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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 −
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${M_o}$ = Mode
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${L}$ = Lower limit of modal class
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${f_1}$ = Frquencey of modal class
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${f_0}$ = Frquencey of pre-modal class
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${f_2}$ = Frquencey of class succeeding modal class
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${i}$ = 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
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${f_1}$ = 30
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${f_0}$ = 15
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${f_2}$ = 20
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${i}$ = 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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