What is Mode – Meaning, Formula, Derivation & Examples

This post is also available in: हिन्दी (Hindi)

The mode, in statistics, is the most occurring value of the given list of data. It is one of the representative figures along with the arithmetic mean and median. These figures, i.e., median, arithmetic mean, and mode are also called measures of central tendencies. While the arithmetic mean is the ratio of the sum of all observations to the total number of observations, and the median is the middlemost value in the list of data items, the mode is a data value having the highest frequency.

Let’s understand how to find mode of grouped data and ungrouped data in statistics, and what are its properties using examples.

What is Mode in Statistics?

Mode means a value or a number that appears most frequently in a dataset. Many practical applications need to find the value, which is occurring more frequently in the dataset. In such cases, we find the mode or the modal value for the set of given data. 

For example, in the given set of data: $2$, $4$, $7$, $4$, $5$, $6$ the mode of the data set is $4$ since it has appeared in the set twice.

A data set can often have no mode, one mode, or more than one mode – it all depends on how many different values repeat most frequently. Your data can be

• without any mode.
• Unimodal: A list of given data with only one mode is called a unimodal list. For example, the list $2$, $4$, $7$, $4$, $5$, $6$ is a unimodal list, as there is only one mode(or modal value) and it is $4$ occurring twice.
• Bimodal: A list of given data with two modes is called a bimodal list. For example, the list $2$, $4$, $7$, $4$, $5$, $7$, $6$ is a bimodal list, as there are two modes(or modal values) and they are $4$ and $7$ both occurring twice.
• Trimodal: A list of given data with three modes is called a trimodal list. For example, the list $2$, $4$, $7$, $2$, $4$, $5$, $7$, $6$ is a trimodal list, as there are three modes(or modal values) and they are $2$, $4$ and $7$ all occurring twice.
• Multimodal: A list of given data with four or more modes is called a multimodal list. For example, the list $2$, $4$, $7$, $2$, $4$, $5$, $7$, $6$, $9$, $5$, $8$, $1$, $9$, $1$ is a trimodal list, as there are six modes(or modal values) and they are $2$, $4$, $7$, $5$, $9$ and $1$ all occurring twice.

How to Find Mode of Ungrouped Data?

The ungrouped data can be presented in either of the two forms.

• Raw Data: Raw data (sometimes called source data, atomic data, or primary data) is data that has not been processed for use. It is recorded in a way it is collected from the source.
• Tabulated Discrete Data: Tabulated data is data that is arranged and classified for use. It is recorded in the form of a table consisting of rows and columns.

Mode of Raw Data

The steps followed to find the mode of raw data are

Step 1: Arrange the data in ascending or descending order. (Ordering is not a prerequisite for mode. It is done to count the number of occurrences easily and quickly).

Step 2: Make a list of occurrences of all the data items in the list.

Step 3: Note down the value(s) having the highest occurrences(or frequencies). There can be more than one such value. There can be a case where none of the values repeats.

Step 4: The value(s) obtained in Step 4 is/are the mode(s) of the given data set.

Examples

Ex 1:  Find the mode of $2$, $4$, $6$, $1$, $3$, $5$, $3$, $6$, $1$, $4$, $4$, $2$, $1$, $3$, $6$, $4$, $2$, $5$.

Given set of data is $2$, $4$, $6$, $1$, $3$, $5$, $3$, $6$, $1$, $4$, $4$, $2$, $1$, $3$, $6$, $4$, $2$, $5$.

Arranging the data is ascending order $1$, $1$, $1$, $2$, $2$, $2$, $3$, $3$, $3$, $4$, $4$, $4$, $4$, $5$, $5$, $6$, $6$, $6$.

$1$ is occurring $3$ times.

$2$ is occurring $3$ times.

$3$ is occurring $3$ times.

$4$ is occurring $4$ times.

$5$ is occurring $2$ times.

$6$ is occurring $3$ times.

Therefore, the mode of the given data set is $4$, which is occurring $4$ times.

Mode of Tabulated Discrete Data

The steps followed to find the mode of tabulated discrete data are

Step 1: The frequency table of a tabulated discrete data consists of two columns – Data Values $\left( x \right)$ and Frequencies $\left(f \right)$.

Step 2: Locate the value of $x$ having the highest frequency. There can be more than one such value of $x$. 

Step 3: Note down the value(s) having the highest occurrences(or frequencies). There can be more than one such value. 

Step 4: The value(s) obtained in Step 3 is/are the mode(s) of the given frequency distribution.

Examples

Ex 1: Find the mode of the following frequency distribution.

The data value with the highest frequency is $13$ with a frequency of $10$. Therefore, the mode of the given frequency table is $13$.

Ex 2: The problem on the raw data set can also be solved by making a frequency table.

Find the mode of $2$, $4$, $6$, $1$, $3$, $5$, $3$, $6$, $1$, $4$, $4$, $2$, $1$, $3$, $6$, $4$, $2$, $5$.

Tabulating the given data in the frequency table, we get:

From the above frequency table, we see that data value $4$ has the highest frequency of $4$. Therefore, the mode(or modal value) of the given data set is $4$.

How to Find Mode of Grouped Data?

The mode for ungrouped data is found by selecting the most frequent item on the list. For finding the mode of the grouped data we use the following formula:

$\text{Mode } = L + h \times \frac{f_{1} – f_{0}}{\left(f_{1} – f_{0} \right) + \left(f_{1} – f_{2} \right)}$

where $L$ is the lower limit of the modal class

$h$ is the size of the class interval 

$f_{1}$  is the frequency of the modal class

$f_{0}$ is the frequency of the class preceding the modal class

$f_{2}$ is the frequency of the class succeeding the modal class

Derivation of Mode Formula

For the grouped data represented on the histogram, there are no individual values, to check for modal values. 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 $f_{m}$ or $f_{1}$. Here, $\text{BC} = h$. 

The frequency of the preceding modal class is $f_{0}$ and the frequency of the class succeeding the modal class be $f_{2}$, the lower limit of the modal class be $\text{L}_{0}$. Thus, the mode is given by $\text{L}_{0} + x$.

From the figure, $\triangle \text{AEB} \sim \triangle \text{DEC}$ 

$\frac{\text{AB}}{\text{CD}} = \frac{\text{BE}}{\text{DE}} = \frac{f_{1} − f_{0}}{f_{1} − f_{2}}$. 

$\triangle \text{BEF} \sim \triangle \text{BDC}$

$\frac {\text{FE}}{\text{BC}} = \frac{\text{BE}}{\text{BD}} = \frac{f_{1} − f_{0}}{\left(f_{1} − f_{0} \right) + \left(f_{1} − f_{2} \right)} = \frac{f_{1} − f_{0}}{2f_{1} − f_{0} − f_{2}}$.

$FE = \frac{f_{1} − f_{0}}{2f_{1} − f_{0} − f_{2}} \times \text{BC} = \left( \frac {f_{1} − f_{0}}{2f_{1} − f_{0} − f_{2}} \right) \times h$.

Therefore, $x = \left( \frac{f_{1} − f_{0}}{2f_{1} − f_{0} − f_{2}} \right) h$.

Thus, $\text{Mode } = L_{0} + x = L_{0} + \left( \frac{f_{1} – f_{0}}{2f_{1} − f_{0} − f_{2}} \right)h$.

Examples

Ex 1: Following is the distribution of height(in cm) of 50 students.

Find the modal height of the students.

Here the maximum frequency is $14$, therefore, the modal class is $130 – 135$. Thus, we have

$\text{L} = 130$, $h = 5$, $f_{1} = 14$, $f_{0} = 7$ and $f_{2} = 10$

$\text{Mode } = L + h \times \frac{f_{1} – f_{0}}{\left(f_{1} – f_{0} \right) + \left(f_{1} – f_{2} \right)}$

$=>\text{Mode } = 130 + 5 \times \frac{14 – 7}{\left(14 – 7 \right) + \left(14 – 10 \right)}$

$=>\text{Mode } = 130 + 5 \times \frac{7}{7 + 4} =>\text{Mode } = 130 + 5 \times \frac{7}{11} = 133.18$ (Rounded off to 2 decimal places).

Properties of Mode

Following are the important properties of mode.

• The mode is not unduly affected by extreme values, that is, values that are extremely high or extremely low. For example if we are given the following set of observations: $1$, $1$, $1$, $1$, $1$, $2$, $2$, $100$. The mean of the given set of data values is $13.625$ which is clearly not representative of the above data values. However, the mode which is equal to $1$ is clearly representative of a typical value from the above data set. This is one advantage of the mode compared to the mean.
• The mode is not calculated on all observations in a data set.
• The value of the mode can be computed graphically whereas the value of the mean cannot be calculated graphically.
• The value of the mode can be calculated in open-end distributions without knowing the class limits.
• The mode can be conveniently found even if the frequency distribution has class intervals of unequal magnitude provided that the modal class and the classes succeeding and preceding it are of the same magnitude.
• Sometimes it may not be possible to calculate the mode. This happens if the data has a bimodal distribution in which there are two possible values for the mode.
• We have the following relationship between the mean, median and the mode: $\text{Mode }= 3 \times \text{ Median } – 2 \times \text{ Mean}$.

Applications of Mode

These are some of the applications of the mode.

• The mode can be used to describe qualitative phenomena. For example, if we want to compare consumer preferences for different types of products such as soap, shampoo, etc. we should find the modal preferences expressed by different groups of people.
• It is the best measure of central tendency for highly skewed or non-normal distributions because it gives the point of maximum concentration of the data.
• It is used when performing non-parametric tests in inferential statistics.

Advantages and Disadvantages of Mode

Following are the advantages and disadvantages of the mode.

Practice Problems

1. Following are the marks obtained by $50$ students in a test.

Find the modal marks of the above distribution.

2. Find the mode of the following frequency distribution.

3. Following frequency distribution that shows the exam scores received by $40$ students in a certain class.

Find the modal marks of the above distribution.

FAQs

Conclusion

The mode of a set of data is the most occurring data item or value with the highest frequency. In this article, we learned how to find the mode of grouped data and raw data and its properties and applications.

Recommended Reading

• Data Collection & Organization(Methods, Tools, Types & Techniques)
• Discrete and Continuous Data(Meaning, Differences & Examples)
• How to Make a Pictograph – Definition, Advantages & Examples
• How to Make a Bar Graph – Definition, Advantages & Examples
• How to Make Double Bar Graph – Definition, Advantages & Examples
• How to Make Stacked Bar Graph – Definition, Advantages & Examples
• What is a Histogram – Definition, Advantages & Examples
• What is Pie Chart? (Definition, Formula & Examples)
• What is a Frequency Polygon Graph – Definition, Plotting & Examples
• How to Draw a Line Graph? (Definition, Plotting & Examples)
• Arithmetic Mean – Definition, Formula, Properties & Examples
• Median – Definition, Formula, Properties & Examples