Measures of Central Tendency I

Definition:

A value which is used in this way to represent the distribution is called an ‘average’.  Since the average lies in the centre of a distribution, they are called ‘measures of central tendency’.  They are also known as ‘measures of location’.

Types of Averages:

(a)    Arithmetic Mean,

(b)   Geometric Mean,

(c)    Harmonic Mean,

(d)   Median, and

(e)    Mode.

 

(a)   Arithmetic Mean (AM):

1.      AM is defined as the value obtained by dividing the sum of the values by their number.

2.      It is expressed as follows for sample data:

3.      For population data:

4.      The above formulae are for ungrouped data and they cannot be applied to grouped data.  For grouped data, the formula for AM is as follows:

 or

Where x (all the values) falling in a class are assumed to be equal to the class mark or mid point of that class.  So on the sum of the values in kth class would be fkxk, and the sum of values in all the k classes would be equal to .

The total number of values is the sum of class frequencies, i.e., .

Example:

Class Boundaries

Frequency

9.5-19.5

5

19.5-29.5

8

29.5-39.5

13

39.5-49.5

19

49.5-59.5

23

59.5-69.5

15

69.5-79.5

7

79.5-89.5

5

89.5-99.5

3

99.5-109.5

2

Total

100

Calculate Mean.

Solution:

Class Boundaries

f

x (Mid-Point)

fx

9.5-19.5

5

14.5

72.5

19.5-29.5

8

24.5

196

29.5-39.5

13

34.5

448.5

39.5-49.5

19

44.5

845.5

49.5-59.5

23

54.5

1253.5

59.5-69.5

15

64.5

967.5

69.5-79.5

7

74.5

521.5

79.5-89.5

5

84.5

422.5

89.5-99.5

3

94.5

283.5

99.5-109.5

2

104.5

209

Total

100

 

5220

Alternate Formulae for Computing Mean:

1.      The computation of AM using the grouped data formula is easily provided that the values x and f are not large.

2.      If the values are large, considerable time can be saved by taking deviations from an assumed or guessed mean.

3.      If A is an assumed or guessed mean and D denotes the deviations of x from A (i.e., D = x – A) then x = A + D.

4.      The AM can be expressed as follows:

 ------- for ungrouped data

 ------- for grouped data

Example:

Class Boundaries

Frequency

9.5-19.5

5

19.5-29.5

8

29.5-39.5

13

39.5-49.5

19

49.5-59.5

23

59.5-69.5

15

69.5-79.5

7

79.5-89.5

5

89.5-99.5

3

99.5-109.5

2

Total

100

Solution:

Class Boundaries

f

X

D = x – A

fD

        9.5-19.5       

5

14.5

-40

-200

19.5-29.5

8

24.5

-30

-240

29.5-39.5

13

34.5

-20

-260

39.5-49.5

19

44.5

-10

-190

49.5-59.5

23

54.5

0

0

59.5-69.5

15

64.5

10

150

69.5-79.5

7

74.5

20

140

79.5-89.5

5

84.5

30

150

89.5-99.5

3

94.5

40

120

99.5-109.5

2

104.5

50

100

Total

100

 

 

-230

Although any class mark can be taken as assumed mean, we take the class mark 54.5 as A, because it corresponds to the largest frequency.  See the above table. We have A = 54.5, n = 100 and  = -230.

Second Alternative Method for Computing Mean:

  1. If all the class intervals are equal size ‘h’, the computation of the mean can be further simplified using a ‘coding variable’ ‘u’, where:

 ,

 -------------------- for ungrouped data

 ---------------------- for grouped data

  1. This method is also known as ‘coding method’.

Example:

Class Boundaries

Frequency

9.5-19.5

5

19.5-29.5

8

29.5-39.5

13

39.5-49.5

19

49.5-59.5

23

59.5-69.5

15

69.5-79.5

7

79.5-89.5

5

89.5-99.5

3

99.5-109.5

2

Total

100

Solution:

Class Boundaries

f

X

D = x – A

fu

9.5-19.5

5

14.5

-40

-4

-20

19.5-29.5

8

24.5

-30

-3

-24

29.5-39.5

13

34.5

-20

-2

-26

39.5-49.5

19

44.5

-10

-1

-19

49.5-59.5

23

54.5

0

0

0

59.5-69.5

15

64.5

10

1

15

69.5-79.5

7

74.5

20

2

14

79.5-89.5

5

84.5

30

3

15

89.5-99.5

3

94.5

40

4

12

99.5-109.5

2

104.5

50

5

10

Total

100

 

 

 

-23

A = 54.5; n = 100; h = 10 and  = -23

Weighted Arithmetic Mean (WAM):

  1. WAM is used to find average of certain values which are not of equal importance.

  2. The numerical values are called ‘weights’, and denoted as w1, w2, ….. wk.

  3. WAM is expressed as follows:

Example:

Sectors

Expenditure

(All figures in Rs. Billion)

Weight

General Public Service

503

46

Development

272

25

Defence

223

20

Public Order Safety

19

3

Education

17

3

Health

4

2

Housing

1

1

Environment Protection

1

1

Calculate Weighted Average Mean.

Solution:

Sectors

X

W

WX

General Public Service

503

46

23138

Development

272

24

6528

Defence

223

20

4460

Public Order Safety

19

3

57

Education

17

3

51

Health

4

2

8

Housing

1

1

1

Environment Protection

1

1

1

Total

 

100

34244

Properties of Arithmetic Mean:

  1. The sum of deviations of values from their mean is equal to zero:

 or

  1. If n1 values have mean ; n2 values have mean ; and so on nk values have mean , the mean of all the values is:

  1. The sum of squares of the deviation of the values x from any value a is minimum if and only if a = :

 is a minimum, if a = .

  1. The AM is affected by change of origin and scale.  By this, we mean that if we add or subtract a constant from all values or multiply or divide all the values by a constant, the mean is affected by these changes:

If x = a (a constant),     then = a

If y = x ± a,                  then =  ± a

If y = bx,                      then = b

If y = ,                      then =

(b)   Geometric Mean (GM):

1.      GM is defined only for non-zero positive values.  It is the nth root of the product of n values in the data.

2.      It can be expressed as follows:

Where  = x1 × x2 × x3 × ………. × xn.

Alternate Method for Computing Geometric Mean:

  1. In practice, it is difficult to extract higher roots.  The GM is, therefore, computed using logarithms:

  1. The GM is used mainly to find the average of ratios, rates of change, economic indices and the like.  It is preferred when data array follows a pattern of ‘geometric progression’.

Example:

Find the GM of the following data:

5,6,7,8,2,3,1,10,13,11

Solution:

Where  = x1 × x2 × x3 × ………. × xn.

Geometric Mean for Grouped Data:

For grouped data, the GM is computed as below:

Taking logarithm of both sides:

Weighted Geometric Mean:

Example:

Class Boundaries

f

9.5-19.5

5

19.5-29.5

8

29.5-39.5

13

39.5-49.5

19

49.5-59.5

23

59.5-69.5

15

69.5-79.5

7

79.5-89.5

5

89.5-99.5

3

99.5-109.5

2

Total

100

Solution:

Class Boundaries

f

x

log x

f log x

9.5-19.5

5

14.5

1.1614

8.07

19.5-29.5

8

24.5

1.3892

11.1136

29.5-39.5

13

34.5

1.5378

19.9914

39.5-49.5

19

44.5

1.6484

31.3196

49.5-59.5

23

54.5

1.7364

39.9372

59.5-69.5

15

64.5

1.8096

27.144

69.5-79.5

7

74.5

1.8722

13.1054

79.5-89.5

5

84.5

1.9269

9.6345

89.5-99.5

3

94.5

1.9754

5.9262

99.5-109.5

2

104.5

2.0191

4.0382

Total

100

 

 

170.2801

(c)   Harmonic Mean (HM):

HM is defined only for non-zero positive values.  It is the reciprocal of mean of reciprocals of values.  More briefly, HM, of a set of n values x1, x2, ….. , xn, is the reciprocal of the AM of the reciprocals of the values.  Thus:

 -------------- for ungrouped data

Harmonic Mean for Grouped Data:

The reciprocal of the class marks (in case of grouped data) will be , ,……., .  Since the reciprocals occur with frequencies f1, f2, ….. , fk, the total value of the reciprocals in the first class is , in second class , ….. , and in the kth class is .  The sum of reciprocals in all the k classes would be:

Weighted Harmonic Mean:

Example:

Find HM of the values 1, 2 and 3.

Solution:

Example:

Class Boundaries

Frequency

9.5-19.5

5

19.5-29.5

8

29.5-39.5

13

39.5-49.5

19

49.5-59.5

23

59.5-69.5

15

69.5-79.5

7

79.5-89.5

5

89.5-99.5

3

99.5-109.5

2

Total

100

Solution:

Class Boundaries

f

x

9.5-19.5

5

14.5

0.06897

0.34485

19.5-29.5

8

24.5

0.04082

0.32656

29.5-39.5

13

34.5

0.02899

0.37687

39.5-49.5

19

44.5

0.02247

0.42693

49.5-59.5

23

54.5

0.01835

0.42205

59.5-69.5

15

64.5

0.01550

0.2325

69.5-79.5

7

74.5

0.01342

0.09394

79.5-89.5

5

84.5

0.01183

0.05915

89.5-99.5

3

94.5

0.01058

0.03174

99.5-109.5

2

104.5

0.00957

0.01914

Total

100

 

 

2.33373

Relation between AM, GM and HM:

  1. The AM is greater than GM, the GM is greater than HM:

  1. AM, GM and HM are equal when all the values are equal, (e.g., 5,5,5,5,….):

  1. Therefore, the relationship between AM, GM and HM is expressed as follows:

Continued

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