Download Statistics - Study Point

6
Statistics
CHAPTER
We are Starting from a Point but want to Make it a Circle of Infinite Radius
ARITHMETIC MEAN OF GROUPED DATA
In order to make life easy and save time and energy following methods are adopted for calculation
purposes.
(i) Direct Method
(ii) Assumed mean deviation method
(iii) Step deviation method.
1.
Mean of Grouped Data
( f i xi )
 fi
Where xi are the varities (class marks) and fi their corresponding frequencies.
x
(i) Direct Method :
xa
(ii) Assumed Mean Method :
( f i d i )
 fi
Where a is the assumed mean, di = xi – a, xi are the varieties (class marks) and fi their
corresponding frequencies.
xa
(iii) Step Deviation Method :
( f i u i ' )
h
 fi
Where a is the assumed mean, xi are the class marks, fi are the corresponding frequencies,
xi  a
, and h is the width of the class with the assumption that the frequency of a class is
h
u
centered at its mid point called its class mark.
Direct Method: Ex 1.1
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 (xi)
Number of student (f i)
10
1
20
1
36
3
40
4
50
3
56
2
60
4
70
4
72
1
80
1
88
2
92
3
95
1
Solution:
Marks obtained (xi)
10
20
36
40
50
56
Number of student
(f i)
1
1
3
4
3
2
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
f x
i i
10
20
108
160
150
112
[82]
CLASS - X
Mathematics
Statistics
60
70
72
80
88
92
95

Total
4
4
1
1
2
3
1
fi
f x
= 30
240
280
72
80
176
276
95
 f x = 1779
i i
1779
 59.3
30
 fi
Therefore, the mean marks obtained is 59.3.
x
i i

Example 2:
Marks (xi)
Number of students ( f i )
2
3
7
6
6
6
17.5
32.5
47.5
62.5
77.5
92.5
Find the mean marks
Solution:
Class mark (xi)
17.5
32.5
47.5
62.5
77.5
92.5
f x
x
f ixi
Number of students ( f i )
2
3
7
6
6
6
f
i
 = 30
i i
 fi

35.0
97.5
332.5
375.0
465.0
555.0
f
x
i
 i = 1860.0
1860.0
 62
30
Assumed Mean Method:
Example 3:
Marks (xi)
17.5
32.5
47.5
62.5
77.5
92.5
Number of students ( f i )
2
3
7
6
6
6
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[83]
CLASS - X
Mathematics
Statistics
Find the mean marks
di = xi – a = xi – 47.5
Number of students
( fi )
2
3
7
6
6
6
f
i
 = 30
Marks (xi)
17.5
32.5
47.5
62.5
77.5
92.5
x = a +
-30
-15
0
15
30
45
fdi
-60
-45
0
90
180
270
f
i
 di = 435
 fi di
 fi
x = 47.5 +
435
30
= 47.5 + 14.5 = 62.
Therefore, the mean of the marks obtained by the students is 62.
Step Deviation Method or Shortcut Method :
Example 4:
xi
20
30
40
50
60
70
80
(fi)
6
11
7
4
4
2
1
Find Mean
Solution:
xi
fi
di = xi – 50
20
30
40
50 = a
60
70
80
6
11
7
4
4
2
1
35
-30
-20
-10
0
10
20
30
Ui =
xi  50
10
fi ui
-3
-2
-1
0
1
2
3
-18
-18
-22
-7
0
4
3
-36
Question based on mean of ungrouped Data
1.
Calculate the mean for the following distribution using direct method .
Variable
Frequency
5
4
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
6
8
7
14
[84]
8
11
9
3
CLASS - X
Mathematics
Statistics
2.
Marks obtained (xi)
No. of students (fi)
15
4
5
8
14
14
10
11
16
3
3.
The following table represents the weight (in kg) of 60 workers in a factory. Find the
mean weight of each worker using assumed mean method.
60
61
62
63
64
65
Weight (kg)
5
8
14
16
10
7
No. of workers
4.
The following table represents the wages of 50 workers in a factory. Find the mean
wages of the worker using step deviation method.
1450
1475
1500
1525
1550
Wages (in Rs.)
12
13
7
10
8
No. of worker
5.
The number of students absent in a section of class X were recorded everyday for 100
days and the information is given in the following frequency table. Find the mean.
0
5
10
15
20
25
No. of students absent
10
20
25
15
20
10
No. of days
GROUPED FREQUENCY DISTRIBUTION
Sometimes the data is so large that it is inconvenient to list every item in the frequency distribution table. Then we
group the items into convenient intervals and the data is presented in a frequency distribution in which each class
interval contains five or ten values of the variate generally. The mid value of each class is the representative of
each item falling in that interval.
Here individual values lose their identity. If we assume that all values falling in a class interval are equal to
the middle value of this class interval. This middle value is called mid-value, mid-point or class mark.
Class mark =
Example 1: Find Mean
Class interval
10−25 25−40 40−55 55−70 70−85 85−100
Number of students
2
3
7
6
6
6
By Direct Method
Class interval
10−25
25.40
40−55
55−70
70−85
85−100
Total
Number of students
2
3
7
6
6
6
The sum of the values in the last column gives us
given by.
=
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[85]
Class mark
(𝓍 ₁)
17.5
32.5
47.5
62.5
77.5
92.5
So, the mean
35.0
97.5
332.5
375.0
465.0
555.0
of the given data is
CLASS - X
Mathematics
Statistics
Example 2: Find Mean
Class interval
10−25 25−40 40−55 55−70 70−85 85−100
Number of Students 2
3
7
6
6
6
By Assumed Mean Method
Class Interval Number
of
students (f₁)
10−25
2
25−40
3
40−55
7
55−70
6
70−85
6
85−100
6
Total
Class
(𝓍₁ )
17.5
32.2
47.5
62.5
77.5
92.5
mark
−30
−15
0
15
30
45
−60
−45
0
90
180
270
Example 3: Find Mean
10−25 25−40 40−55 55−70 70−85 85−100
Class interval
3
7
6
6
6
Number of students 2
By Step deviation Method\
Class interval
10−25
25−40
40−55
55−70
70 – 85
85 – 100
2
3
7
6
6
6
17.5
32.5
47.5
62.5
77.5
92.5
−30
−15
0
15
30
45
−2
−1
0
1
2
3
−4
−3
0
6
12
18
Total
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[86]
CLASS - X
Mathematics
Statistics
Example 4: The table below gives the percentage distribution of female teacher in the
primary schools of rural areas of various states and union (U.T) of India. Find the mean
percentage of female teachers by all the three methods discussed in this section.
Percentage of Female teachers 15−25 25−35 35−45 45−55 55−65 65−75 75−85
6
11
7
4
4
2
1
Number of State/U.T.
Number of State/U.T.
Percentage of
teachers
15−25
25−35
35−45
45−55
55−65
65−75
75−85
Total
6
11
7
4
4
2
1
35
20
30
40
50
60
70
80
−30
−20
−10
0
10
20
30
−3
−2
−1
0
1
2
3
120
330
280
200
240
140
80
1390
−180
−220
−70
0
40
40
30
−360
−18
−22
−7
0
4
4
3
−36
From the table above, we obtain
Using the direct method,
Using the assumed mean method,
Using the assumed mean method,
Using the step−deviation method,
Example 5 : 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 60−100 100−150 150−250 250−350 350−450
7
5
16
12
2
3
Number of bowlers
Solution : Here, the class size varies, and the
are large. Let us still apply the step
deviation method with a = 200 and h = 20. Then, we obtain the data as in Table.
Number of
wickets
taken
Number of
bowlers (
20−60
60−100
100−150
150−250
250 – 350
350−450
7
5
16
12
2
3
Total
40
80
125
200
300
400
−160
−120
−75
0
100
200
−8
−6
−3.75
0
5
10
−56
−30
−60
0
10
30
−106
45
Therefore,
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[87]
CLASS - X
Mathematics
Statistics
Question based on mean
Direct Methods
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.
No. 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.
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
3.
4.
Find the mean concentration of SO2 in the air.
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.
0 – 6 6 – 10 10 – 14
14 – 20
20 – 28
28 – 38 38 – 40
Number of days
11
10
7
4
4
3
1
Number of students
The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
45 – 55
55 – 65
65 – 75
75 – 85
85 – 95
Literacy rate (in %)
3
10
11
8
3
Number of cities
Assumed Mean Method
5.
Consider the following distribution of daily wages of 50 workers of a factory.
100 – 120 120 – 140
140 – 160
160 – 180 180 – 200
Daily wages (in Rs.)
12
14
8
6
10
Number of Workers
Find the mean daily wages of the workers of the factory by using an appropriate method.
6.
Find the mean of the following data:
Class interval
Frequency
0 – 10
12
10 – 20
16
20 – 30
6
30 – 40
7
40 – 50
9
7.
Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were
recorded and summarized as follows. Find the mean heart beats per minute for these women, choosing a
suitable method.
No. of heart beats per 65 – 68 68 – 71 71 – 74 74 – 77 77 – 80 80 – 83 83 – 86
minutes
2
4
3
8
7
4
2
Number of Women
8.
The table below shows the daily expenditure on food of 25 households in a locality.
100 – 150 150 – 200 200 – 250 250 – 300
Daily Expenditure (in Rs.)
4
5
12
2
Number of households
Find the mean daily expenditure on food by a suitable method.
9.
300 – 350
2
The following table shows the age distribution of cases of a certain disease reported during a year in a particular
city.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[88]
CLASS - X
Mathematics
Statistics
Age (in years)
5 – 14
15 – 24
25 – 34
35 – 44
45 – 54
55 – 64
Total
No of cases
6
11
21
23
14
5
80
Find the average (in years) per case reported.
10.
Calculate the Arithmetic mean of the following frequency distribution:
Class interval
0 – 20
20 – 40
40 – 60
60 – 80
80 – 100
100 – 120
Frequency
20
35
52
44
38
31
11. 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.
50 – 52 53 – 55
56 – 58
59 – 61 62 – 64
No. of mangoes
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?
12. Find the mean of the following data by assumed mean method
Class-interval
Frequency
150 – 200
14
200 – 250
26
250 – 300
17
300 – 350
23
350 – 400
20
Missing frequency:
13.
The arithmetic mean of the following data is 11. Find the value of p.
x1
f1
14.
5
2
8
4
11
8
14
P
17
2
If the mean of the following data is 18, find the missing frequency p:
X
F
10
5
15
10
20
P
25
8
15. 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.
11 – 13
13 – 15
15 – 17 17 – 19 19 – 21
21 – 23 23 – 25
Daily pocket allowance
(in Rs.)
7
6
9
13
F
5
4
Number of children
16. The mean of the following distribution is 132 and sum of observation is 50. Find the values of x and y.
0 – 40
40 – 80
80 – 120
120 – 160
160 – 200
200 – 240
Class
4
7
x
12
Y
9
Frequency
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[89]
CLASS - X
Mathematics
Statistics
17. The mean of the following data is 46.2. Find the missing frequencies f1 and f2.
20 – 30
30 – 40
40 – 50
50 – 60
60 – 70
Classes
6
f1
16
13
f2
Frequencies
70 – 80
2
Total
50
MODE :
Mode is the size of the variable which occurs most frequently. In discrete or ungrouped case, the mode is
the variate having maximum frequency.
For Example ,In the series 6,5,3,4,3,7,8,5,9,5,4, we notice that 5 occurs most frequently therefore 5 is the
mode.
TYPES OF MODAL SERIES
(i)
Unimodal series. The series which contains one mode.
(ii) Bimodal series. The series which contains two modes.
(iii) Trimodal series. The series which contains three modes.
MERITS OF MODE
(i)
Mode is readily comprehensively and easy to calculate. It can be located in some case morely by
inspection.
(ii) Mode is not at all affected by extreme values.
(iii) Mode can be conveniently even class interval of unequal magnitude.
DEMERITS OF MODE
(i)
Mode is ill-defined. In some cases we may come across two modes.
(ii) It is not based upon all the observations.
(iii) No further mathematical treatment is possible in case of mode.
(iv) Mode is affected to a greater extant by fluctuations of sampling.
USES OF MODE
Mode is used to find the ideal size e.g., Readymade garments, shoes etc.
WORKING RULE TO FIND MODE
Step 1.
Find out the maximum frequency (f0) from the given table. Class of this maximum frequency is
the modal class.
Step 2.
Find the lower limit (l) of the modal class.
Step 3.
Write down the preceding frequency (f1) and succeeding frequency (f2) of modal class
frequency.
Step 4.
Write down the width (h) of modal class.
Step 5.
Apply the following formula to find out the mode :
Mode = l + h

f1  f 0

 2 f1  f 0  f 2



Example 1 :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
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[90]
CLASS - X
Mathematics
Statistics
Frequency (f)0 of class preceding the modal class = 7,
Frequency (f1) of the modal class = 8,
Frequency (f2) of class succeeding the modal class = 2,
Now, let us substitute these values is the formula.
 f1  f 0 
Mode = l  
h
 2f1  f 0  f 2 
87
2


= 3
  2  3   3.286
7
 28  7  2 
Therefore, the mode of the data above is 3.286.
Example 2 :
The marks distribution of 30 students in a mathematics examination are given below.
Class Interval
10-25
25-40
40-55
55-70
7-85
85-100
Total
Number of students (fi)
2
3
7
6
6
6
 f1  30
Find the mode of this data.
Solution: Refer to Table 14.3 of Example 1. Since the maximum number of students (i.e., 7) have got
marks in the interval 4 – 55, the modal class is 40 – 55. Therefore,
The lower limit (l) of the modal class = 40,
The class size (h) = 15,
The frequency (f0) of the class preceding the modal class = 3,
The frequency (f1) of modal class = 7,
The frequency (f2) of the class succeeding the modal class = 6,
Now, using the formula:
 f1  f 0 
Mode  l  
 h ,
 2f1  f 0  f 2 
 73 
We get Mode  40  
  15  52
 14  6  3 
So, the mode marks is 52.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[91]
CLASS - X
Mathematics
Statistics
Example 3 : 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
15-20
20-25
25-30
30-35
35-40
40-45
45-50
50-55
Number of states/U.T.
3
8
9
10
3
0
0
2
Solution: Mode: Since greatest frequency 10 corresponds to class 30-35.

Modal class = 30–35
And
h = 5, l = 30, f1 = 10, f0 = 9, f2 = 3
 f1  f 0 
 10  9 
 Mode: l + 
  h  30  
 5
 20  9  3 
 2f1  f 0  f 2 
1
8
 30   5  300  0.625  30.6(approx.)
Mean:
Let the assumed mean, a = 37.5 and class size, h = 5
 We have the following table:
x i  37.5
5
Number of students
per teacher
Class
mark(xi)
Frequency
(fi)
15-20
17.5
3
17.5  37.5 20

 4
5
5
3 × (-4) = –12
20-25
22.5
8
22.5  37.5 15

 3
2
5
8 × (-3) = –24
25-30
27.5
9
27.5  37.5 10

 2
2
5
9 × (-2) = –18
30-35
32.5
10
32.5  37.5 5

 1
2
5
10 × (–1) = –
10
35-40
37.5
3
37.5  37.5 0
 0
2
5
3×0=0
40-45
42.5
0
42.5  37.5 5
 1
2
5
0×1=0
45-50
47.5
0
47.5  37.5 10
 2
2
5
0×2=0
50-55
52.5
2
52.5  37.5 15
 3
2
5
2×3=6
Total
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
 fi  35
[92]
ui 
(fiui)
 fi ui  58
CLASS - X
Mathematics
Statistics
  fi u i 
 58 
 37.5  5  
 37.5  8.3  29.2

 35 
  fi 
x  ah
Interpretation.
The maximum teacher-student ratio is 30.6 while average teacher-student ratio is 29.
Question based on Mode.
1.
2.
The wickets taken by a bowler in 10 cricket matches are as follows:
2
6
4
5
0
2
1
3
2
Find the mode of the data.
3
The following table shows the ages of the patients admitted in a hospital during a year:
Age (in years)
Number of patients
5 – 15
6
15 – 25
11
25 – 35
21
35 – 45
23
45 – 55
14
55 – 65
5
Find the mode and the mean of the data given above. Compare and interpret the two measures of central
tendency.
3.
The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:
Life time (in hours)
Frequency
0 – 20
10
20 – 40
35
40 – 60
52
60 – 80
61
80 – 100
38
100 – 120
29
Determine the modal lifetimes of the components.
4.
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.)
1000 – 1500
1500 - 2000
2000 - 2500
2500 - 3000
3000 - 3500
3500 - 4000
4000 - 4500
4500 – 5000
5.
Number of families
24
40
33
28
30
22
16
7
The given distribution shows the number of runs scored by some top batsmen of the world in one-day
international cricket matches.
Runs Scored
3000 - 4000
4000 - 5000
5000 - 6000
6000 - 7000
7000 - 8000
8000 - 9000
9000 - 10000
10000 - 11000
Find the mode of the data.
6.
Number of batsman
4
18
9
7
6
3
1
1
A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and
summarized it in the table given below. Find the mode of the data
Number of cars
Frequency
0 – 10
7
10 – 20
14
20 – 30
13
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
30 – 40
12
[93]
40 – 50
20
50 – 60
11
60 – 70
15
CLASS - X
Mathematics
70 – 80
8
Statistics
MEDIAN
Median is defined as the measure of central term when they are arranged in ascending or descending order
1
of magnitude. When the total number of terms is odd and equal to say n then the value of (n  1) th term
2
give the median. When the total number of terms is even, say n, then there are two middle terms and so the
1
n 
mean of the values of nth and   1 th term is the median.
2
2 
(i)
 n 1
Median = Value of 
 th term, if n is odd.
 2 
(ii)
Median = Value of

1  n 
n 
 th term    1 th term  , if n is even.
2  2 
2 

MERITS OF MEDIAN
(i)
(ii)
(iii)
(iv)
(v)
(vi)
It is rigidly defined
It is easily understood and easy to calculate.
It is not at all affected by extreme values.
It can be located graphically.
It can be determined even by inspection in some cases.
It can be located if the class interval be unequal.
DEMERITS OF MEDIAN
(i)
(ii)
(iii)
(iv)
In case of even number of observations median can not be determined exactly.
It is not based on all the observations.
It is not subject to algebraic treatment.
It is much affected by fluctuations of sampling.
USES OF MEDIAN
(i)
(ii)
Median is the only average to be used while dealing with qualitative data which can not be measured
quantitively but can be arranged in ascending or descending order of magnitude.
It is used for determining the typical value in problems concerning wages; distribution of wealth etc.
MEDIAN OF GROUPED FREQUENCY DISTRIBUTION
N
N
In order to find the median first we have to find out
then the class in which
lies. This class will be
2
2
called the median class. Median lies in this class. Median can be calculated by the following formula.
where,
N
C
h
Median = l  2
f
l = lower limit of median class interval
C = cumulative frequency preceding to the median class frequency.
F = frequency of the class interval to which median belongs.
H = width of the class interval.
N = f1 + f2 + f3 + ….. + fn.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[94]
CLASS - X
Mathematics
Statistics
Example 1 : 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 medina height, we need to find the class intervals and their corresponding
frequencies.
So, our frequency distribution table with the given cumulative frequencies becomes:
Class intervals
Frequency
Cumulative
frequency
Below 140
4
4
140 – 145
7
11
145 – 150
18
29
150 – 155
11
40
155 – 160
6
46
160 – 165
5
51
Now 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
  cf
Using the formula, Median = l   2
 f


  h, we have

 25.5  11 
Median = 145  
5
 18 
== 145 
72.5
 149.03
18
So, the median height of the girls is 149.03cm.
This means that the height of about 50% of the girls is less than this height, and 50% are teller than
this height.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[95]
CLASS - X
Mathematics
Statistics
Example 2 : The median of the following data is 525. Find the values of x and y, if the total
frequency is 100.
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 + 7
900-1000
4
76 + x + y
It is given that n = 100
So, 76 + x + y = 100, i.e., x + y = 24
The median is 525, which lies in the class 500 – 600
So, l = 500, f = 20, cf = 36 + x, h = 100
n
  cf
Using the formula: Median  l   2
 f


 h, we get \

 50  36  x 
525  500  
 100
20


525 – 500 = (14 – x)  5
i.e.,
25 = 70 – 5x
i.e.,
5x = 70 – 25 = 45
i.e.,
x=9
Therefore, from (1), we get 9 + y = 24
y = 15
Example 3 : 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
1-4
4-7
7-1
10-13
13-16
16-19
Number of
surnames
6
30
40
16
4
4
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[96]
CLASS - X
Mathematics
Statistics
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.
Solution: The cumulative frequency distribution table is as follows:
Number of letters
Frequency (f)
Cumulative frequencies
1-4
6
6+0=6
4-7
30
6 + 30 = 36
7-10
40
40 + 36 = 76
10-13
16
16 + 76 = 92
13-16
4
4 + 92 = 96
16-19
4
4 + 96 = 100
Total
 fi  100
n = 100
n  100 
n 100

 50 Since, the cumulative frequency just greater than 50 is 76.
2
2
 The class 7-10 is the median class,
We have
n
 50, l  7,cf  36,f  40 and h  3 and
2
n

 2  cf 
 50  36 
 Median  l  
h  7  
3
 f 
 40 
 7
14
42
3  7 
 7  1.05  8.05 Mean :
40
40
Class intervals
Class Marks (x)
Frequency (fi)
(fixi)
1-4
2.5
6
6 × 2.5 = 15
4-7
5.5
30
30 × 5.5 = 165
7-10
8.5
40
40 × 8.5 = 340
10-13
11.5
16
16 × 11.5 = 184
13-16
14.5
4
4 × 14.5 = 58
16-19
17.5
4
4 × 17.5 = 70
 f i  100
 f i x i  832
Total
  f i x i  832
 8.32

  fi  100
 Mean : x  
Mode:
Since the class 7-10 has the maximum frequency.
 The modal class is 7-10
So, we have l = 7, h = 3, f1 = 40, f0 = 30, f2 = 16
 f1  f 0 
 40  30 
 Mode = l + 
h  7  
 3
 80  30 16 
 2f1  f 0  f 2 
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[97]
CLASS - X
Mathematics
Statistics
30
 10 
 7  0.88  7.88
3  7 
34
 34 
 7
Thus, the required
Median = 8.05, Mean = 8.32 and Mode = 7.88.
Question based on median
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)
65 - 85
85 - 105
105 - 125
125 - 145
145 - 165
165 - 185
185 - 205
2.
Number of consumers
4
5
13
20
14
8
4
If the median of the distribution given below is 28.5, find the values of x and y.
Class Interval
0 - 10
10 - 20
20 - 30
30 - 40
40 – 50
50 - 60
Total
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.
Age (in years)
Below 20
Below 25
Below 30
Below 35
Below 40
Below 45
Below 50
Below 55
Below 60
4.
Frequency
5
x
20
15
y
5
60
Number of Policy holder
2
6
24
45
78
89
92
98
100
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)
118 - 126
127 - 135
136 - 144
145 - 153
154 - 162
163 - 171
172 - 180
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
Number of leaves
3
5
9
12
5
4
2
[98]
CLASS - X
Mathematics
Statistics
Find the median length of the leaves.
5.
The following table gives the distribution of the life time of 400 neon lamps :
Life time (in hours)
1500 - 2000
2000 - 2500
2500 - 3000
3000 - 3500
3500 - 4000
4000 - 4500
4500 - 5000
Number of lamps
14
56
60
86
74
62
48
Find the median life time of a lamp.
6.
The distribution below gives the weights of 30 students of a class. Find the median weight of the
students.
Weight (in Kg)
No. of Students
40 – 45
2
45 – 50
3
50 – 55
8
55 – 60
6
60 – 65
6
65 – 70
3
70 – 75
2
Graphical representation of Cumulative frequency Distribution.
In previous classes we have represented the data through bar graph, histograms and frequency polygon. In this
section we represent a cumulative frequency distribution graphically.
First we prepare the cumulative frequency table then, the cumulative frequency are plotted against th upper of
lower limits of the corresponding class−interval. By joining the point the curve so obtained is called cumulative
frequency curve or ogive.
There are two types of ogives:
Less than ogive : Plot the points with the upper limits of the class as abscissa and the corresponding less than
cumulative frequencies as ordinates. The points are joined by free hand smooth curve to give less than cumulative
frequency curve or the less than Ogive. It is a rising curve.
Greater than ogive : Plot the points with the lower limits of the classes as abscissa and the corresponding Greater
then cumulative frequencies as ordinates. Join the points by a free hand smooth curve to get the “More than
Ogive”. It is a falling curve.
When the points obtained are joined by straight lines, the picture obtained is called cumulative frequency polygon.
Example 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 that type cumulative frequency distribution, and draw its
ogive.
Solution: We have the cumulative frequency distribution as:
Daily Income (In Rs.)
Cumulative Frequencies
Less than 120
12 + 0 = 12
Less than 140
12 + 14 = 26
Less than 160
26 + 8 = 34
Less than 180
34 + 6 = 40
Less than 200
40 + 10 = 50
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[99]
CLASS - X
Mathematics
Statistics
Now, we plot the points corresponding to the ordered pair (120, 12), (140, 26), (160, 34), (180, 40)
and (200, 50) on a graph paper and join them by a free hand to get a smooth curve as shown below:
The curve so obtained is called the less than ogive.
Example-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.
Solution: Here, the values 38, 40, 42, 44, 46, 48, 50 and 52 are the upper limits of the respective
class-intervals.
We plot the points (ordered pairs) (38, 0), (40, 3), (42, 5), (44, 9), (46, 14), (48, 28), (50, 32) and (52,
35) on a graph paper and join them by a free hand to get a smooth curve. The curve so obtained is the
less than type ogive.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[100]
CLASS - X
Mathematics
Statistics
n
2
n  35,  
35
 17.5
2
The point 17.5 is on y-axis.
From this point (i.e., from 17.5) we draw a line parallel to the x-axis which cuts the curve at P. From
this point (p, draw a perpendicular to the x-axis, meeting the x-axis at Q. The point Q represents the
median of the data which is 47.5.
Verification:
To verify the result using the formula, let us make the following table in order to find mode using the
formula:
Weight (in
kg)
Frequency
Number of Students
(Cumulative Frequency)
38-40
3
3
40-42
2
5
42-44
4
9
44-46
5
14
46-48
14
28
48-50
4
32
50-52
3
35
Here, n  35 
n 35

 17.5
2 2
Since, this observation lies in the class 46-48.
 The median class is 46-48 such that l = 46, h = 2, f = 14, cf = 14
n

 2  cf 
17.5 14
3.5
 Median = l + 
 2  46 
2
  h  46  9
 f 
14
14
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[101]
CLASS - X
Mathematics
Statistics
1
2
 46   46.5kg
Thus, the median = 46.5 kg is approximately verified.
Example 3:
The following table gives production yield per hectare of wheat of 100 farms of a village.
50-55
55-60
60-65
65-70
70-75
-75-80
Production yield (in
kg/ha)
2
8
12
24
38
16
Number of farms
Change the distribution to a more than type distribution, and draw its ogive.
Solution: For more than type distribution we have:
Production yield (in kg/ha)
Number of Farms
More than 50
100
More than 55
98
More than 60
90
More than 65
78
More than 70
54
More than 75
16
Now, we plot the points (50, 100), (55, 98), (60, 90), (65, 78), (70, 54) and (75, 16) and join the point with a free
hand to get a smooth curve.
The curve so obtained is the „more than type ogive‟.
Example 4 : The annual profits earned by 30 shops of a shopping complex in a locality give rise to the
following distribution:
Profit (in lakhs Rs.)
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
Number of shops (frequency)
30
28
16
14
10
7
3
Draw both ogive for the data above. Hence obtain the medina profit.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[102]
CLASS - X
Mathematics
Statistics
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.
Now let us obtain the classes, their frequencies and the cumulative frequency from the table above.
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
Classes
Using these values, we plot the points (10, 2), (15, 14), (20, 16), (30, 23), (35, 27), (40, 30) on the same axes as
in fig. to get the „less than‟ ogive, as show in fig.
The abscissa 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.
Question based on ogives.
1.
Draw a more than ogive for the following frequency distribution :
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
2.
Draw a less than ogive for the following frequency distribution :
I. Q.
60 – 70 70 – 80 80 – 90 90 – 100 100 – 110 110 – 120 120 – 130
No. of Students
2
5
12
31
39
10
4
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[103]
CLASS - X
Mathematics
Statistics
3.
The following table shows the daily sales of 230 footpath sellers of Chandni Chowk.
Sales (in Rs.)
0 – 500
500 – 1000
1000 – 1500
1500 – 2000
2000 – 2500
2500 – 3000
3000 – 3500
3500 – 4000
No. of Sellers
12
18
35
42
50
45
20
8
Locate the median of the above data using only the less than type ogive.
4.
Draw the two ogives for the following frequency distribution of the weekly wages of (less than and more
than) number of workers.
Weekly wages
No. of Workers
5.
0 – 20
41
20 – 40
51
40 – 60
64
60 – 65
65 – 70
70 – 75
75 – 80
12
24
38
16
Following table gives the cumulative frequency of the age of a group of 199 teachers.
Draw the less then ogive and greater than ogive and find the median.
Age in years
20 – 25
25 – 30
30 – 35
35 – 40
40 – 45
45 – 50
50 – 55
55 – 60
60 – 65
65 – 70
7.
80 – 100
7
The following table gives production yield per hectare of wheat of 100 farms of a village.
Production yield (in
50 – 55
55 – 60
kg/ha)
Number of farms
2
8
Draw the less then ogive and greater than ogive.
6.
60 – 80
38
No. of Sellers
21
40
90
130
146
166
176
186
195
199
The distribution below gives the weights of 30 students of a calss.
Weight (in kg)
40-45
45-50
50-55
55-60
Number of
2
3
8
6
students
Draw the less then ogive and greater than ogive and find the median.
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[104]
60-65
6
65-70
3
CLASS - X
Mathematics
70-75
2
Statistics
ANSWER
Question based on mean of ungroup data.
1. 7.025
2. 1.5
3.62.65
4. 1494.5
5. 12.25
Question based on mean of group data.
1. 8.1
2. 0.099
3. 12.38
4.69.73
8. 211 9. 34.88
10. 62.54
11.5719
14. P=7 15. F=20
16. x=10, y = 8
17. F₁ = 9, f₂ = 4
5. 145.20
12. 27.95
6. 22
13.P=4
7.75.9
Question based on mode.
1.2
5. 4608.7 runs.
2.mode = 36.8, mean =
35.37
6. 44.7 cars.
3. 65.625
4. Modal monthly expenditure = Rs. 1847.83, mean = 2662.5
Question based on median.
1. median = 137, mean = 137.05, mode = 135.76. The three measure
are approximately same.
4. Median = 146.75 mm
2. x = 8, y = 7
5. median life =
3406.98 hours
3. Median age = 35.76
years.
6. median = 56.67 kg.
Question based on ogive.
1. median profit = Rs. 17.5
4. Median = 37.375
5.
Production yield (kg/ha)
More than or equal to 50
More than or equal to 55
More than or equal to 60
More than or equal to 65
More than or equal to 70
More than or equal to 75
Cumulative
100
98
90
78
54
16
Now, draw the ogive by plotting the points : (50,100), (55, 98), (60,90), (65,78), (70,54)
6.
7.
37.375
56.67
011-26925013/14
NTSE, NSO
+91-9811134008 Diploma, XI Entrance
+91-9582231489
[105]
CLASS - X
Mathematics
Statistics