Measures of Central Tendency: Mode

Hello! In the third article of this series on the basics of statistics, we will understand what mode is and how we calculate the mode for a particular type of data.

The mode is the simplest measure of central tendency. It is defined as the value that appears most frequently in a dataset. Unlike the mean (average) or median (middle value), the mode can be used for both numerical and categorical (non-numerical) data, making it uniquely versatile in statistical analysis. A dataset may have one mode (unimodal), two modes (bimodal), multiple modes (multimodal), or no mode if all values occur with the same frequency.

Feature	Description
Data types	Nominal, ordinal, interval, ratio
Uniqueness	Not necessarily unique (can have 0, 1, or many modes)
Effect of outliers	Not affected at all
Ease of understanding	Very intuitive
Best used when	Finding the most common category, value, or response

In this article, we will understand how to find the mode in various types of data provided.

For Ungrouped Data (Raw Data)

1. List all distinct values in the dataset.
2. Count how many times each value occurs (frequency).
3. Identify the value(s) with the highest frequency.
   • Example: Dataset: 4, 1, 2, 4, 3, 4, 5 → Mode = 4
   • Example: Dataset: 4, 1, 2, 4, 3, 4, 5, 3, 3 → Modes = 4, 3 (since they both occur with the same frequency, 3 times) (also called ‘bimodal’)

For Grouped Data (Continuous distribution)

1. Arrange the data in groups of equal intervals.
2. Find the class with the highest frequency (modal class).
3. If a single mode is needed, use the following formula:
   • $$ Mode = L + (\frac{d_1}{d_1 + d_2})\times {h} $$
   • Where,
     • L = lower limit of the modal class
     • 
     • 
     • 
     • 
     • 
     • h = width of the modal class
   • Let’s work out an example to understand this:

Class Interval (Marks)	Frequency (Students)
0 – 10	4
10 – 20	8
20 – 30	12
30 – 40	18
40 – 50	6
50 – 60	2

For this data, following are the values:

• L = 30
• 
• 
• 
• 
• 
• h = 10
• $$ Mode = 30 + (\frac{6}{6 + 12})\times {10} $$
• $$ Mode = 30 + (0.3333\times {10}) = 33.333 $$

For data groups with unequal widths

1. We have to find the frequency density of each group.
   • Frequency density = Frequency/(Class width)
2. The modal class is the group with highest frequency density.
3. Then use the same formula as above to find the mode:
   • $$ Mode = L + (\frac{d_1}{d_1 + d_2})\times {h} $$
   • Where,
     • L = lower limit of the modal class
     • 
     • 
     • 
     • 
     • 
     • h = width of the modal class

Sometimes in the given data, you cannot find either the frequency before the modal class or the frequency after the modal class. Then you have to assume their value as zero and then calculate the rest using the interpolation formula for calculating the mode (given above).

Assuming a missing neighbor frequency as zero means the following:

• There is no data beyond the extreme class.
• The frequency “before” the first class or “after” the last class is effectively zero.
• This pulls the estimated mode away from the extreme boundary toward the interior—which is reasonable because density drops off when you have data only on one side.

Special case (only one class):
When there is only one class, both $f_{m-1}=0$ and $f_{m+1}=0$. The formula reduces to the following: $$\text{Mode}=L+\left(\frac{f_m}{f_m+f_m}\right)\times h=L+\frac{h}{2}$$​This gives the midpoint of the only class—the best you can do.

We will work an example for unequal class widths.

The following table shows the distribution of weekly wages (in dollars) of 100 workers in a factory. The class intervals have unequal widths.

Class Interval (Wages in $)	Number of Workers (Frequency)
50 – 70	8
70 – 90	12
90 – 100	15
100 – 120	30
120 – 150	20
150 – 200	10
200 – 250	5

Class Interval	Width	Frequency (f)	Frequency Density = f/width
50 – 70	20	8	8 ÷ 20 = 0.40
70 – 90	20	12	12 ÷ 20 = 0.60
90 – 100	10	15	15 ÷ 10 = 1.50
100 – 120	20	30	30 ÷ 20 = 1.50
120 – 150	30	20	20 ÷ 30 ≈ 0.667
150 – 200	50	10	10 ÷ 50 = 0.20
200 – 250	50	5	5 ÷ 50 = 0.10

This is a bimodal distribution where two classes have highest frequency density.

So we can use the same formula for two modal classes (90-100) and (100-120), using the same values as given in the above table.

1. For (90-100)
   • L = 90 (lower boundary of 90–100)
   • $f_m = 15$ (frequency of modal class)
   • $f_{m-1} = 12$ (frequency of previous class: 70–90)
   • $f_{m+1} = 30$ (frequency of next class: 100–120)
   • $h=10$ (width of modal class: 100 − 90)
   • d1 = 15 -12 = 3
   • d2 = 15 – 30 = -15
   • $$ Mode = 90 + (\frac{3}{3-15})\times{10} $$
   • $$ Mode = 90 + (\frac{3}{-12})\times{10} = 90 – 2.5 = 87.5 $$
2. For (100-120)
   • L = 100 (lower boundary of 100–120)
   • $f_m = 30$ (frequency of modal class)
   • $f_{m-1} = 15$ (frequency of previous class: 90–100)
   • $f_{m+1} = 20$ (frequency of next class: 120–150)
   • $h=10$ (width of modal class: 120 − 100)
   • d1 = 30 – 15 = 15
   • d2 = 30 – 20 = 10
   • $$ Mode = 100 + (\frac{15}{15 + 10})\times{20} $$
   • $$ Mode = 100 + (\frac{15}{25})\times{20} = 100 + 12 = 112 $$

Empirical Mode (with rough approximation)

Mode ≈ (3×Median) − (2×Mean)
Use only when distribution is moderately skewed.

Method of finding the mode by using bar graphs

1. Draw the histogram.
2. The rectangle with the highest height is the modal class for that data.
3. Draw a line joining the upper left corner of modal rectangle to the upper left corner of the next rectangle.
4. Draw a line joining the upper right corner of modal rectangle to the upper right corner of the previous rectangle.
5. Draw a vertical line from the point of intersection to the x-axis.
6. The point on the x-axis is the mode.
7. Why we should not use the midpoint of modal class:
   • Using the midpoint of the modal class is a rough approximation but less accurate because it ignores the frequencies of neighboring classes. The interpolation formula above is the standard statistical method.

No mode case

When: All values occur with equal frequency.
Example: 1, 2, 3, 4 → No mode. (Sometimes called “multimodal” if every value is a mode, but most texts say no mode.)

Advantages of Using Mode

• Applicable to all data types (including nominal/categorical data like colors, brands, or yes/no responses)
• Unaffected by extreme outliers (e.g., in income data, the mode is not distorted by billionaires)
• Easy to understand and explain to non-technical audiences
• Useful for discrete data (like number of children per family)
• Can be identified without calculations (just by inspection or from a frequency table)

Field	Example
Retail / Business	Most frequently sold product size or color
Education	Most common exam score
Healthcare	Most frequently reported symptom
Market Research	Most preferred brand or feature
Manufacturing	Most common defect type
Demographics	Most common household size or age group

The mode is an essential, intuitive, and often overlooked measure of central tendency. While it may not be as mathematically sophisticated as the mean or median, its ability to handle categorical data and resist outliers makes it important in fields ranging from market research to public health.

Here it is! We have learned through examples about the mode as a measure of central tendency. I will be posting about other concepts regarding statistics, which are used in physics and experiments conducted in laboratories. Stay curious, and keep learning!