Data handling - Statistics & Probability for All.

Introduction

Find the mean of first five odd multiples of 5.

If the mean of 5 observations x, x + 2, x + 4, x + 6 and x + 8 is 11,

find the value of x.

If the mean of 25 observations is 27 and each observation is decreased by 7, what will be the new mean?

The mean of the marks secured by 15 students of section A of class X is 40, that of 25 students of section B is 50 and that of 20 students of section C is 60. Find the combined mean of the marks of students of three sections of class X.

Method of Calculating Mean of Grouped Data by Direct Method:

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

Number of students (fi) 1 1 3 4 3 2 4 4 1 1 2 3 1

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 byDirect method.

Percentages of female teacher 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85

Number of states/U.T. 6 11 7 4 4 2 1

If the mean of the following frequency distribution is 24, find the value of p.

Class 0 - 10 10 - 20 20 - 30 30  - 40 40 - 50

Frequency 3 4 p 3 2

Mean of Grouped Data by Assumed Mean Method:

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.

Percentages of female teacher 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85

Number of states/U.T. 6 11 7 4 4 2 1

For the following distribution, calculate mean by using direct assumed mean method.

Class interval 1 – 4 4 – 9 9 – 16 16 – 27

Frequency 6 12 26 20

Find the mean of the following frequency distribution using a suitable method:

Class 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70

Frequency 25 40 42 33 10

Mean of Grouped Data by Step Deviation Method:

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 Step Deviation Method.

Percentages of female teacher 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85

Number of states/U.T. 6 11 7 4 4 2 1

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

Number of bowlers 7 5 16 12 2 3

The mean of the following distribution is 31.4. Determine the missing frequency x.

Class 0 - 10 10 -20 20 -30 30 -40 40 -50 50 - 60

Frequency 5 x 10 12 7 8

Mode of Grouped Data

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.

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 ahousehold:

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.

Following frequency distribution shows the daily expenditure on milk of 30 households in a locality:

Daily expenditure on milk (in Rs) 0 - 30 30 -60 60 -90 90 -120 120-150

Number of households 5 6 9 6 4

Find the mode for the above data.

On Sports Day of a school, agewise participation of students is shown in the following distribution:

Age(in years) 5-7 7-9 9-11 11-13 13-15 15-17 17-19

Number of students x 15 18 30 50 48 x

Find the mode of the data. Also, find the missing frequencies when sum of frequencies is 181.

The mode of the following data is 36. Find the missing frequency x in it.

Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70

Frequency 8 10 x 16 12 6 7

Median of ungrouped &Grouped Data:

Find the median of the first ten prime numbers.

The median of the observations 21, 24, 27, 30, (x – 1), (x + 1), 35, 38, 48 and 50 arranged in ascending order is 33. Then, find the value of x.

Find the median of the following data.

Find the unknown values in the following table :

Median for Grouped Data & Missing frequency

The weekly expenditure of 500 families is tabulated below:

Weekly Expenditure (Rs) Number of families

0-1000 150

1000-2000 200

2000-3000 75

3000-4000 60

4000-5000 15

Find the median expenditure.

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

Less than 145

Less than 150

Less than 155

Less than 160

Less than 165 4

11

29

40

46

51

Find the median height.

Find the median of the following data :

Profit (in lakh of rupee) Number of shops

More than or equal to 5 30

More than or equal to 10 28

More than or equal to 15 16

More than or equal to 20 14

More than or equal to 25 10

More than or equal to 30 7

More than or equal to 35 3

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

Relationship between Mean, Median and Mode

Find the mean of the data using an empirical formula when it is given that mode is 50.5 and median is 45.5.

If the median of a series exceeds the mean by 3, find by what number the mode exceeds its mean?

If the median of a series exceeds the mean by 3, find by what number the mode exceeds its mean?

Recently the half-yearly examination was conducted in DAV public school. The mathematics teacher maintains a record of the marks of 100 students. On the basis of the recorded data of the marks obtained in Mathematics, the histogram is given below:

On the basis of the above histogram, answer the following questions:

1. Identify the modal class from the given graph.

2. Find the mode of the following distribution of marks obtained by the students in an examination.

Given the mean of the above distribution is 53, using empirical relationship estimate the value of its median.

Graphical Representation of Cumulative Frequency Distribution: 

Less than Type Ogive

The marks scored by 750 students in an examination are given in the form of a frequency distribution table:

Marks Number of students

600 – 640 16

640 – 680 45

680 – 720 156

720 – 760 284

760 – 800 172

800 – 840 59

840 – 880 18

Prepare a cumulative frequency table of less than type and draw an ogive.

The following distribution gives the daily income of 50 workers of a factory

Daily income (in Rs) 200-250 250-300 300 - 350 350-400 400-450 450-500

Number of workers 10 5 11 8 6 10

Convert the distribution to a 'less than type' cumulative frequency distribution and draw its ogive. Hence obtain the median of daily income.

In annual day of a school, age-wise participation of students is shown in the following frequency distribution:

Age of student (in years) 5-7 7-9 9-11 11-13 13-15 15-17 17-19

Number of students 20 18 22 25 20 15 10

Draw a less than type’ ogive for the above data and from it find the median age.

Graphical Representation of Cumulative Frequency Distribution:

More than Type Ogive

Following distribution shows the marks obtained by a class of 100 students

Marks 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70

Frequency 10 15 30 32 8 5

Draw a 'more than' ogive for the above data.

In an orchard, the numbers of apples on trees are given below :

Number of apples 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 more

than or

equal

to 100 more

than or

equal

to 110

Number of trees 60 55 39 29 10 6 2

Draw a 'more than type' ogive and hence obtain median from the curve.

The annual profits earned by 30 shops of a shopping complex in a locality give rise to the following distribution :

Profit (₹ in lakhs) 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

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

A survey was conducted by a group of students as a part of their environment awareness program, 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.

Which method did you use for finding the mean, and why?

Consider the following distribution of daily wages of 50 workers of a factory.

Find the mean daily wages of the workers of the factory by using an appropriate method.

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs18. Find the missing frequency f.

 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 heartbeats per minute for these women, choosing a suitable method.

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.

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

The table below shows the daily expenditure on food of 25 households in a locality.

Daily expenditure

(in ₹)

100 – 150

150 – 200

200 – 250

250 - 300

300 - 350

Number of households

4

5

12

2

2

Find the mean daily expenditure on food by a suitable method.

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

0.04 – 0.08

0.08 – 0.12

0.12 – 0.16

0.16 – 0.20

0.20 – 0.24

4

9

9

2

4

2

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.

Number of days

0 – 6

6 – 10

10 – 14

14 – 20

20 - 28

28 – 38

38 - 40

Number of students

11

10

7

4

4

3

1

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

Number if cities

3

10

11

8

3

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

Number of patients 6 11 21 23 14 5

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

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

Frequency 10 35 52 61 38 29

Determine the modal lifetimes of the components.

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 ₹) 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

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

20 – 25

25 – 30

30 – 35

35 – 40

40 – 45

45 – 50

50 - 55 3

8

9

10

3

0

0

2

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

4000 – 5000

5000 – 6000

6000 – 7000

7000 – 8000

8000 – 9000

9000 – 10000

10000 - 11000 4

18

9

7

6

3

1

1

Find the mode of the data

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

Frequency 7 14 13 12 20 11 15 8

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

85 – 105

105 – 125

125 – 145

145 – 165

165 – 185

185 - 205 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 Frequency

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60 5

X

20

15

Y

5

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) Number of policy holders

Below 20

Below 25

Below 30

Below 35

Below 40

Below 45

Below 50

Below 55

Below 60 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) Number of leaves

118 – 126

127 – 135

136 – 144

145 – 153

154 – 162

163 – 171

172 – 180 3

5

9

12

5

4

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.)

The following table gives the distribution of the life time of 400 neon lamps

Life time (in hours) Number of lamps

1500 – 2000

2000 – 2500

2500 – 3000

3000 – 3500

3500 – 4000

4000 – 4500

4500 - 5000 14

56

60

86

74

62

48

Find the median life time of a lamp.

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 – 10 10 – 13 13 – 16 16 – 19

Number of surnames 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.

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Weight (in kg) 40 – 45 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 70 – 75

Number of students 2 3 8 6 6 3 2

The following distribution gives the daily income of 50 workers of a factory.

Daily income

(in ₹) 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.

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

Less than 40

Less than 42

Less than 44

Less than 46

Less than 48

Less than 50

Less than 52 0

3

5

9

14

28

32

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.

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.