14 STATISTICS CHAPTER

CHAPTER
14
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STATISTICS
Points to Remember :
1. Facts or figures, collected with a definite pupose, are called Data.
2. Statistics is the area of study dealing with the collection, presentantion, analysis and interpretation of
data.
3. The data collected by the investigator himself with a definite objective in mind are known as Primary
data.
4. The data collected by somone else, other than the investigator, are known as Secondary data.
5. Any character which is capable of taking reversal different values is called a variable.
6. Each group into which the raw data are condensed is known as class-interval. Each class is bounded by
two figures known as its limits. The figure on the left is lower limit and figure on the right is upper limit.
7. The difference between true upper limit and true lower limit of a class is known as its class-size.
8. Mid-value of a class (or class mark) =
9.
10.
11.
12.
13.
B
upper limit  lower limit
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Class size is the difference between any two successive class marks (mid-values).
The difference between the maximum value and the minimum value of the variable is known as Range.
The count of number of observations in a particular class is known as its Frequency.
The data can be presented graphically in the form of bar graphs, histograms and frequency polygons.
The three measures of central tendency for an ungrouped data are :
(i) Mean : It is found by adding all the values of the observations and dividing it by the total number of
observations. It is denoted by x .
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n
 xi
x1  x2  ........  xn i 1 .

n
n
For an ungrouped frequency distribution,
So,
x
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x
n
f1 x1  f 2 x2  ...... f n x n

f1  f 2  ........ f n
 f i xi
i 1
n
 fi
i 1
(ii) Median : It is the value of the middle-most observation(s).
 n 1 

If n is an odd number, then median = value of the 
 2 
th
observation.
th
n
n 
and, if n is an even number, then median = mean of the values of   and   1
 2
2 
(iii) Mode : The mode is the most frequently occurring observation.
Empirical formula for calculating mode is given by, Mode = 3 (Median) – 2 (Mean)
MATHEMATICS–IX
STATISTICS
th
observations.
173
ILLUSTRATIVE EXAMPLES
Example 1. The relative humidity (in %) of a certain city for a month of 30 days was as follows :
98.1 98.6 99.2 90.3 86.5 95.3 92.9 96.3 94.2 95.1
89.2 92.3 97.1 93.5 82.7 95.1 97.2 93.3 95.2 97.3
96.2 92.1 84.9 90.2 95.7 98.3 97.3 96.1 92.1 89.0
(i) Construct a grouped frequency distribution table with classes 84-86, 86-88 etc.
(ii) Which month or season do you think this data is about ?
(iii) What is the range of this data?
—NCERT
Solution.
(i) Frequency distribution table
86  88
|
88  90
90  92
||
||
92  94
|||| ||
94  96
96  98
|||| |
|||| ||
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Relative humidity (in %) Tally Marks Frequency
84  86
|
1
1
2
2
7
6
7
98  100
||||
4
Total
30
(ii) Month-June or season-Monsoon
(iii) Range = Maximum observation – minimum observation
= 99.2 – 84.9 = 14.3
Example 2. The value of  upto 50 decimal places is given below :
3.14159265358979323846264338327950288419716939937510
(i) List the digits from 0 to 9 and make a frequency distribution of the digits after the decimal point.
(ii) What are the most and the least frequency occurring digits?
—NCERT
Solution.
(i) Frequency distribution table
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Digit Tally Marks Frequency
0
||
2
1
2
||||
||||
5
5
3
4
|||| |||
||||
8
4
5
6
||||
||||
5
4
7
8
||||
||||
4
5
9
Total
|||| |||
8
50
(ii) Most frequency occuring digits are 3 and 9, and least occurring digit is 0.
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STATISTICS
MATHEMATICS–IX
Example 3. The following table gives the life times of 400 neon lamps:
Life time (in hours)
No. of lamps
300 - 400
400 - 500
14
56
500 - 600
600 - 700
60
86
700 - 800
74
800 - 900
900 - 1000
62
48
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(i) Represent the given information with the help of a histogram.
(ii) How many lamps have a life time of more than 700 hours?
Solution.
(i)
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—NCERT
(ii) No. of lamps having life time more than 700 hours
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= 74 + 62 + 48 = 184
Example 4. The following two tables give the distribution of students of two sections according to the marks
obtained by them :
Section A
Marks
0-10
10-20
20-30
30-40
40-50
Section B
Frequency
Marks
Frequency
3
9
17
12
9
0-10
10-20
20-30
30-40
40-50
5
19
15
10
1
Represent the marks of the students of both the sections on the same graph by frequency
polygon.
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Solution.
Required frquency polygon is as follows :
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Example 5. Draw histogram of the weekly pocket expenses of 125 students of a school given below :
No. of students
10
15
30 - 50
50 - 60
60 - 90
40
25
30
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90 - 100
Solution.
B
Weekly pocket expenses (in Rs.)
10 - 20
20 - 30
05
Here, we observe that class intervals are unequal, so we will first adjust the frequencies of each
class interval. Here, the minimum class size is 10.
We know, Adjusted frequency of a class interval
Minimum class size

 frequency of the class
class size
 The adjusted frequency of each class interval is given below :
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Weekly pocket expenses Frequency Adjusted frequency
10
10 - 20
10
10  10
10
10
20 - 30
15
 15  15
10
10
30 - 50
40
 40  20
20
10
50 - 60
25
 25  25
10
10
60 - 90
30
 30  10
30
10
90 - 100
05
 5  05
10
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STATISTICS
MATHEMATICS–IX
So, required histogram is given below.
B
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Example 6. In a mathematics test given to 15 students, the following marks (out of 100) are recorded :
41, 48, 39, 46, 52, 54, 62, 40, 96, 52, 98, 40, 42, 52, 60.
Find the mean, median and mode of the above marks.
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15
 xi
Solution.
(i) Mean ( x ) 

i 1
15
41  48  39  46  52  54  56  62  40  96  52  98  40  42  52  60
15
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822
 54.8
15
(ii) Arranging the data in the ascending order :

39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96, 98
here,

n = 15, which is odd.
 n 1
median = value of 

 2 
 15  1 


 2 
th
observation
th
observation = 8th observation = 52
(iii) Since, 52 occurs most frequently i.e. 3 times, so mode is 52.
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177
Example 7. Find the mean salary of 60 workers of a factory from the following table:
Salary (in Rs)
3000
4000
No. of workers
16
12
5000
6000
7000
10
8
6
8000
9000
4
3
10000
1
Solution.
Salary (in Rs.) xi
3000
No. of workers f i
16
4000
12
48000
5000
6000
10
8
50000
48000
7000
6
8000
9000
4
3
10000
Total
1
 f i  60
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f i xi
48000
B
—NCERT
42000
32000
27000
10000
 f i xi  305000
 f i xi 305000
 Mean ( x ) 

 Rs. 5083.33 Ans.
 fi
60
Example 8. The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Find the excluded
number.
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Solution.
Here, n = 5, x  18.
Now, x 
 xi
  xi  5  18  90
n
So, total of 5 numbers is 90.
Let the excluded number be a. Then, total of 4 numbers is 90 – a.
Mean of 4 numbers 
90  a
4
90  a
 16
( Given, new mean = 16)
4
 90 – a = 64  a = 26
 the excluded number is 26. Ans.

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Example 9. The median of the observations 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 arranged in ascending order
is 24. Find x.
Solution.
Here, n = 10. Since n is even :
th
th
n
n 
observatio
n


1
 

 observation
2 
Median   2 
2
5th observation  6th observation
 24 
2
( x  2)  ( x  4)
2x  6
 24 

 24  24  x  3
2
2
 x = 21 Ans.
Example 10. Find the mode for the following data :
14, 25, 28, 14, 18, 17, 18, 14, 23, 22, 14, 18.
Solution.
Arranging the given data in ascending order:
14, 14, 14, 14, 17, 18, 18, 18, 22, 23, 25, 28
Since, 14 occurs maximum number of times (4 times),  14 is the required mode.
PRACTICE EXERCISE
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1. Construct a frequency table for the following ages (in years) of 30 students using equal class intervals,
one of them being 9-12, where 12 is not included.
18, 12, 7, 6, 11, 15, 21, 9, 13, 8, 15, 17, 19, 22, 14, 21, 8, 23, 12, 17, 6, 18, 15, 23, 16,
22, 9, 21, 16, 11
2. The electricity bills (in Rs.) of 40 houses in a locality are given below :
116, 127, 100, 107, 80, 82, 65, 91, 101, 95, 87, 105, 81, 129, 92, 75, 78, 89, 61, 121, 128, 63, 76, 84, 62, 98, 65, 95,
108, 115, 101, 65, 52, 59, 81, 87, 130, 118, 108, 116
Construct a grouped frequency table.
3. For the following data of weekly wages (in Rs.) received by 30 workers in a factory, construct a grouped
frequency distribution table.
258, 215, 320, 300, 290, 311, 242, 272, 268, 210, 242, 258, 268, 220, 210, 240, 280, 316, 306,
215, 236, 319, 304, 278, 254, 292, 306, 332, 318, 300
4. Construct a frequency table, with equal class-intervals from the following data on the weekly wages
(in Rs.) of 25 labourers working in a factory, taking one of the class-intervals as 460-500 (500 not
included).
580, 625, 485, 537, 540, 425, 637, 605, 607, 430, 611, 632, 600, 640, 638, 612, 584, 440, 536,
515, 449, 480, 556, 561, 508
5. Given below are two cumulative frequency distribution tables. Form a frequency distribution table for
each of these.
(i)
(ii)
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Ages ( in years ) No . of persons
Below 10
15
Below 20
28
B
Marks obtained
More than 10
More than 20
No. of Students
0
17
Below 30
Below 40
Below 50
39
60
73
More than 30
More than 40
More than 50
27
39
52
Below 60
80
More than 60
60
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6. On a certain day, the temperature in a city was recorded as under :
Time
5 a.m. 8 a.m. 11a.m. 3 p.m. 6 p.m.
Temperature (in C )
20
24
26
22
18
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Draw a bar graph to represent the above data.
7. Read the bar graph given below and answer the questions given below :
(i) What information is given by the bar graph?
(ii) In which year was the production maximum?
(iii) After which year was there a sudden fall in the production?
(iv) Find the ratio between the maximum and minimum production during the given period.
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8. The table given below shows the number of blinds in a village :
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Age group
0  20
20  40
No . of blinds
5
9
40  60
60  80
80  100
10
4
2
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Total
30
Draw a histogram to represent the above data.
9. The following table shows the average daily earnings of 40 general stores in a market, during a certain
week.
Daily earning (in Rs.) No. of stores
180
600 - 650
650 - 700
5
10
700 - 750
750 - 800
2
7
800 - 850
12
850 - 900
Total
4
40
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MATHEMATICS–IX
Draw a histogram to represent the above data.
10. The following table gives the heights of 50 students of a class. Draw a frequency polygon to represent
this.
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Height (in cm) No . of Students
120 -135
1
135 - 150
18
150 -165
165 - 180
180 -195
23
7
1
Total
50
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11. In a study of diabetic patients in a village, the following observations were noted. Represent the given
data by a frequency polygon.
Age (in Years) No. of Patients
10 - 20
20 - 30
30 - 40
40 - 50
50 - 60
60 - 70
Total
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6
B
12
20
9
4
53
12. Draw a histogram and a frequency polygon on the same graph to represent the following data :
Weight (in cm) No . of Persons
40 - 50
30
50 - 60
25
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60 - 70
70 - 80
80 - 90
40
30
10
Total
135
13. Draw a histogram to represent the following frequency distribution.
MATHEMATICS–IX
Class Interval Frequency
10 - 15
6
15 - 20
20 - 30
9
10
30 - 50
50 - 80
8
18
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181
14. Draw a histogram for the marks of students given below :
Marks No . of Students
0 - 10
8
10 - 30
32
30 - 45
45 - 50
50 - 60
18
10
6
Total
74
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15. The runs scored by two teams A and B on the first 120 balls in a cricket match are given below :
Number of balls Team A Team B
0 - 12
10
4
12 - 24
12
2
24 - 36
36 - 48
4
20
16
18
48 - 60
60 - 72
72 - 84
10
12
6
8
10
12
84 - 96
96 - 108
8
16
20
12
108 - 120
20
4
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Represent the data of both the teams on the same graph with the help of frequency polygons.
16. Find the mean for each of the following sets of numbers :
(i) 25, 12, 37, 19, 43, 40, 11
(ii) 6.2, 4.9, 7.1, 2.9, 5.7, 8.3
2
2
2
2
2
2
(iii) 1 , 2 , 3 , 4 , 5 , 6
(iv) 13, 23, 33, 43, 53
17. Calculate the mean ( x ) for each of the following distribution :
(i)
x
f
2 4 6 8 10
1 3 5 2 6
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(ii)
x
f
5 10 15 20 25 30
6 3 2 5 1 3
18. The following table shows the number of accidents met by 120 workers in a factory during a month :
No. of accidents 0 1 2 3 4 5
No. of workers 36 34 21 25 3 1
Find the average number of accidents per workers.
19. The marks obtained out of 50 by 80 students in a test are given below :
Marks
15 20 22 24 25 30 33 38 35
No. of Students 5 8 7 16 12 18 7 3 4
Calculate the average marks.
20. If the mean of the following data is 18.75, find the value of p.
xi 10 15 p 25 30
f i 5 10 7 8 2
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21. The average of height of 30 boys out of a class of 50, is 160 cm. If the average height of the remaining
boys is 165 cm, find the average height of the whole class.
22. The average of six numbers is 30. If the average of first four is 25 and that of last three is 35, find the
fourth number.
23. The mean of 100 observations was calculated as 40. It was found later on that one of the observations
was misread as 83 instead of 53. Find the corrected mean.
24. The mean of 10 numbers is 18. If 3 is subtracted from every number, what will be the new mean?
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25. If x is the mean of n observations x1, x2, ...., xn, then prove that
 ( xi  x )  0 i.e. the algebraic sum of
i 1
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deviations from mean is zero.
26. Find the median of following data :
(i) 14, 6, 18, 9, 23, 22, 10, 19, 24
(ii) 17, 13, 28, 19, 23, 22, 12, 32
27. The numbers 5, 7, 10, 12, 2x – 8, 2x + 10, 35, 41, 42, 50 are arranged in ascending order. If their median is
25, find the value of x.
28. Find the median of the following observations :
46, 64, 58, 87, 41, 77, 35, 55, 90, 92, 33. If 92 is replaced by 99 and 41 by 43 in the above data, find the new
median.
29. Find out the mode of the following data :
(i) 14, 28, 19, 25, 14, 31, 17, 14, 12, 27
(ii) 8.3, 8.9, 8.1, 8.7, 8.9, 7.9, 8.7, 8.9, 8.1
30. Given below is the number of pairs of shoes of different sizes sold in a day by the owner of the shop.
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Size of shoe
1 2 3 4 5 6 7 8 9
No. of pairs sold 2 2 3 4 5 5 6 9 1
What is the modal shoe size?
PRACTICE TEST
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General Instructions :
Each question carry 3 marks.
Time :
1
2
hour
1. Three coins were tossed 30 times simultaneously. Each time the number of heads occuring was noted
down as follows:
0
1
2
2
1
2
3
1
3
0
1
3
1
1
2
2
0
1
2
1
3
0
0
1
1
2
3
2
2
0
Prepare a frequency distribution table for this data.
2. A random survey of the number of children of various age groups playing in a park was found as follows:
Age ( in years ) 1 - 2 2 - 3 3 - 5 5 - 7 7 - 10 10 - 15 15 -17
No. of children 5
3
6
12
9
10
4
Draw a histogram to represent the data above.
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183
3. Find mean ( x ) for the following distribution:
x 10 15 20 25 30 35
f 3 2 4 7 3 1
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4. The following observations have been arranged in ascending order. If the median of the data is 63, find
the value of x.
29, 32, 48, 50, x, x + 2, 72, 78, 84, 95
5. The marks obtained by 80 students in a test are given. Find the modal marks.
Marks
4 12 20 28 36 44
No. of students 8 10 15 24 15 8
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ANSWERS OF PRACTICE EXERCISE
1.
2.
3.
Class
Frequency
6 - 9 9 -12 12 - 15 15 -18 18 - 21 21 - 24
5
4
4
7
B
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Weekly Wages
200 - 220 220 - 240 240 - 260 260 - 280 280 - 300 300 - 320 320 - 340
( in Rs.)
No. of
4
2
6
4
3
9
2
workers
Weekly Wages
420 - 460 460 - 500 500 - 540 540 - 580 580 - 620 620 - 660
( in Rs.)
No. of
4
2
4
3
7
5
workers
5. (i)
(ii)
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3
Class
50 - 60 60 - 70 70 - 80 80 - 90 90 - 100 100 - 110 110 - 120 120 - 130 130 -140
Frequency
2
6
3
8
5
7
4
4
1
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4.
7
Age (in year ) 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60
No. of persons 15
13
11
21
13
7
Marks obtained 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
No. of Students
8
13
12
10
17
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6.
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7. (i) The given bar graph shows the production (in million tonnes) of food grains during the period from
2000 to 2004.
(ii) 2002
(iii) 2000
(iv) 5 : 2
8.
9.
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11.
B
STATISTICS
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9
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6
5
12.
13.
3
2
0
14.
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B
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15.
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16. (i) 26.71
(ii) 5.85
(iii) 15.16
(iv) 45
17. (i) 7.05
(ii) 15.25
18. 1.4
19. 26.7 (approx)
21. 162 cm
22. 25
23. 39.7
24. 15
26. (i) 18
(ii) 20.5
27. 12
28. 58, 58
29. (i) 14
(ii) 8.9
30. 8
20. 20
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ANSWERS OF PRACTICE TEST
3. 22
4. 62
5. 28
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B
STATISTICS
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187