 3 is known as
Cumulative Frequency Table. The median marks 28.5 conveys the information that
about 50% students obtained marks less than 28.5 and another 50% students obtained
marks more than 28.5.
Now, let us see how to obtain the median of grouped data, through the following
situation.
Consider a grouped frequency distribution of marks obtained, out of 100, by 53
students, in a certain examination, as follows:
Table 14.12
Marks

Number of students

0 - 10

5

10 - 20

3

20 - 30

4

30 - 40

3

40 - 50

3

50 - 60

4

60 - 70

7

70 - 80

9

80 - 90

7

90 - 100

8

From the table above, try to answer the following questions:
How many students have scored marks less than 10? The answer is clearly 5.

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MATHEMATICS

How many students have scored less than 20 marks? Observe that the number
of students who have scored less than 20 include the number of students who have
scored marks from 0 - 10 as well as the number of students who have scored marks
from 10 - 20. So, the total number of students with marks less than 20 is 5 + 3, i.e., 8.
We say that the cumulative frequency of the class 10 -20 is 8.
Similarly, we can compute the cumulative frequencies of the other classes, i.e.,
the number of students with marks less than 30, less than 40, . . ., less than 100. We
give them in Table 14.13 given below:
Table 14.13
Marks obtained

Number of students
(Cumulative frequency)

Less than 10

5

Less than 20

5+3=8

Less than 30

8 + 4 = 12

Less than 40

12 + 3 = 15

Less than 50

15 + 3 = 18

Less than 60

18 + 4 = 22

Less than 70

22 + 7 = 29

Less than 80

29 + 9 = 38

Less than 90

38 + 7 = 45

Less than 100

45 + 8 = 53

The distribution given above is called the cumulative frequency distribution of
the less than type. Here 10, 20, 30, . . . 100, are the upper limits of the respective
class intervals.
We can similarly make the table for the number of students with scores, more
than or equal to 0, more than or equal to 10, more than or equal to 20, and so on. From
Table 14.12, we observe that all 53 students have scored marks more than or equal to
0. Since there are 5 students scoring marks in the interval 0 - 10, this means that there
are 53 – 5 = 48 students getting more than or equal to 10 marks. Continuing in the
same manner, we get the number of students scoring 20 or above as 48 – 3 = 45, 30 or
above as 45 – 4 = 41, and so on, as shown in Table 14.14.

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281

Table 14.14
Marks obtained

Number of students
(Cumulative frequency)

More than or equal to 0
More than or equal to 10
More than or equal to 20
More than or equal to 30
More than or equal to 40
More than or equal to 50
More than or equal to 60
More than or equal to 70
More than or equal to 80
More than or equal to 90

53
53 – 5 = 48
48 – 3 = 45
45 – 4 = 41
41 – 3 = 38
38 – 3 = 35
35 – 4 = 31
31 – 7 = 24
24 – 9 = 15
15 – 7 = 8

The table above is called a cumulative frequency distribution of the more
than type. Here 0, 10, 20, . . ., 90 give the lower limits of the respective class intervals.
Now, to find the median of grouped data, we can make use of any of these
cumulative frequency distributions.
Let us combine Tables 14.12 and 14.13 to get Table 14.15 given below:
Table 14.15
Marks

Number of students ( f )

Cumulative frequency (cf )

0 - 10
10 - 20
20 - 30
30 - 40
40 - 50
50 - 60
60 - 70
70 - 80
80 - 90
90 - 100

5
3
4
3
3
4
7
9
7
8

5
8
12
15
18
22
29
38
45
53

Now in a grouped data, we may not be able to find the middle observation by
looking at the cumulative frequencies as the middle observation will be some value in

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MATHEMATICS

a class interval. It is, therefore, necessary to find the value inside a class that divides
the whole distribution into two halves. But which class should this be?

n
.
2
We now locate the class whose cumulative frequency is greater than (and nearest to)
n
n
⋅ This is called the median class. In the distribution above, n = 53. So,
= 26.5.
2
2
Now 60 – 70 is the class whose cumulative frequency 29 is greater than (and nearest
n
, i.e., 26.5.
to)
2
To find this class, we find the cumulative frequencies of all the classes and

Therefore, 60 – 70 is the median class.
After finding the median class, we use the following formula for calculating the
median.
⎛n
⎞
⎜ 2 − cf ⎟
Median = l + ⎜
⎟ × h,
⎜⎜ f ⎟⎟
⎝
⎠

where

l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).

n
= 26.5, l = 60, cf = 22, f = 7, h = 10
2
in the formula above, we get

Substituting the values

⎛ 26.5 − 22 ⎞
Median = 60 + ⎜
⎟ × 10
7
⎝
⎠
= 60 +

45
7

= 66.4
So, about half the students have scored marks less than 66.4, and the other half have
scored marks more 66.4.

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283

Example 7 : A survey regarding the heights (in cm) of 51 girls of Class X of a school
was conducted and the following data was obtained:
Height (in cm)

Number of girls

Less than 140

4

Less than 145

11

Less than 150

29

Less than 155

40

Less than 160

46

Less than 165

51

Find the median height.
Solution : To calculate the median height, we need to find the class intervals and their
corresponding frequencies.
The given distribution being of the less than type, 140, 145, 150, . . ., 165 give the
upper limits of the corresponding class intervals. So, the classes should be below 140,
140 - 145, 145 - 150, . . ., 160 - 165. Observe that from the given distribution, we find
that there are 4 girls with height less than 140, i.e., the frequency of class interval
below 140 is 4. Now, there are 11 girls with heights less than 145 and 4 girls with
height less than 140. Therefore, the number of girls with height in the interval
140 - 145 is 11 – 4 = 7. Similarly, the frequency of 145 - 150 is 29 – 11 = 18, for
150 - 155, it is 40 – 29 = 11, and so on. So, our frequency distribution table with the
given cumulative frequencies becomes:
Table 14.16
Class intervals

Frequency

Cumulative frequency

Below 140
140 - 145
145 - 150
150 - 155
155 - 160
160 - 165

4
7
18
11
6
5

4
11
29
40
46
51

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Now

MATHEMATICS

n = 51. So,

n 51
= = 25.5 . This observation lies in the class 145 - 150. Then,
2 2

l (the lower limit) = 145,
cf (the cumulative frequency of the class preceding 145 - 150) = 11,
f (the frequency of the median class 145 - 150) = 18,
h (the class size) = 5.
⎛n
⎞
⎜ 2 − cf ⎟
⎟ × h , we have
Using the formula, Median = l + ⎜
f
⎜
⎟
⎝
⎠

⎛ 25.5 − 11 ⎞
Median = 145 + ⎜
⎟×5
⎝ 18 ⎠

72.5
= 149.03.
18
So, the median height of the girls is 149.03 cm.
= 145 +

This means that the height of about 50% of the girls is less than this height, and
50% are taller than this height.
Example 8 : The median of the following data is 525. Find the values of x and y, if the
total frequency is 100.
Class interval

Frequency

0 - 100
100 - 200
200 - 300
300 - 400
400 - 500
500 - 600
600 - 700
700 - 800
800 - 900
900 - 1000

2
5
x
12
17
20
y
9
7
4

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285

Solution :
Class intervals

Frequency

Cumulative frequency

0 - 100

2

2

100 - 200

5

7

200 - 300

x

7+x

300 - 400

12

19 + x

400 - 500

17

36 + x

500 - 600

20

56 + x

600 - 700

y

56 + x + y

700 - 800

9

65 + x + y

800 - 900

7

72 + x + y

900 - 1000

4

76 + x + y

It is given that n = 100
So,

76 + x + y = 100,

i.e.,

x + y = 24

(1)

The median is 525, which lies in the class 500 – 600
So,

l = 500,

f = 20,

cf = 36 + x, h = 100

⎛n
⎜ 2 − cf
+
l
Median =
⎜
⎜ f
⎝

Using the formula :

⎞
⎟
⎟ h, we get
⎟
⎠

⎛ 50 − 36 − x ⎞
525 = 500 + ⎜
⎟ × 100
20
⎝
⎠
i.e.,

525 – 500 = (14 – x) × 5

i.e.,

25 = 70 – 5x

i.e.,

5x = 70 – 25 = 45

So,

x= 9
Therefore, from (1), we get

i.e.,

9 + y = 24
y = 15

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MATHEMATICS

Now, that you have studied about all the three measures of central tendency, let
us discuss which measure would be best suited for a particular requirement.
The mean is the most frequently used measure of central tendency because it
takes into account all the observations, and lies between the extremes, i.e., the largest
and the smallest observations of the entire data. It also enables us to compare two or
more distributions. For example, by comparing the average (mean) results of students
of different schools of a particular examination, we can conclude which school has a
better performance.
However, extreme values in the data affect the mean. For example, the mean of
classes having frequencies more or less the same is a good representative of the data.
But, if one class has frequency, say 2, and the five others have frequency 20, 25, 20,
21, 18, then the mean will certainly not reflect the way the data behaves. So, in such
cases, the mean is not a good representative of the data.
In problems where individual observations are not important, and we wish to find
out a ‘typical’ observation, the median is more appropriate, e.g., finding the typical
productivity rate of workers, average wage in a country, etc. These are situations
where extreme values may be there. So, rather than the mean, we take the median as
a better measure of central tendency.
In situations which require establishing the most frequent value or most popular
item, the mode is the best choice, e.g., to find the most popular T.V. programme being
watched, the consumer item in greatest demand, the colour of the vehicle used by
most of the people, etc.
Remarks :
1. There is a empirical relationship between the three measures of central tendency :
3 Median = Mode + 2 Mean
2. The median of grouped data with unequal class sizes can also be calculated. However,
we shall not discuss it here.

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287

EXERCISE 14.3
1. The following frequency distribution gives the monthly consumption of electricity of
68 consumers of a locality. Find the median, mean and mode of the data and compare
them.
Monthly consumption (in units)

Number of consumers

65 - 85

4

85 - 105

5

105 - 125

13

125 - 145

20

145 - 165

14

165 - 185

8

185 - 205

4

2. If the median of the distribution given below is 28.5, find the values of x and y.
Class interval

Frequency

0 - 10

5

10 - 20

x

20 - 30

20

30 - 40

15

40 - 50

y

50 - 60

5

Total

60

3. A life insurance agent found the following data for distribution of ages of 100 policy
holders. Calculate the median age, if policies are given only to persons having age 18
years onwards but less than 60 year.

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MATHEMATICS
Age (in years)

Number of policy holders

Below 20

2

Below 25

6

Below 30

24

Below 35

45

Below 40

78

Below 45

89

Below 50

92

Below 55

98

Below 60

100

4. The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and
the data obtained is represented in the following table :
Length (in mm)

Number of leaves

118 - 126

3

127 - 135

5

136 - 144

9

145 - 153

12

154 - 162

5

163 - 171

4

172 - 180

2

Find the median length of the leaves.
(Hint : The data needs to be converted to continuous classes for finding the median,
since the formula assumes continuous classes. The classes then change to
117.5 - 126.5, 126.5 - 135.5, . . ., 171.5 - 180.5.)

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289

5. The following table gives the distribution of the life time of 400 neon lamps :
Life time (in hours)

Number of lamps

1500 - 2000

14

2000 - 2500

56

2500 - 3000

60

3000 - 3500

86

3500 - 4000

74

4000 - 4500

62

4500 - 5000

48

Find the median life time of a lamp.
6. 100 surnames were randomly picked up from a local telephone directory and the
frequency distribution of the number of letters in the English alphabets in the surnames
was obtained as follows:
Number of letters
Number of surnames

1-4

4-7

7 - 10

10 - 13

13 - 16

16 - 19

6

30

40

16

4

4

Determine the median number of letters in the surnames. Find the mean number of
letters in the surnames? Also, find the modal size of the surnames.
7. The distribution below gives the weights of 30 students of a class. Find the median
weight of the students.
Weight (in kg)
Number of students

40 - 45 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75
2

3

8

6

6

3

2

14.5 Graphical Representation of Cumulative Frequency Distribution
As we all know, pictures speak better than words. A graphical representation helps us
in understanding given data at a glance. In Class IX, we have represented the data
through bar graphs, histograms and frequency polygons. Let us now represent a
cumulative frequency distribution graphically.
For example, let us consider the cumulative frequency distribution given in
Table 14.13.

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MATHEMATICS

Recall that the values 10, 20, 30,
. . ., 100 are the upper limits of the
respective class intervals. To represent
the data in the table graphically, we mark
the upper limits of the class intervals on
the horizontal axis (x-axis) and their
corresponding cumulative frequencies
on the vertical axis ( y-axis), choosing a
convenient scale. The scale may not be
the same on both the axis. Let us now
plot the points corresponding to the
ordered pairs given by (upper limit,
corresponding cumulative frequency),
Fig. 14.1
i.e., (10, 5), (20, 8), (30, 12), (40, 15),
(50, 18), (60, 22), (70, 29), (80, 38), (90, 45), (100, 53) on a graph paper and join them
by a free hand smooth curve. The curve we get is called a cumulative frequency
curve, or an ogive (of the less than type). (See Fig. 14.1)
The term ‘ogive’ is pronounced as ‘ojeev’ and is derived from the word ogee.
An ogee is a shape consisting of a concave arc flowing into a convex arc, so
forming an S-shaped curve with vertical ends. In architecture, the ogee shape
is one of the characteristics of the 14th and 15th century Gothic styles.
Next, again we consider the cumulative frequency distribution given in
Table 14.14 and draw its ogive (of the more than type).
Recall that, here 0, 10, 20, . . ., 90
are the lower limits of the respective class
intervals 0 - 10, 10 - 20, . . ., 90 - 100. To
represent ‘the more than type’ graphically,
we plot the lower limits on the x-axis and
the corresponding cumulative frequencies
on the y-axis. Then we plot the points
(lower limit, corresponding cumulative
frequency), i.e., (0, 53), (10, 48), (20, 45),
(30, 41), (40, 38), (50, 35), (60, 31),
(70, 24), (80, 15), (90, 8), on a graph paper,
Fig. 14.2
and join them by a free hand smooth curve.
The curve we get is a cumulative frequency curve, or an ogive (of the more than
type). (See Fig. 14.2)

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291

Remark : Note that both the ogives (in Fig. 14.1 and Fig. 14.2) correspond to the
same data, which is given in Table 14.12.
Now, are the ogives related to the median in any way? Is it possible to obtain the
median from these two cumulative frequency curves corresponding to the data in
Table 14.12? Let us see.
One obvious way is to locate

n 53
on
the
y-axis
=
= 26.5
2
2
(see Fig. 14.3). From this point, draw a
line parallel to the x-axis cutting the curve
at a point. From this point, draw a
perpendicular to the x-axis. The point of
intersection of this perpendicular with the
x-axis determines the median of the data
(see Fig. 14.3).

Fig. 14.3

Another way of obtaining the
median is the following :
Draw both ogives (i.e., of the less
than type and of the more than type) on
the same axis. The two ogives will
intersect each other at a point. From this
point, if we draw a perpendicular on the
x-axis, the point at which it cuts the
x-axis gives us the median (see Fig. 14.4).

Fig. 14.4
Example 9 : The annual profits earned by 30 shops of a shopping complex in a
locality give rise to the following distribution :
Profit (in lakhs Rs)

Number of shops (frequency)

More than or equal to 5
More than or equal to 10
More than or equal to 15
More than or equal to 20
More than or equal to 25
More than or equal to 30
More than or equal to 35

30
28
16
14
10
7
3

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MATHEMATICS

Solution : We first draw the coordinate
axes, with lower limits of the profit along
the horizontal axis, and the cumulative
frequency along the vertical axes. Then,
we plot the points (5, 30), (10, 28), (15, 16),
(20, 14), (25, 10), (30, 7) and (35, 3). We
join these points with a smooth curve to
get the ‘more than’ ogive, as shown in
Fig. 14.5.

Cumulative frequency

Draw both ogives for the data above.
Hence obtain the median profit.

50
40
30
20
10
10 20 30 40 50

Lower limits of profit
(in lakhs Rs)

Now, let us obtain the classes, their
frequencies and the cumulative frequency
from the table above.

Fig. 14.5

Table 14.17
Classes

5 - 10

10 - 15 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40

No. of shops

2

12

2

4

3

4

3

Cumulative
frequency

2

14

16

20

23

27

30

Using these values, we plot the points
(10, 2), (15, 14), (20, 16), (25, 20), (30, 23),
(35, 27), (40, 30) on the same axes as in
Fig. 14.5 to get the ‘less than’ ogive, as
shown in Fig. 14.6.
The abcissa of their point of intersection is
nearly 17.5, which is the median. This can
also be verified by using the formula.
Hence, the median profit (in lakhs) is
Rs 17.5.
Remark : In the above examples, it may
be noted that the class intervals were
continuous. For drawing ogives, it should
be ensured that the class intervals are
continuous. (Also see constructions of
histograms in Class IX)

Fig. 14.6

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293

EXERCISE 14.4
1. The following distribution gives the daily income of 50 workers of a factory.
Daily income (in Rs)

100 - 120

120 - 140

140 - 160

160 - 180

180 - 200

Number of workers

12

14

8

6

10

Convert the distribution above to a less than type cumulative frequency distribution,
and draw its ogive.
2. During the medical check-up of 35 students of a class, their weights were recorded as
follows:
Weight (in kg)

Number of students

Less than 38

0

Less than 40

3

Less than 42

5

Less than 44

9

Less than 46

14

Less than 48

28

Less than 50

32

Less than 52

35

Draw a less than type ogive for the given data. Hence obtain the median weight from
the graph and verify the result by using the formula.
3. The following table gives production yield per hectare of wheat of 100 farms of a village.
Production yield
(in kg/ha)

50 - 55

55 - 60

60 - 65

65 - 70

70 - 75

75 - 80

Number of farms

2

8

12

24

38

16

Change the distribution to a more than type distribution, and draw its ogive.

14.6 Summary
In this chapter, you have studied the following points:
1. The mean for grouped data can be found by :
(i) the direct method : x =

Σfi xi
Σfi

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MATHEMATICS
(ii) the assumed mean method : x = a +

Σ f i di
Σf i

⎛ Σf u
(iii) the step deviation method : x = a + ⎜ i i
⎝ Σf i

⎞
⎟× h,
⎠

with the assumption that the frequency of a class is centred at its mid-point, called its
class mark.
2. The mode for grouped data can be found by using the formula:

⎛
⎞
f1 − f 0
Mode = l + ⎜
⎟×h
⎝ 2 f1 − f 0 − f 2 ⎠
where symbols have their usual meanings.
3. The cumulative frequency of a class is the frequency obtained by adding the frequencies
of all the classes preceding the given class.
4. The median for grouped data is formed by using the formula:

⎛n
⎞
⎜ 2 − cf ⎟
Median = l + ⎜
⎟× h,
f
⎜⎜
⎟⎟
⎝
⎠
where symbols have their usual meanings.
5. Representing a cumulative frequency distribution graphically as a cumulative frequency
curve, or an ogive of the less than type and of the more than type.
6. The median of grouped data can be obtained graphically as the x-coordinate of the point
of intersection of the two ogives for this data.

A NOTE TO THE READER
For calculating mode and median for grouped data, it should be
ensured that the class intervals are continuous before applying the
formulae. Same condition also apply for construction of an ogive.
Further, in case of ogives, the scale may not be the same on both the axes.