What is the mode of 1, 2, 3, 4 and 5?

In statistics, for a given data distribution the mode is the value or number that occurs most frequently. It is representative of the value or integer that occurs the maximum number of times. However, there may or may not be a modal value for a given set of datasets. This is because the given data set may have repeating or non-repeating values. In addition to this, a given data set may have one, two, or multiple modes. A dataset influences the value of the mode. 

Formula for Mode

[Tex]Mode=L+h\frac{(f_m-f_1)}{(f_m-f_1)+(f_m-f_2)}[/Tex]

Here, we have,

L – lower limit of the modal class, 

h – size of the class interval, 

fm – frequency of the modal class, 

f1 – frequency of the class preceding the modal class, and 

f2 –  frequency of the class succeeding the modal class.

Mode Formula of Ungrouped Data

In the case of ungrouped data, the data distribution is first arranged in either ascending or descending order. The repeated values are then depicted along with their frequency. The observation that corresponds to the highest frequency is known as the modal value for the given data. 

Mode Formula of Grouped Data

The mode formula for grouped data is given by, 

[Tex]Mode=L+h\frac{(f_m-f_1)}{(f_m-f_1)+(f_m-f_2)}[/Tex]

Here, we have, 

L – lower limit of the modal class,

h – size of the class interval,

f_m – frequency of the modal class,

f₁ – frequency of the class preceding the modal class and

f₂ – frequency of the class succeeding the modal class.

How to find Mode of Grouped and Ungrouped Data

Mode for Ungrouped Data

The data that does not appear in groups is known as ungrouped data. To illustrate we have an example where let us assume that there is a garment company that manufactures winter coats. The following tabular data with the shirts along with the sizes are mentioned in the shown frequency distribution table:

Size of the winter coat	38	39	40	42	43	44	45
Total number of shirts	33	11	22	55	44	11	22

Since it is evident that the size 42 has the greatest frequency. Therefore, the mode for the size of the winter coats is 42. 

The computation of mode for ungrouped data is different from that of the grouped data. 

Mode for Grouped Data

The following steps correspond to the computation of mode for grouped data: 

Step 1: Compute the class interval that corresponds to the maximum frequency. This value is also called modal class.

Step 2: Compute the size of the class by subtracting the upper limit from the corresponding lower limit.

Step 3: Calculate the value of mode using the mode formula:

[Tex]Mode=L+h\frac{(f_m-f_1)}{(f_m-f_1)+(f_m-f_2)}[/Tex]

What is the mode of 1, 2, 3, 4, and 5?

Solution: 

Step 1: The data is already arranged in ascending order. Therefore, there is no need to arrange the data in increasing order. 

Step 2: Constructing the frequency table: 

Number	1	2	3	4	5
Frequency	1	1	1	1	1

Step 3: Since the frequency of all the numbers are same, therefore, there is no specific mode for the given set of data distribution. 

Conclusively, if no numbers in the data distribution are repeated, then there is no mode for such observation. 

Sample Problems

Question 1: What is the mode of first n natural numbers?

Solution: 

Since, all the numbers occur just once, therefore, there is no mode of the distribution. 

Question 2: There are 10 random numbers comprising a dataset. Let us assume 3 to be the mode of this distribution. How many times minimum the number 3 should occur to be the mode?

Solution: 

Since, if the frequency of occurrence of mode value is equivalent to 1, then the other 9 numbers also would somehow be occurring once. In that case, there would be no mode for the given data. Therefore, minimum of the times that number 3 should occur is 2. 

Question 3: Illustrate one such example where Question 2 is satisfied. 

Solution: 

For instance, let us assume the data given to be 

1, 2, 3, 3, 4, 5, 6, 7, 8 and 9. In this case, the number 3 occurs 2 times and has the highest frequency. Therefore, it becomes the mode of the given dataset. 

Question 4: Why can’t be the mode of a dataset determinant of the central tendency of data?

Solution: 

Since the mode value of any given data set may occur either as the first value of the increasing order data arrangement or at last. Therefore, it cannot be used as a measure to determine what the other numbers are. Also, the mode of a dataset may or may be defined based on the repetition of values. 

Question 5: Where can’t the mode of data be defined?

Solution: 

The mode of data holds no relevance where all the values that comprise the data set occur an equal number of times. It may include the case where: 

• Repetition of every number occurs the same number of times
• No repetition occurs