 An average should be rigidly defined; otherwise its value will be affected by the bias of the person who calculates it. It cannot be a good representative if it is not a fixed value.
 It should be based on all the observations. It sill not be a good representative if some observations are left out of calculation.
 It should be easy to calculate and easy to understand. If the calculations requires tedious mathematical process, it will not be understood by many and it use will be limited.
 It should be capable of further algebraic treatment. This makes the average more useful.
 It should not be affected much by sampling fluctuations. If two independent samples are taken from the same population, the average should not differ significantly.
 Find the arithmetic mean for the following sets of observations:
 125, 132, 127, 139, 140, 142, 137, 122, 120 and 130
 13.1, 15.2, 11.9, 10.2, 12.5, 14.3, 11.2, 10.8
 1357, 1454, 1389, 1405, 1485
 53, 31, 35, 25, 100, 60, 16, 13, 3, 95
xi:

12

14

16

18

20

22

fi:

5

10

15

12

8

3

Size of Shoe:

6

7

8

9

10

11

No. of pairs:

32

40

52

40

32

25

xi:

5

6

7

8

9

10

11

fi:

11

15

20

16

12

9

4

Wt. in Kgs.

11

12

13

14

15

16

No. of Children

7

11

15

13

9

4

Marks:

0

1

2

3

4

5

6

7

8

9

10

No. of Students:

2

4

5

7

11

15

13

10

7

3

1

Marks:

010

1020

2030

3040

4050

No. of students

6

11

15

8

3

Wages in Rs.:

2040

4060

6080

80100

100120

120140

No. of workers:

7

12

16

13

13

4

Age in years

No. of persons

Less than 10
Less than 20
Less than 30
Less than 40
Less than 50
Less than 60

15
33
54
80
97
100

Daily wages:

515

1525

2535

3545

4455

Ans:
31.54

No. of persons:

3

8

13

10

5

Salary in Rs.:

700800

800900

9001100

1101500

15001600

No. of persons:

32

43

55

22

18

Class Interval:

2030

3050

5070

7090

90100

Frequency:

9

14

20

12

5

Sales in ‘000 Rs.:

510

1015

1525

2535

3545

4550

No. of shops:

2

8

12

15

11

5

Yield per acre:

700750

750800

800850

850900

900950

9501000

No. of farms:

32

43

55

22

17

18

Height in cms:

140145

145150

150155

155160

160165

No. of students:

7

10

15

13

5

Height in cms:

148152

152156

156160

160164

164168

168172

172176

No. of persons:

3

5

9

15

10

6

2

Amount in Rs.:

500599

600699

700799

800899

900999

10001099

11001199

12001299

No. of accounts:

25

42

55

70

62

50

35

11

Tax in Rs.:

100399

400699

700999

10001299

13001599

16001899

No. of Employees:

12

20

25

35

15

8

No. of units produced:

5099

100149

150199

200249

250299

300349

350399

No. of factories:

4

9

11

15

12

8

2

Salary in Rs.:

No. of Employees.

900 – 1000
1000 – 1200
1200 – 1400
1400 – 1600
1600 – 1800
1800 – 1900
1900 – 2000

4
11
19
22
18
9
3

Wages in Rs.:

3040

4050

5060

6070

7080

8090

No. of persons:

10

20

40

…

8

6

Marks:

0  10

1020

2030

3040

4050

No. of students:

3

…

15

10

5

Wages in Rs.:

2040

4060

6080

80100

100120

120140

No. of persons:

10

18

22

…

11

5

Marks:

010

1020

2030

3040

4050

5060

No. of students:

5

10

25

30

…

10

Class Interval:

05

510

1015

1520

2025

2530

3035

3540

Frequencies:

2

5

7

…

…

16

8

3

 The average marks of a group of 100 students in Accountancy are 60 and for another group of 50 students, the average marks are 90. Find the average marks of the combined group of 150 students. [ 70 marks ]
 The average daily wages for 90 workers in a factory is Rs.59/, the average wages for 50 male workers out of them is Rs.63/. Find the average wages for the remaining female workers. [ Rs.54/ ]
 The average marks of a class of students are 76. The average marks of boys and girls are 69 and 83 respectively. If there are 100 boys in the class find the number of girls in the class. [ No. of girls = 100 ]
 The mean daily wages of a group of employees are Rs.180/ The mean daily wages of men and women are Rs.186/ and Rs.175/ respectively. Find the ratio of men and women in the group. [ 5 : 6 ]
 If the average marks in a certain test of boys and girls in a class are 80 and 85 respectively and if the average marks for the entire class are 83.75, find the percentage of boys in the class. [ 25% ]
 The mean weight of a group of 70 workers is 60 kgs. The second group consists of 80 workers with average weight 57 kgs and there are 50 workers in the third group with average weight 62 kgs. Find the average weight of the combined group of 200 workers. [ 59.3 kgs ]
 The mean marks of 100 boys in a class are 45. The mean marks of the entire class of 150 students are 50. Find the mean marks of the remaining group of girls. [ 60 ]
 There are three groups in a class of 100 students. The first contains 25 students with average pocket money Rs.62/, the second group consists of 50 students with average pocket money Rs.55/. Find the average pocket money of the students from the third group if the average for the entire class is Rs.58/. [ Rs.60/ ]
 The average monthly salary of employees of a firm is Rs.5200/. The average salaries of gents and ladies from the firm are Rs.6000/ and Rs.4800/ . Find the percentage of gents and ladies in the firm. [ 1 : 2 ]
 A garment factory makes both men’s and women’s shirts. The average profit of the factory is 8% of sales. Average profit on men’s shirt is 10%. Women’s shirts form 60% of the total sales. What is the average profit on sales of women’s shirts? [ 6.67% ]
 There are men, women and children working in a factory. The total number of workers is 500. The average daily wages of 250 male workers is Rs.100/. The average daily wages of 150 women workers is Rs.80/. What is the average daily wages of children working in that factory, given that the average daily wages of all the 500 workers taken together is Rs.82/ [ Rs.40/ ]
 The sum of the deviations of a certain number of observations measured from 4 is 72 and the sum of the deviations of the same observations from 7 is 3. Find the number of observations and their mean. [ 25 & 6.88 ]
 The mean of a certain number of observations is 40. If two more observations with values 50 and 64 are added to the data, the mean rises to 42. Find the number of items in the original data. [ 15 ]
 The mean weight of 98 students as calculated from a frequency distribution is found to be 50 kgs. It is later discovered that the frequency of the class interval 30 – 40 was wrongly taken as 8 instead of 10. Calculate the correct arithmetic mean. [ 49.7 kgs ]
 The mean monthly salaries paid to all 77 employees in a company was Rs.78/. The mean monthly salaries of 32 of them was Rs.75/ and that of the other 25 was Rs.82/. What was the mean salary of the remaining? [ Rs.77.80 ]
 n is ODD, then there will be only one middle term and the value of the middle term is the median. i.e., Median = .
 n is EVEN, then there will be two middle terms, the average of the values of the two middle terms is the median. i.e., Median = .
 The very first cumulative frequency value for which becomes less than determines the median class.
 To calculate the median the class intervals must be continuous. If they are not continuous then we have to make them continuous by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.
 The value of median thus calculated must be a value in the C.I. of the median class.
 43, 30, 44, 67, 35, 40, 59 [ 43 ]
 16, 19, 27, 10, 5, 7, 12, 15 [ 13.5 ]
 Calculate the median of the following distribution:
xi:

15

20

25

30

35

40

45

50

fi:

7

15

19

23

20

15

8

5

Ht. in cms:

158162

162166

166170

170174

174178

178182

No. of students:

3

7

12

15

6

2

Wages in Rs.:

Less than 35

35 – 40

40 – 45

45 – 50

Over 50

No. of workers:

24

62

99

18

15

Age in years:

Below 35

35 – 50

50 – 65

65 – 80

Above 80

No. of persons:

20

18

32

18

12

Rainfall in cms:

2025

2530

3035

3540

4045

4550

5055

No. of Years:

2

5

8

12

10

7

6

Weights in Kgs:

3034

3539

4044

4549

5054

5559

6064

No. of students:

3

5

12

18

14

6

2

Commission in Rs.

100140

140180

180200

200220

220240

240260

260300

No. of Salesmen:

14

45

52

80

32

23

24

Height in cms:

150 – 155

155 – 160

160 – 165

165 – 170

170 – 175

175 – 180

No. of students:

3

6

8

…

3

1

Age in years:

1019

2029

3039

4049

5059

6069

7079

No. of Persons:

2

5

10

8

…

5

3

Wages in Rs.:

No. of persons
 
12.5 – 17.5
17.5 – 22.5
22.5 – 27.5
27.5 – 32.5
32.5 – 37.5
37.5 – 42.5
42.5 – 47.5
47.5 – 52.5
52.5 – 57.5

2
22
10
…
3
4
6
1
1

[ 16 ]

Total

63

Marks:

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

No. of Students:

5

5

10

…

3

Weekly wages in Rs.

6075

7590

90105

105120

120135

135150

No. of workers:

3

3

6

5

…

6

Weekly wages in Rs.

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

No. of workers:

5

…

20

30

…

10

Weekly wages in Rs.

100200

200300

300400

400500

500600

600700

No. of workers:

7

10

…

20

16

…

C.I.:

110120

120130

130140

140150

150160

160170

170180

Frequency:

12

…

34

65

46

…

18

C.I.:

1020

2030

3040

4050

5060

6070

7080

Frequency:

13

30

…

65

…

25

18

Income in Rs.

200 – 400

400 – 600

600 – 800

8001000

10001200

No. of persons:

16

34

60

37

13

No. of Units:

0 100

100200

200300

300400

400500

500600

No. of consumers:

9

18

35

32

28

10

Marks:

1030

3050

5070

7090

90110

110130

No. of students:

4

10

14

12

8

6

Wages in Rs.:

1015

1520

2025

2530

3035

3540

No. of Employees:

3

5

15

20

8

7

Class Interval:

6075

7590

90105

105120

120135

135150

Frequency:

3

3

6

…

7

6

Life in hrs.:

200400

400600

600800

8001000

10001200

12001400

14001600

No. of bulbs:

10

…

50

45

30

…

5

Age (below)

10

20

30

40

50

60

No. of persons:

11

35

50

79

89

100

Class Interval:

6075

7590

90105

105120

120135

135150

Frequency:

3

3

6

5

7

6

Class Interval:

1030

3050

5070

7090

90110

110130

Frequency:

4

10

14

12

8

6

Wages in Rs.:

010

1020

2030

3040

4050

5060

6070

Total

No. of Workers:

4

16

…

…

…

6

4

230

Wages in Rs.:

100200

200300

300400

400500

500600

600700

No. of Workers:

5

12

…

…

14

11

 Find the missing frequencies for the following data, given that the modal marks are 53.25 and median is 52.5. Find the arithmetic mean of the completed data.
Marks:

2029

3039

4049

5059

6069

7079

8089

No. of Students:

10

18

25

…

15

12

…

Age in years:

1020

2030

3040

4050

5060

6070

7080

8090

No. of persons:

10

10

…

50

29

15

…

10

Weekly Expenditure:

50 – 70

70 – 90

90 – 110

110 – 130

130 – 150

No. of Families:

…

60

70

…

10

Marks:

1030

3050

5070

7090

90110

110130

130150

No. of students:

5

…

25

30

…

10

5

Age in years:

2530

3035

3540

4045

4550

5055

5560

No.of persons:

8

…

24

30

…

20

14

Life in hours:

9501150

11501350

13501550

15501750

17501950

19502150

No. of bulbs:

..

43

100

…

23

13

Marks:

05

510

1015

1520

2025

2530

3035

3540

Total

Frequency:

4

8

…

19

…

10

5

…

72

Life in hours:

10002000

20003000

30004000

40005000

50006000

60007000

70008000

No. of bulbs:

100

…

200

…

150

50

50

Age in years:

0 – 20

20 – 40

40 – 60

60 – 80

80 – 100

No. of persons:

4

5

…

11

5
