How do you find the mode if no number is repeated?

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 dataset. 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

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, 

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 the Mode?

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 which 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:

How do you find the mode if no number is repeated?

Solution: 

For a data distribution that has no repeated numbers, there exists no mode. To prove this, let us assume a data distribution given by : A, B, C, D and E. 

Constructing the frequency distribution table for the given set of observations, we have, 

Observation	A	B	C	D	E
Frequency	1	1	1	1	1

We can clearly observe that each of the observation is repeated just once, thereby having a frequency equivalent to 1. Hence, there is no maximum occurrence in this data set and hence, no mode. 

Sample Questions

Question 1. Find the mode of the given ungrouped data.

Bike color	Red	Blue	Green	Black	Golden	Silver	Grey
Number of cars sold	15	25	42	32	35	55	22

Solution:

Here as to find the mode of this ungrouped data

Observe the bike color with the highest frequency

As we can see that ‘Silver’ color bike has the highest frequency 

Therefore,

The mode is 55.

Question 2. Calculate the mode of the grouped data given below

Marks obtained	Number of Students
10-20	4
20-30	8
30-40	16
40-50	12

Solution:

Here we have to find the mode of marks

First find the maximum frequency

f_m = 16

The corresponding class interval to f_m is 30-40

The lower limit of this class ‘L’ is 30

Size of the class interval = 10

Frequency of the preceding class f₁ = 8

Frequency of the succeeding class f₂ = 12

Substitute the value in the mode formula

i.e,

Question 3. Calculate the mode of the following data

15, 14, 19, 25, 26, 58, 109, 15, 14, 59, 58, 15, 17, 14, 19, 20, 25, 26, 109, 15, 109, 25, 59, 14, 17, 15

Solution:

To find the mode of the following data

First arrange the data in the ascending order

14, 14, 14, 14, 15, 15, 15, 15, 15, 17, 17, 19, 19, 20, 25, 25, 25, 26, 26, 58, 58, 59, 59, 109, 109, 109

The repeating numbers in the data

14 = 4 times

15 = 5 times

17 = 2 times

19 = 2 times

25 = 3 times

26 = 2 times

58 = 2 times

59 = 2 times

109 = 3 times

Here as we can see that 15 occurs most of the times

Therefore,

Mode of the data is 15.

Question 4. Find the mode of the following frequency distribution

Age group of teachers in the school	20-30	30-40	40-50	50-60
Number of teachers	40	66	55	20

Solution:

Highest frequency f_m = 66

Lower level of frequency (L) = 30

Modal class = 30-40

Frequency of the interval class preceding f₁ = 40

Frequency of the class succeeding f₂ = 55

Width of the class h = 10

Thus,

Question 5. Find the mode of the data

14, 15, 5, 3, 18, 19, 5, 16, 25, 33, 5, 3, 14, 18

Solution:

To find the mode first arrange the data in the ascending order

3, 5, 5, 5, 14, 14, 15, 16, 18, 18, 19, 25, 33 

Here as we can observe that 5 is occurring maximum times

Therefore,

The mode of the data is 5.