Statistics - Continuous Series Arithmetic Mode


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When data is given based on ranges along with their frequencies. Following is an example of continous series −

Items 0-5 5-10 10-20 20-30 30-40
Frequency 2 5 1 3 12

Formula

$M_o = {L} + \frac{f_1-f0}{2f_1-f_0-f_2} \times {i}$

Where −

  • ${M_o}$ = Mode

  • ${L}$ = Lower limit of modal class

  • ${f_1}$ = Frquencey of modal class

  • ${f_0}$ = Frquencey of pre-modal class

  • ${f_2}$ = Frquencey of class succeeding modal class

  • ${i}$ = Class interval.

In case there are two values of variable which have equal highest frequency, then the series is bi-modal and mode is said to be ill-defined. In such situations mode is calculated by the following formula −

Mode = 3 Median - 2 Mean

Arithmetic Mode can be used to describe qualitative phenomenon e.g. consumer preferences, brand preference etc. It is preferred as a measure of central tendency when the distribution is not normal because it is not affected by extreme values.

Example

Problem Statement

Calculate the Arithmetic Mode from the following data −

Wages

(in Rs.)

No.of workers
0-5 3
5-10 7
10-15 15
15-20 30
20-25 20
25-30 10
30-35 5

Solution −

Using following formula

$M_o = {L} + \frac{f_1-f0}{2f_1-f_0-f_2} \times {i}$

  • ${L}$ = 15

  • ${f_1}$ = 30

  • ${f_0}$ = 15

  • ${f_2}$ = 20

  • ${i}$ = 5

Substituting the values, we get

$M_o = {15} + \frac{30-15}{2 \times 30-15-20} \times {5} \\[7pt] \, = {15+3} \\[7pt] \, = {18}$

Thus Arithmetic Mode is 18.

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