Mode in Statistics | Definition, Formula, How to Calculate Mode

In statistics, the mode is the data point that comes the most frequently among a group of data members. It is one of three measures of central tendency, alongside the mean and median. To determine the mode, count how frequently each number appears. The number that comes the most frequently is the mode. Mode can be calculated for both numerical and categorical data. It is symbolised as Z or M₀.

Example: In the given set of data: 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, the mode of the data set is 4 since it has appeared in the set four times.

Unlike mean and median, which calculate the average and middle value of a dataset respectively, mode simply identifies the value that appears most frequently.

Types of Mode in Statistics

Depending upon the number of modal solutions, mode is classified into the following categories:

• Unimodal
• Bimodal
• Trimodal
• Multimodal

Note : A dataset without recurring values, however, lacks a mode.

Mode for Ungrouped Data

In case of ungrouped data we can simply find the mode by finding the most frequent observation.

Example : For this data 8, 7, 8, 6, 7, 7, 10, 8, 9, 7, 8, 8, 8, 7, 7, 7, 9, 8, 7, 7, 10, 7, 8, 8, 7, 8, 7, 8, 8, 8, 6, 7 the mode for this data is 8 as it is the most frequent value.

How to Calculate Mode of Ungrouped Data

To find the mode of the ungrouped dataset, we observe the most occurring value in the dataset. The values in the dataset must be rearranged either in increasing or decreasing order and their frequency should be noted.

The value which is appearing the most number of times has the highest frequency in the dataset and it is the Mode of the data.

Steps for Calculating Mode for Ungrouped Data

To calculate the mode of any given ungrouped data set, we use the following steps:

Step 1: Sort the data in ascending or descending order, whichever is more convenient.

Step 2: Determine the value that occurs most frequently in the data set. This value is the mode.

Step 3: If there are two or more values that occur with the same highest frequency, then the data set has multiple modes.

Let's consider an example for better understanding.

Example: Find the mode in the given set of data: 4, 6, 8, 16, 22, 24, 41, 24, 42, 24, 15, 13, 61, 24, 29.

Solution:

Arrange the given set of data in ascending order,
4, 7, 8, 13, 15, 16, 22, 24, 24, 24, 24, 29, 41, 42, 61.

The mode of the data set is 24 as it appeared in the given most.

Example : Imagine a shoe store that tracks the sizes of shoes sold over a month. The sizes are recorded as:

6, 7, 8, 7, 9, 7, 8, 8, 7, 6, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 6, 7, 7, 10, 8, 9, 7, 8, 8, 8, 7, 7, 7, 9, 8, 7, 7, 10, 7, 8, 8, 7, 8, 7, 8, 8, 8, 6, 7, 9, 8, 7, 6, 8, 8, 7, 7, 9, 8, 10, 7, 7, 7, 8, 8, 7, 7, 6, 8, 8, 9, 7, 7, 8, 10

• Size 6: 6 times
• Size 7: 26 times
• Size 8: 27 times
• Size 9: 8 times
• Size 10: 4 times

Here, the most frequently sold shoe size is 8, which occurs 27 times. Therefore, the mode of this data set is 8.

Mode for Grouped Data

The mode for grouped data is calculated by using the formula :

Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

where,

• l is the lower limit of the modal class.
• h is the size of the class interval,
• f₁ is the frequency of the modal class,
• f₀ is the frequency of the class preceding the modal class, and
• f₂ is the frequency of the class succeeding the modal class.

How to Calculate Mode of Grouped Data

For grouped data, calculation of mode just by simple observation of frequency is not possible. To determine the mode of data in such cases we calculate the modal class and the Mode lies inside the modal class.

Modal Class

The modal class refers to the class interval (or group) in a frequency distribution or groped data that has the highest frequency. In other words, it’s the class with the most data points.

Example : In a frequency distribution of students' scores on a test, grouped into class intervals:

In this example, the class interval 31 - 40 has the highest frequency, with 18 students scoring within this range.

Therefore, 31 - 40 is the modal class.

Steps for Calculating Mode for Grouped Data

Follow the given steps to calculate the mode of grouped data :

Step 1: Organize the data into a frequency distribution table if not given, which includes the class intervals and their corresponding frequencies.

Step 2: Identify the class interval with the highest frequency i.e., modal class.

Step 3: Observe all the values required in the formula for mode using modal class i.e., l , f₁, f₀, f₂, and h.

Step 4: Put all the values observed in the formula for mode given as follows:

Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)]×h

where:

• l is the lower limit of the modal class.
• h is the size of the class interval,
• f₁ is the frequency of the modal class,
• f₀ is the frequency of the class preceding the modal class, and
• f₂ is the frequency of the class succeeding the modal class.

Step 5: Calculate the Mode and round the mode to the nearest value, depending on the nature of the data and the context of the problem.

Example : Calculate the mode of the following data:

Solution:

To find the mode, we need to identify the class interval with the highest frequency. In this case, the class interval with the highest frequency is 30-40, which has a frequency of 12.

Modal class is 30-40

Lower limit of the modal class (l) = 30
Size of the class interval (h) = 10
Frequency of the modal class (f₁) = 12
Frequency of the class preceding the modal class (f₀) = 8
Frequency of the class succeeding the modal class (f₂)= 9

Using these values in the formula 

Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)]×h
⇒ Mode = 30 + [(12 - 8)/(2×12 - 8 - 9)] × 10
⇒ Mode = 30 + (4/7) × 10
⇒ Mode = 30 +40/7
⇒ Mode ≈ 30 + 5.71 = 35.71

So, the mode for this set of data is approximately 35.71.

Mean, Median and Mode

The relationship between Mean, Median, and Mode is given by the formula :

Mode = 3 Median – 2 Mean

The key differences between mean, median, and mode are tabulated below :

Points To Remember

Some important points about mode are discussed below:

• For any given data set, mean, median, and mode all three can have the same value sometimes.
• Mode can be easily calculated when the given set of values is arranged in ascending or descending order.
• For ungrouped data, the mode can be found by observation, whereas for grouped data mode is found using the mode formula.
• Mode is used to find Categorical Data.

Merits and Demerits of Mode

Merits and Demerits of Mode are discussed below:

Merits of Using Mode

• Mode is the most frequently occurring term in a series, unlike the isolated Median or the variable Mean.
• It remains stable against extreme values, making it a reliable representation.
• Mode can be identified graphically.
• Knowing the length of open intervals is unnecessary for determining the mode in open-end intervals.
• It is applicable in quantitative phenomena.
• Mode is easily identifiable with just a quick glance at the data, making it the simplest average.

Demerits of Mode

• Mode cannot be determined if the series has multiple modes, like being bimodal or multimodal.
• Mode only considers concentrated values, ignoring others even if they significantly differ from the mode. In continuous series, only the lengths of class intervals are taken into account.
• Mode is highly influenced by fluctuations in sampling.
• Mode's definition is not as strict. Different methods may yield different results compared to the mean.
• Mode lacks further algebraic treatment. Unlike the mean, it's impossible to find the combined mode of some series.
• Total series value cannot be derived from the mode alone, unlike the mean.
• Mode can be considered a representative value only when the number of terms is sufficiently large.
• Sometimes, mode is described as ill-defined, ill-definite, and indeterminate.

Solved Examples on Mode in Statistics

Example 1: Find the mode in the given set of data: 3, 6, 7, 15, 21, 23, 40, 23, 41, 23, 14, 12, 60, 23, 28
Solution:

First arrange the given set of data in ascending order: 
3, 6, 7, 12, 14, 15, 21, 23, 23, 23, 23, 28, 40, 41, 60

Therefore, the mode of the data set is 23 since it has appeared in the set four times.

Example 2: Find the mode in the given set of data: 1, 3, 3, 3, 6, 6, 6, 4, 4, 10
Solution:

First arrange the given set of data in ascending order: 
1, 3, 3, 3, 4, 4, 6, 6, 6, 10

Therefore, the mode of the data set is 3 and 6, because both 3 and 6 is repeated three times in the given set.

Example 3: For a class of 40 students marks obtained by them in math out of 50 are given below in the table. Find the mode of data given.

Solution:

Maximum Class Frequency = 23 
Class Interval corresponding to maximum frequency = 30-40
Modal class is 30-40

Lower limit of the modal class (l) = 30
Size of the class interval (h) = 10
Frequency of the modal class (f₁) = 23
Frequency of the class preceding the modal class (f₀) = 7
Frequency of the class succeeding the modal class (f₂)= 10

Using these values in the formula 
Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)]×h
⇒ Mode = 30 + [(23-7) / (2×23 - 7- 10)]×10
⇒ Mode = 35.51

Thus, mode of the dataset is 35.51

Also Read:

• Statistics Formulas
• What is Mean?
• What is Median?
• Variance