Mean, Median, and Mode

Mean, median, and mode are three key measures that help us understand the central tendency of a data set. Instead of examining each value, these measures allow us to summarize the entire data set with a single representative value that shows where the values tend to cluster. The mean gives us the average, the median shows the middle value when the data is ordered, and the mode tells us which value occurs most often. Together, these measures help us analyze patterns, compare groups, and make sense of real-world information more simply.

What is Mean?

The mean, also called the arithmetic average, is obtained by adding all the values in a data set and dividing by the total number of values. It is used when the data is evenly distributed. The symbol represents it ‘μ’.

What is Median?

The median is the middle value of a data set when the data set is arranged in ascending or descending order. If the number of values is odd, the median is the middle value; if the number of values is even, the median is the average of the two middle values. ‘M’ represents it.

What is Mode?

The mode is the value that appears most often in a data set. The mode can be applied to both numerical data and categorical data. The mode is handy for identifying the most common values in a given data set. 'Z' represents the mode.

Let us know more about mean, median, and mode using an example:
A group of students scored the following marks in a quiz:
8, 6, 7, 7, 9, 5, 7
Let’s find the mean, median, and mode of this data.

Solution:

To find the mean, add all the numbers and divide them by the total number.
Therefore,
Mean = \(\frac{8+6+7+7+9+5+7}{7}\)
\(=\frac{49}{7}\)
\(=7\)
 

To find the median, arrange the data in ascending order; the middle number is the median.
5, 6, 7, 7, 7, 8, 9
The 4th value is the middle number.
Therefore, median = 7.
 

To find the mode, find the number that appears most often.
Here, 7 is the most frequently occurring number, appearing 3 times.
Therefore, the mode = 7.
 

For this dataset:
Mean = 7,
Median = 7
Mode = 7.
 

Mean of ungrouped data: Ungrouped data is data that is not arranged in classes or groups. These are individual numbers, such as:
5, 8, 12, 7, 9.

To find the mean (average) of ungrouped data, follow these steps:
Step 1: Add all the data values
Find the sum of all numbers.
Example: \(5 + 8 + 12 + 7 + 9 = 41\).

Step 2: Count how many data values are present.
There are five numbers.

Step 3: Apply the mean formula
Sum / Total number.
Mean \(= \frac{41}{5} ​= 8.2\)
Mean of the ungrouped data = 8.2

Mean of grouped data: To find the mean of a grouped data, we use three methods, which are mentioned below:

• Direct method: To calculate the mean of a grouped data, we use the formula mentioned below:
  \(x = fᵢ × xᵢ ÷ fi\)
  
  Where \(fi \) is the sum of all the frequencies.
   

• Assumed mean method: To calculate the mean using assumed mean method, we use the following formula:

  \(x = a +fᵢ × xᵢ ÷fi\)
  
  Where \(a\) is the assumed mean

  
  \(di \) stands for deviation, where \(di = xi - a\)

  
  \(fi\) is the sum of all the frequencies.
   

• Step deviation method: To calculate the mean using step deviation method, we use the following formula:
  
  \(\bar{x} = a + h \cdot \frac{\sum f_i u_i}{\sum f_i}\)
   

  Where \(a\) is the assumed mean

  
  \(u_i\) is called as the reduced deviation, where \(u_i = (x_i - a) ÷ h\)

  
  \(h\) is the class size

  
  \(fi\) is the sum of all the frequencies.

   

Median of ungrouped data: To find the median of ungrouped data, follow these steps.

Step 1: Sort the data in ascending or descending order.

Step 2: Consider n to be the total number of observations. If the n is an odd number, then
the median \(=\frac{n+1}{2}\)th observation in the sorted list. 
If n is even, then the median is the average of the \(\frac{n}{2}\)th and the \(\frac{n}{2}+1\)th observation. 

Example 1: 
Find the median of the data 56, 67, 54, 34, 78, 43, 23. 

Solution: 

Sort the data : 23, 34, 43, 54, 56, 67, 78.
Here, the number of observations n = 7.

Then, the median = \(\frac{n+1}{2}\)
\(=\frac{7+1}{2}\)
\(=\frac{8}{2}=4\)

Therefor, the 4th observation is the median.
That is 54.

Example 2: 
Find the median of the data 50, 67, 24, 34, 78, 43.

Solution: 

By sorting the data: 24, 34, 43, 50, 67, 78.
Here, the number of observations n = 6
\(\frac{n}{2}=\frac{6}{2}=\) 3rd term.

\(\frac{n}{2}+1=\frac{6}{2}+1=4\)th term.

Median \(=\frac{43+50}{2}\)

\(=46.5\)

Median of grouped data: When data are grouped into class intervals (with frequencies), use these steps:

Step 1: Compute the cumulative frequency of each class, to determine where \(\frac{n}{2}\) lies. Here, \(n=∑fi (n = \sum f_i) \) (total number of observations). 

Step 2: Identify the median class: the class interval in which \(\frac{n}{2}\) falls.

Step 3: Use the formula,
Median \(= l+\frac{(\frac{n​}{2}−c​)}{f}×h\)

Where, 
l = lower limit of the median class.
c = cumulative frequency of the class just before the median class.
f = frequency of the median class.
h = class width (size of the class interval).
 

Example: 
Classes: 0–10, 10–20, 20–30, 30–40, 40–50
frequencies : 2, 12, 22, 8, 6
Total n = 50
So, \(\frac{n}{2}\) = 25.

The cumulative frequencies are: 
0–10 → 2
10–20 → 14
20–30 → 36, ……

Since 25 lies in the class 20–30, that’s the median class.
Here, l = 20, c = 14, f = 22, h = 10. 

By applying the formula: 

Median \( = 20 + \left(\frac{25 - 14}{22}\right) \times 10 = 20 + \left(\frac{11}{22}\right) \cdot 10 = 20 + 5 = 25 \)

Mode of ungrouped data: The mode of ungrouped data is the number (or value) that appears most frequently in the dataset. To calculate the mode of ungrouped data:

Step 1: List all observations clearly.

Step 2: The value with the highest frequency is the mode.

• If one value has the highest frequency, → unimodal.
• Two values have the same highest frequency → bimodal.
• More than two values frequently repeat → multimodal.
• No repetition → no mode.

Let us look into a few examples:

Data: 7, 8, 8, 9, 6, 8, 5 → Here, 8 occurs 3 times, more than any other value.
Therefore, mode = 8 (Unimodal).

Data: 4, 6, 4, 9, 6, 7 → 4 appears 2 times and 6 appears 2 times.
Therefore, mode = 4 and 6 (bimodal).

Mode of grouped data: When data are grouped into class intervals (e.g., 0–10, 10–20, etc.), the mode is found using a specific formula. To see the mode of grouped data, follow the steps below:

Step 1: Identify the modal class, that is, the class with the highest frequency.

Step 2: Use the formula: 

Mode = \(l + (\frac{f_1 - f_0}{2f_1 - f_2}) ×h\)

\(l \)= lower limit of the modal class,
\(f_1\)​ = frequency of the modal class,
\(f_0\)​ = frequency of the class before the modal class,
\(f_2​\) = frequency of the class after the modal class,
\(h \) = class width (upper limit − lower limit).

Example: 
Find the mode for the below data.
 

Solution: 

Step 1: Identify the modal class.
Here, the highest frequency is 20.
So, the modal class is 20-30.

Step 2: Now identify the values.
\(l = 20, f_1 = 20, f_0 ​= 12\) (previous class frequency), \(f_2​=14\) (next class frequency), h = 10.
 

Step 3: Apply the formula.

Mode \(= 20 + (\frac{20 -12}{(2 × 20}) - 12 - 14)×10 \)

\(= 20 + (\frac{8}{40}-26) × 10 \)

\(= 20 +(\frac{8}{14}) × 10 \)

\(= 20 +5.71 ≈25.71 \)
 

Therefore, the mode \(≈ 25.71\)

Mean: 

• Formula for finding the mean of ungrouped data \(= \frac{x_1 + x_2 + . . . .x_n}{n}\)
   
• Formula for finding the mean of ungrouped data \(= \frac{x_1f_1+x_2f_2+.......x_nf_n}{f_1+f_2++....f_n} \)

Median: 

• Formula for finding the median of ungrouped data:
  If n is odd, median \(= \frac{n+1}{2}\)th observation.
  If n is even, median = average of \(\frac{n}{2}\)th and \(\frac{n}{2} +1\) th observation. 
   
• Formula for finding the median of grouped data \(= l + (\frac{\frac{n}{2} -c}{f})×h\).

  Where, l = lower limit of median class, c = cumulative frequency of class before median class, f = frequency of median class, h = class width. 

Mode: 

• Mode \(= l + (\frac{f_m - f_1 }{ 2f_m - f_1 - f_2})×h\),
  
  Where, l = lower limit of modal class, \(f_m \)= frequency of modal class, \(f_1\)= frequency of class before modal class, \(f_2\) = frequency of class after modal class, h = class width. 
   

In a grouped data, the relation between the measures of the three tendencies which are mean, median and mode is shown below:

Mode = 3 Median - 2 Mean

The relationship between these three tendencies helps us to understand how to find the other tendency. Say for example we know what is the mean and mode, we can convert the formula by solving for median. 

For example, Median = 20 and Mode = 45, find the Mean.

Substituting the values in the formula:

45 = 3 median - 2(20)

45 = 3(20) - 2 mean

2 mean = 45 + 60

2 mean = 105

Mean = 105÷2 = 52.5

The differences between mean, median, and mode are given below:
 

There are a lot of ways to master mean, median, and mode. Some ways to master mean, median, and mode are mentioned below:

• Understanding Concepts Clearly: We have to understand the concepts of mean, median, and mode properly to master it. Mean is the average of all the values in the dataset, median is the middle value of the dataset, and mode is the most frequently occurring value.

• Handling Grouped Data (Class intervals): When calculating for grouped data for mean, we use assumed mean or step deviation method; for median, we use the median formula with cumulative frequency tables; and for mode, use the modal class and the formula:
  
  Mode = L + f1 - f0 ÷ 2f1 - f0 -f2 x h
  
  Where, L = lower boundary of modal class
  \(f_1\) = frequency of modal class 
  \(f_0\) = frequency before modal class
  \(f_2\) = frequency after modal class
  \(h\) = class width

• Practice Strategies: To master the concept of mean, median and mode, we can use real-life examples, solve previous year exam question papers and practice timed quizzes for a quick recall of the concepts.
  
   
• Use visual representations: Drawing bar graphs, histograms, or frequency polygons can help you see the distribution of data, making it easier to identify mean, median, and mode.
  
   
• Group study and discussions: Explaining concepts to peers or solving problems together reinforces understanding and helps uncover different methods to approach questions.
  
   
• Practice with real-life examples at home: Encourage children to apply mean, median, and mode in daily situations. For example, calculate the average daily screen time, find the median of weekly test scores, or identify the mode of most used snacks or toys. This builds a strong understanding of the concept through familiar activities.
  
  
   
• Use visual tools: Parents and teachers can encourage children to draw simple bar graphs, dot plots, or tally charts for small datasets. Visuals make it easier and more transparent to identify the median (middle value), mode (tallest bar), and mean (balancing point).
  
   
• Create learning games for students: Use household items such as beads, crayons, or toys to create data sets. Ask your child to group them, count frequencies, or find the middle item. This fosters conceptual clarity through fun, simple experiences.
  
   
• Discuss strategies and thinking: Have conversations with children about why they chose a method. For example, ask, “How did you decide which interval is the modal class?” or “Why is the median better for this data?” This develops their reasoning skills and clarifies concepts.
  
   
• Use online learning tools and visualizers: Introduce children to educational resources that generate graphs, frequency tables, or allow interactive manipulation of data. Parents and teachers can make use of mean, median, and mode worksheets and mean median mode calculators for students. 
   

There are a lot of real-world applications of mean, median and mode. Let us now see what kind of applications and fields we use mean, median and mode in:

• Education and Academic Performance: We use mean to calculate the average exam scores in a class, median is used to report the middle score from the very high scores to the lower scores, and mode is used to identify the most common grade that is achieved by students.

• Business and Finance: In business and finance, we use mean to determine the average sales per month, median is used for analyzing the median salary to represent typical employee earnings without being skewed by executives’ high salaries, and mode is used to find the most frequently sold product.

• Healthcare and Medicine: In healthcare and medicine, we use means to calculate the average blood pressure and cholesterol levels in a patient, we use median to find the median survival time in clinical trials, and we use mode to find the most common symptoms of a disease.

• Sports Analytics: Mean is used to calculate average scores, Median for ranking performances, Mode to find most common results.
  
   
• Analyzing exam results: Schools and colleges use mean, median, and mode to summarize student scores, identify overall performance, and understand the most common grades in a class.