Mode

The mode is one of the values of the measures 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 are 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.

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 modal 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.

A list can be unimodal, bimodal, trimodal, or multimodal, depending upon the number of modes it has.

Unimodal List: A list of given data with only one mode is called a unimodal list.

Bimodal List: A list of given data with two modes is called a bimodal list.

In statistics, the mode formula is used to calculate the mode or modal value of a given set of data. It is defined as the value that is repeatedly occurring in a given set. That means, the value or number in a data set, which has a high frequency or appears more frequently is called mode or modal value. Mode is one of the three measures of central tendency, apart from mean and median.

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, '\((f)_{m}\)' is the frequency of the modal class, '\((f)_{1}\)' is the frequency of the class preceding the modal class, and '\((f)_{2}\)' 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:

Mode Formula of Ungrouped Data

To find the mode for ungrouped data, it just requires the data values to be arranged either in ascending or descending order, then finding the repeated values and their frequency. The observation with the highest frequency is the modal value for the given data is here referred to as the modal value.

Mode Formula of Grouped Data

Mode formula for grouped data is given as,

Mode = \(L+h \frac{\left(f_{m}-f_{1}\right)}{\left(f_{m}-f_{1}\right)+\left(f_{m}-f_{2}\right)}\)

where,

• L is the lower limit of the modal class
• h is the size of the class interval
• \(\mathrm{f}_{\mathrm{m}}\) is the frequency of the modal class
• \(\mathrm{f}_{\mathrm{1}}\) is the frequency of the class preceding the modal class
• \(\mathrm{f}_{\mathrm{2}}\) is the frequency of the class succeeding the modal class

We might find the above formula written in different forms in some references, as given below,

Mode = \(I_{0}\) + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h\)

Here,

• I\(_0\) is the lower limit of the modal class
• h is the size of the class interval
• \(\mathrm{f}_{\mathrm{1}}\) is the frequency of the modal class
• \(\mathrm{f}_{\mathrm{0}}\) is the frequency of the class preceding the modal class
• \(\mathrm{f}_{\mathrm{2}}\) is the frequency of the class succeeding the modal class

For the grouped data represented on the histogram, there are not individual values, to check for modal value. Thus, we take up the modal class of size h, and then find out the mode based on that. Consider the graph given below. Let the frequency of the modal class be \(\mathrm{f}_{\mathrm{m}}\) or \(\mathrm{f}_{\mathrm{1}}\). Here, BC = h. The frequency of the preceding modal class be \(\mathrm{f}_{\mathrm{0}}\) and the frequency of the class succeeding the modal class be \(\mathrm{f}_{\mathrm{2}}\), the lower limit of the modal class be \(I_{0}\). Thus, the mode is given by \(I_{0}\)+x. Let's have a look!

From the figure, △AEB ~ △DEC
AB/CD = BE/DE = \(\frac{f_{1}-f_{0}}{f_{1}-f_{2}}\)
△BEF ~ △BDC
FE/BC = BE/BD = \(\frac{f_{1}-f_{0}}{\left(f_{1}-f_{0}\right)+\left(f_{1}-f_{2}\right)}=\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\)
FE = \(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}} \times B C=\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h\)
x = \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h\)
Thus, mode = \(I_{0}+x=I_{0}\) + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) h\)

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

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:

We can clearly see that size 42 has the greatest frequency. Hence, the mode for the size of the winter coats 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:

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 = \((f)_{m}\) = 7, Frequency of the preceding modal class = \((f)_{1}\) = 3, Frequency of the next modal class = \((f)_{2}\) = 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:

Listed below are a few important points that help to summarize our learning on this concept of mode.

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

☛ Related Topics to Mode and Mode Formula:

• Categorical Data
• Range in Statistics
• Geometric Mean
• Sample Standard Deviation Formula

What is Meant By Mode in Statistics?

Mode is defined as the value that is repeatedly occurring in a given set. It is one of the three measures of central tendency, apart from mean and median. That means, mode or modal value is the value or number in a data set, which has a high frequency or appears more frequently.

Can There be Two Modes For a Set of Data?

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 is Mode Formula in Statistics?

In statistics, the mode formula is defined as the formula to calculate the mode of a given set of data. Mode refers to the value that is repeatedly occurring in a given set and mode is different for grouped and ungrouped data sets. Mode = \(L+h \frac{\left(f_{m}-f_{1}\right)}{\left(f_{m}-f_{1}\right)-\left(f_{m}-f_{2}\right)}\)

What is h in Mode Formula?

In the mode formula,Mode = \(L+h \frac{\left(f_{m}-f_{1}\right)}{\left(f_{m}-f_{1}\right)-\left(f_{m}-f_{2}\right)}\), h refers to the size of the class interval.

What Happens if there is No Mode For a Set of Data?

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, has 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.

Is the Mode Formula Same for Grouped and Ungrouped Data?

No, The mode formula is not the same for grouped and ungrouped data. To find the mode for ungrouped data,

• Step 1: Arrange the data values either in ascending or descending order
• Step 2: Identify the repeated values and find their frequency.
• Step 3: The observation with the highest frequency will be the mode of the given data.

To find the mode for grouped data, use the mode formula which is given as, Mode = \(L+h \frac{\left(f_{m}-f_{1}\right)}{\left(f_{m}-f_{1}\right)-\left(f_{m}-f_{2}\right)}\)

How To Find Mode Using Mode Formula?

We use the mode formula only in the case of grouped data which involves the following steps:

• Step 1: Identify the modal class.
• Step 2: Check for the frequency of the modal class, which is represented as \(\mathrm{f}_{\mathrm{m}}\) or \(\mathrm{f}_{\mathrm{1}}\).
• Step 3: Find the size of the class interval, represented as h.
• Step 4: Find the lower limit of the modal class, that is l.
• Step 5: Identify \(\mathrm{f}_{\mathrm{0}}\) (the frequency of the class preceding the modal class) and \(\mathrm{f}_{\mathrm{2}}\) (the frequency of the class succeeding the modal class)
• Step 6: Put the above value in the mode formula, Mode = \(L+h \frac{\left(f_{m}-f_{1}\right)}{\left(f_{m}-f_{1}\right)-\left(f_{m}-f_{2}\right)}\)