# Mode

The mode is one of the values of the measure of central tendency. This value gives us a rough idea about which of the items in a data set tend to occur most frequently. For example, you know that a college is offering 10 different courses for students. Now, out of these, the course that has the highest number of registrations from the students will be counted as the mode of our given data (number of students taking each course). So overall, mode tells us about the highest frequency of any given item in the data set.

There a lot of real-life uses and importance of using the value of mode. There are a lot of aspects wherein just finding the average (or mean) will not work. For instance, refer to the example given above. In order to find the highest number of students enrolled in a course, finding any of the mean or median won't work. Hence, we tend to use the Mode in such cases.

## What Is Mode?

Mode means a value or a number that appears most frequently in a dataset. Sometimes we may need to find the value, which is occurring more frequently in the dataset.  In such cases, we find the mode for the set of given data. There may or may not be a mode value for a given set of data. For data without any repeating values, there might be no mode at all. Also, we can find data with only one mode, two modes, three modes, or multiple modes. This depends on the given dataset.

## General Formula to Calculate Mode

Mode for ungrouped data is found by selecting the most frequent item on the list. Now, for any given data range, let us consider 'L' is the lower limit of the modal class, 'h' is the size of the class interval, 'fm' is the frequency of the modal class, 'f1' is the frequency of the class preceding the modal class, and 'f2' is the frequency of the class succeeding the modal class. Here, the modal class is the data interval with the highest frequency. Thus, the mode can be calculated by the formula:

## How to Find the Mode?

We know that the value occurring most frequently in a set of data is called mode. Now depending on the data given (grouped or ungrouped), the method to find the mode can be changed. As the name suggests, a grouped data is the data that is shown in intervals. Such data is often shown in the form of a graph. On the other hand, ungrouped data is data that can be shown in the form of a table. Hence, we categorize the data we have into two groups: Grouped and Ungrouped. Let us now look to into the method to find mode for ungrouped data, and grouped data.

### Mode for Ungrouped Data

Any data that does not appear in groups are called ungrouped data. Let us take an example to understand how to find the mode of ungrouped data. Let us say a garment company manufactured winter coats with the sizes as mentioned in the frequency distribution table:

 Size of the winter coat Total number of shirts 3838 3939 4040 4242 4343 4444 4545 33 11 22 55 44 11 22

We can clearly see that size 42 has the greatest frequency. Hence, the mode for the size of the winter coat is 42. However, the same does not hold good for grouped data.

### Mode for Grouped Data

To find the mode for grouped data, follow the steps shown below.

• Step 1:  Find the class interval with the maximum frequency. This is also called modal class.
• Step 2: Find the size of the class. This is calculated by subtracting the upper limit from the lower limit.
• Step 3: Calculate the mode using the mode formula:

Mode = L + h \begin{align}\dfrac{(f_m-f_1)}{(f_m-f_1)+(f_m-f_2)}\end{align}

Let us understand this with an example. Below given is the data representing the scores of the students in a particular exam. Let us try to find the mode for this:

 Class Interval Frequency 0−5 5−10 10−15 15−20 20−25 5 3 7 2 6

Modal class = 10 - 15  (This is the class with the highest frequency). The Lower limit of the modal class = (L) = 10, Frequency of the modal class = (fm) = 7, Frequency of the preceding modal class = (f1) = 3, Frequency of the next modal class = (f2) = 2, and Size of the class interval = (h) = 5. Thus, the mode can be found by substituting the above values in the formula: Mode = L + h \begin{align}\dfrac{(f_m-f_1)}{(f_m-f_1)+(f_m-f_2)}\end{align} . Thus, Mode = 10 + 5 \begin{align}\dfrac{(7-3)}{(7-3)+(7-2)}\end{align} = 10 + 5 × 4/9 = 10 + 20/9 = 10 + 2.22 = 12.22.

Therefore the mode for the above dataset is 12.22.

### Important Notes and Tips on Mode

The below list of a few important points helps to summarize our learning on this concept of mode.

1. Mode value can sometimes be the same as mean and/or median, but not always.
2. The mode is very useful to find out categorical data.
3. There can be no mode for data that does not have any repeating numbers.
4. Mode can also be found out for data sets that do not have any numbers.
5. It is easy to find the mode when the given set of numbers are arranged in ascending order.
6. Mode for ungrouped data can be found by observation, whereas mode for grouped data can be found using the formula.

### Important Topics

Given below is the list of topics that are closely connected to Mode. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

## FAQs on Mode

### Can There Be Two Modes?

Yes, there can be two modes for a given set of data. A data set having two modes is called a bimodal data set. For example, the data set 1, 4, 7, 1, 7, 5, 6 has two modes, 1 and 7.

### What Happens if There is No Mode?

Yes, it is possible to have no mode for a set of data. If this is the case, we cannot use mode as a measure of central tendency. For example, the dataset: 2, 4, 6, 8, 10, 12, 14, 16, does not have any repeating number, and hence it does not have any mode.

### Can 0 Be a Mode?

Yes, 0 can be a mode if it occurs more than once in a given set of values. For example, the dataset: 6, 3, 0, 7, 7, 0, 3, 0, 0, had 0 as its mode.

### What Are the Modes in Math?

Modes are the numbers in the given list that occur with the highest frequency in a given data set. For example, the dataset: 4, 5, 8, 11, 12, 4, 2, 1, 4, has 4 as its mode.

### What if There Are 2 Modes?

If there are two modes, then it means that the two numbers (modes) are the most commonly found numbers in the data set; And the dataset having two such numbers is called a bimodal dataset.

### What is The Mode if There are No Repeating Numbers?

In case there are no repeating numbers in the list, then it means that no number can be called a mode. In such cases, zero modes are found for the given data set. Hence, no modes are found.

## Solved Examples on Mode

Example 1: Find the mode of the ungrouped data from the following table.

 Car color Number of cars sold Red Blue Silver Black Yellow 10 12 20 15 11

Solution:

To find the mode of ungrouped data, we will observe the class that has the highest frequency. Here, the class of the car with the color "Silver" has the highest frequency. Hence, the mode is 20.

∴ Mode = 20.

Example 2: Find the mode for the ungrouped data for the class interval and frequency mentioned in the table.

 The age group of employees Number of people of that age 20−30 30−40 40−50 50−60 30 55 44 25

Solution:

Modal class = 30 - 40  (This is the class with the highest frequency). Lower limit of the modal class = (L) = 30, Frequency of the modal class = (fm) = 55, Frequency of the preceding modal class = (f1) = 30, Frequency of the next modal class = (f2) = 44, and Size of the class interval = (h) = 10. Thus, the mode can be found as:

Mode = L + h $$\dfrac{(f_m-f_1)}{(f_m-f_1)+(f_m-f_2)}$$

Thus, Mode = 30 + 10 $$\times \frac{(55-30)}{(55-30) + (55-44)}$$ = 30 + 10{25/(25 + 11)} = 30 + 250/26 = 30 + 6.94 = 36.94

∴ Mode = 36.94

## Practice Questions on Mode

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

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