What is Mode?

The most frequently occurring observation in a data set is called Mode (also known as modal value).This is a measure of the tendency of the data values.

Example

Find the mode of 2, 5, 5, 8 and 9

In this case, the most frequently occurring value is 5. So, the mode is 5.


How to Find Mode of a Data?


Mode of Ungrouped Data

Mode of an ungrouped data is equal to the most frequent observation in the data. Data can consists of more than one mode.


A data distribution with one mode value is called unimodal whereas distributions with more than one mode values is called multimodal (they can be bimodal, trimodal etc.)


Example 1
 The mode is 8, and the data distribution is unimodal.

Explanation:

First we arrange the data in the ascending order as shown below :

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9

Next we try to determine the most frequently occurring observation, which in this case is 8.

=> Mode = 8

Since there is only one most frequently occurring value (which is 8), the data distribution is unimodal.

Example 2
 The modal values are 1, 3 and 4, and the data distribution is trimodal.

Explanation:

First we arrange the data in the ascending order as shown below:

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6

Next we try to determine the most frequently occurring observation. Note that each of 1, 3 and 4 appear 4 times.

Therefore 1, 3, 4 are modes of the given set of data.

=> Modal values = 1, 3, 4

Since there are 3 modal values in the given data set, the data distribution is trimodal.


Mode of Discrete Grouped Data


The method to determine the mode of discrete grouped data is the same as that of an ungrouped data. The observation that occurs maximum number of times in the given data set is the mode.


Example 1
Number on the Dice (X) Frequency (f)
1 4
2 6
3 2
4 2
5 5
6 1
  `2`

Explanation:

The value on the dice that has the maximum frequency is the mode of the data.


=> Mode = 2 (Having maximum frequency.)


Mode of Continuous Grouped Data


In order to determine the mode of continuous grouped data, we need the following steps:

  •  1.  Find the class boundaries of all the groups
  •  2.  Determine the group with the maximum frequency
  •  3.  Use the following formula to determine the mode:

`Mode` = `l + ((f_m-f_1)/(2f_m-f_1-f_2)) h`


Here,


l : lower class limit of the modal group

h : interval size of the modal group

`f_(m )`: frequency of the modal group

`f_1` : frequency of the group before the modal group

`f_2` : frequency of the group after the modal group


Example 1
Classes (X) Frequency (f)
0 – 9 7
10 - 19 4
20 - 29 8
30 – 39 5
40 - 49 9
50 - 59 7
60 - 69 10
Total = 50
  `45.75`

Explanation:

First of all, we need to determine the class boundaries and record them in a separate column as shown below:


Classes (X) Class Boundaries Frequency (f)
0 – 9 0.5 – 9.5 7
10 - 19 9.5 – 19.5 4
20 - 29 19.5 – 29.5 8
30 – 39 29.5 – 39.5 5
40 - 49 39.5 – 49.5 10
50 - 59 49.5 – 59.5 7
60 - 69 59.5 – 69.5 9
Total = 50

Next we find the class with the maximum frequency: 39.5 – 49.5 is the class with the maximum frequency of 10.


Now we use the formula for mode:


`Mode ` = ` l + (f_(m )- f_(1 ))/(2f_(m )- f_(1 )- f_(2 ) ) h`


Here,


l = lower limit of the modal group = 39.5

h = modal group interval size = 10

`f_(m )` = frequency of the modal group = 10

`f_(1 ) ` = frequency of the group before the model group = 5

`f_(2 )` = frequency of the group after the model group = 7


Mode = ` 39.5+ (10-5 )/(2(10)– 5- 7 ) ⨯10`


Mode = `39.5+ 50/( 8) = 39.5+6.25 = 45.75`


Hence 45.75 is the modal value of the given continuous grouped data.






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