Lesson Notes By Weeks and Term - Senior Secondary School 1

Mean Deviation, Variance and standard Deviation of Grouped Data

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SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 3rd TERM

REFERENCE BOOKS

  • New General Mathematics SSS 1 by M.F. Macrae et al 
  • Essential Mathematics SS 1

 

 
WEEK ELEVEN

TOPIC: Mean Deviation, Variance and standard Deviation of Grouped Data

 

Mean Deviation of Grouped Data

Example 1

The speeds of 40 cars in a certain road are tabulated as follows: 

Speed (km/h)

50 – 54

55 – 59

60 – 64

65 – 69

60 – 74

75 – 80

80 - 84

Frequency

5

10

15

12

10

6

2

For this distribution, calculate 

  1. The mean
  2. The mean deviation 

 

Solution 

The complete table of the distribution is shown below. 

Class interval

Mid – value (xm)

f

fxm

xm - x

fxm - x

50 – 54

52

5

260

13.17

65.85

55 – 59

57

10

570

8.17

81.5

60 – 64

62

15

930

3.17

47.55

65 – 69

67

12

804

1.83

21.96

60 – 74

72

10

720

6.83

68.3

75 – 80

77

6

462

11.83

70.98

80 – 84

82

2

164

16.83

33.66

Total 

 

Ef=60

Efxm=3910

 

Efxm - x

 

  1. Mean, x = fxmf =  391060 =  65.17

The mean is 65.2km/h to 1 d.p.

  1. Mean deviation = fxm - xf = 39060 =  6.5

The mean deviation is 6.5km/h 

 

EVALUATION 

  1. Calculate the mean and the mean deviation of the following:
  1. 8, 5, 12, 8, 13, 4, 9, 5, 4, 7
  2. 9.25, 8.04, 12.08, 9.82, 10.05, 2.05, 8.25, 7.64, 7.02, 8.02

 

Variance and Standard Deviation of a Grouped Data 

Example 1

The table shows the time to the nearest hours of television watched by a group of students in a week. 

Time

1 – 5

6 – 10

11 – 15

16 – 20

21 – 25

26 – 30

31 – 35

36 – 40

Frequency

2

5

8

10

14

6

4

1

Calculate 

  1. The mean
  2. The variance 
  3. The standard deviation

 

Solution 

Let xm represents the mid-value (or class mark) of the interval.

  1. x = fxmf = 99050=19.8h

Now subtract 19.8 from each value in the 2nd column to obtain the results in the 5th column. Then complete the other two columns as shown in the table.

  1. S2 = fx - x2f = 323850=64.76

Variance = 64.8h to 3 s.f.

  1. S = 64.76 = 8.047h

Standard deviation is 8.05h to 3 s.f.

 

Alternative method 

Interval

xm

f

fxm

x2m

fx2

1 – 5 

3

2

6

9

18

6 – 10

8

5

40

64

320

11 – 15

13

8

104

169

1352

16 – 20

18

10

180

324

3240

21 – 25

23

14

322

529

7406

26 – 30

28

6

168

784

4704

31 – 35

33

4

132

1089

4356

36 – 40

38

1

38

1444

1444

Total 

 

f=50

fxm=990

 

fx2m=22 840

 

EVALUATION

  1. Calculate to 1 d.p the mean and standard deviation of the following numbers: 
  1. 5, 7, 12, 10, 5, 15, 14, 9, 7, 8
  2. 6.5, 8.5, 6.5, 8.4, 6.9, 2.5, 6.2, 5.5

 

GENERAL EVALUATION 

  1. The table bellows shows the age distributions of a group of people. 

Age (yrs)

20 – 29

30 – 39

40 – 49

50 – 59

60 – 69

70 – 79

Frequency

3

5

10

13

7

2

Calculate: 

  1. The mean age
  2. The variance
  3. The standard deviation 

 

READING ASSIGNMENT 

Essential Mathematics for Senior Secondary 1 pgs 237 - 248

 

WEEKEND ASSIGNMENT 

  1. The lowest temperatures of a city in Asia for 10 consecutive days are recorded as: - 5oC, - 6oC, -5oC, 4oC, 0oC, 1oC, 2oC, 3oC, 4oC, 7oC. Find the mean deviation.    A. 3.9     B. 4.0     C. 3.6      D. 6.4

Use the table below to answer question 2 to 4

A dice is thrown 100 times. The results are recorded as shown in the following table 

Score

1 

2

3

4

5

6

Frequency

15

18

17

21

14

15

Calculate:

  1. The mean score    A. 4.0     B. 3.5      C. 1.0       D. 5.6 
  2. The variance    A. 2.7      B. 3.7       C. 2.1        D. 1 
  3. The standard deviation     A. 4       B. 5.1       C. 1.6       D. 7
  4. Find the variance of x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x and 10x.    A. 8.25x29x2     B. 10x2     7.25x2

 

THEORY 

  1. The shoe sizes of a group of people are as follows: 

Shoe size

5

6

7

8

9

10

11

12

13

Frequency

3

8

14

16

20

10

5

3

1

For this distribution, calculate the mean deviation 

  1. The table below show the age distributions of a group of people.

Age (yrs)

20 – 29

30 – 39

40 – 49

50 – 59

60 -69

70 – 79 

Frequency

3

5

10

13

7

2

Calculate (a) the mean age    (b) the variance    (c) the standard deviation 








 



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