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MEASURE OF CENTRAL TENDENCY: MODE

Dr. Anshul Singh Thapa

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�MODE

  • Sometimes, you may be interested in knowing the most typical value of series or the value around which maximum concentration of items occurs. For example, a manufacturer would like to know the size of shoes that has maximum demand or style of the shirt that is more frequently demanded. Here, Mode is the most appropriate measure.
  • The word mode has been derived from the French word “la Mode” which signifies the most fashionable values of a distribution, because it is repeated the highest number of times in the series. Mode is the most frequently observed data value. It is denoted by Mo.

​

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MODE

MODE

FOR UNGROUPED DATA

MODE

FOR GROUPED DATA

EVEN NUMBER

ODD NUMBER

DISCRETE SERIES

CONTINUOUS SERIES

Value which occurs with the highest frequency

Grouping Table and Analysis Table

L + (D1) × i

D1 – D2

Unimodal, Bimodal and Multimodal

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�COMPUTATION OF MODE

  • For determining mode count the number of times the various values repeat themselves and the value occurring maximum number of times is the modal value. The process of determining mode in case of ungrouped data essentially involves grouping of data.

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EXAMPLE

Sr. No.

Marks Obtained

1

10

2

27

3

24

4

12

5

27

6

27

7

20

8

18

9

15

10

30

Since the item 27 occurs the maximum number of times, i.e., 3, hence the modal marks are 27

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EXAMPLE

Sr. No.

Marks Obtained

1

10

2

27

3

24

4

12

5

27

6

27

7

20

8

18

9

15

10

30

Since the item 27 occurs the maximum number of times, i.e., 3, hence the modal marks are 27

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�DISCRETE SERIES

Consider the data set 1, 2, 3, 4, 4, 5. The mode for this data is 4 because 4 occurs most frequently (twice) in the data.

Example

Look at the following discrete series:

Marks 10 15 20 25 30 35 40

Frequency 08 12 36 25 28 18 09

  • Here, as you can see the maximum frequency is 36, the value of mode is 20. In this case, as there is a unique value of mode, the data is Unimodal.
  • But, the mode is not necessarily unique, unlike arithmetic mean and median. You can have data with two modes (bi-modal) or more than two modes (multi-modal). It may be possible that there may be no mode if no value appears more frequent than any other value in the distribution. For example, in a series 1, 1, 2, 2, 3, 3, 4, 4, there is no mode.

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  • From the above data we can clearly say that the modal size is 20 because the value 20 has occurred the maximum numbers of times, i.e., 36. However, where the mode is determined just by inspection, if maximum frequency is repeated or the gap between the frequency preceding it or succeeding it is very small (if the difference is 3-4 points) and items are heavily concentrated on either side. In such cases it is desirable to prepare a grouping table and an analysis table.
  • A grouping table has six columns. In column 1 the maximum frequency is marked or put in a circle; in column 2 frequency are grouped in two’s; in column 3 leave the first frequency and then grouped the remaining in two’s; in column 4 group the frequencies in three’s; in column 5 leave the first frequency and group the frequencies in three’s; and in column 6 leave the first two frequencies and then group the remaining in three’s. in each of these cases take the maximum total and mark it in a circle or by bold type.

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  • After preparing the grouping table, prepare an analysis table. While preparing this table put column number on the left hand side and the various probable values of mode on the right hand side. The values against which frequencies are the highest are marked in the grouping table and then entered by means of a bar in the relevant ‘box’ corresponding to the value they represent.

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�GROUPING TABLE

x

f

II

III

IV

V

VI

10

20

15

35

25

30

40

12

08

35

09

36

28

18

20

71

46

48

63

27

56

81

83

55

99

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�GROUPING TABLE

x

f

II

III

IV

V

VI

10

20

15

35

25

30

40

12

08

35

09

36

28

18

20

71

46

48

63

27

56

81

83

55

99

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ANALYSIS TABLE

Col. No.

I

V

II

IV

VI

III

20

25

30

1

1

1

1

1

1

1

1

1

1

1

1

3

5

4

Corresponding to the maximum total 5, the value of the variable is 25

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NUMERICAL

X

f

15

2

20

3

25

4

30

10

35

11

40

12

45

3

50

2

55

1

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�CONTINUOUS SERIES

  • In case of continuous frequency distribution, modal class is the class with largest frequency. Mode can be calculated by using the formula:
  • Mo = L + D1 x i

D1 + D2

  • Where L = lower limit of the modal class, D1= difference between the frequency of the modal class and the frequency of the class preceding the modal class (ignoring signs). D2 = difference between the frequency of the modal class and the frequency of the class succeeding the modal class (ignoring signs), i = class interval of the distribution.

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Marks

No. of Students

0 – 10

3

10 – 20

5

20 – 30

7

30 – 40

10

40 – 50

10

50 – 60

15

60 – 70

10

70 – 80

6

80 – 90

2

90 – 100

8

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  • By inspection the model class is 50 – 60.

Mo = L + D1 x i

D1 + D2

L = 50, D1 = (15 - 10) = 5, D2 = (15 - 10) = 5, i = 10

Mo = 50 + 5 x 10 = 50 + 5 = 55

5 + 5

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Weight

No. of person

100 – 110

4

110 – 120

6

120 – 130

20

130 – 140

25

140 – 150

33

150 – 160

17

160 – 170

8

170 – 180

2

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Weight

No. of person

100 – 110

4

110 – 120

6

120 – 130

20

130 – 140

32

140 – 150

33

150 – 160

17

160 – 170

8

170 – 180

2

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GROUPING TABLE

x

f

II

III

IV

V

VI

100 - 110

120 - 130

110 - 120

150 -160

130 -140

140 -150

160 - 170

06

04

32

08

20

33

17

10

52

50

26

65

25

30

82

58

58

85

170 - 180

02

10

27

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�ANALYSIS TABLE

Col. No.

I

V

II

IV

VI

III

120 - 130

130 - 140

140 - 150

1

1

1

1

1

1

1

1

1

1

1

1

5

5

3

This is a bi-modal series. Hence mode has to be determined by applying the formula

1

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�EMPIRICAL MODE

Mode = 3 Median – 2 Mean

​

Mode = (3 X 139.69) – (2 X 139.51)

​

Mode = 419.07 – 279.02 = 140.05

​

Hence the modal weight is 140.05

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EMPIRICAL MODE

  • Where mode is ill-defined, its value may be ascertained by the following formula base upon the relationship between mean, median and mode.

Mode = 3 Median – 2 Mean

  • This measure is called the empirical mode.

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Types of Series

Method of Calculation

1. Ungrouped Data (individual Series)

  1. The value that occurs the most in the series is identified as Mode of the series by inspection method.
  2. If the frequency of all values is equal the same are changed into discrete frequency distribution and then mode is calculated.

2. Discrete Series/ Frequency Array

  1. The value that occurs the most in the series is identified as Mode of the series by inspection method.
  2. If the items with highest frequency are more than one, then the grouping method is used.

3. Continuous Series

  1. Exclusive: Series with highest frequency is called modal class. The actual value of mode is determined using the following formula:

Mo = L + D1 x i

D1 + D2

(b) Inclusive: Inclusive series are converted into exclusive.

4.Moderately asymmetrical distributions

Mode = 3 Median – 2 Mean

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RELATIVE POSITION OF ARITHMETIC�MEAN, MEDIAN AND MODE

  • Suppose we express, Arithmetic Mean = Me, Median = Mi, Mode = Mo
  • The relative magnitude of the three are Me>Mi>Mo or Me<Mi<Mo. The median is always between the arithmetic mean and the mode.

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RELATIONSHIP AMONG MEAN, MEDIAN AND MODE

  • A distribution in which the values of mean, median and mode coincide (i.e., mean = median = mode)is known as a symmetrical distribution.
  • Conversely stated, when the values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed.
  • in case of asymmetrical distribution, value of mean, mode and median will be different. As such, frequency curve will not be bell shaped. It may be skewed either to the right or to the left.
  • If the distribution is skewed more to the right i.e. positive, the mean and the median will be grater than mode.. In order words mode is the minimum.
  • If the distribution is skewed more to the left, i.e., negative, the mean and the median will be less than mode. In other words mode will be maximum.

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