Mean, Median, and Mode are measures of the central tendency. These values are used to define the various parameters of the given data set. The measure of central tendency (Mean, Median, and Mode) gives useful insights about the data studied, these are used to study any type of data such as the average salary of employees in an organization, the median age of any class, the number of people who plays cricket in a sports club, etc.
Let’s learn more about the Mean, Median, and Mode Formulas, Examples, and FAQs in this article.
Table of Content
Measures of Central Tendency
Measure of central tendency is the representation of various values of the given data set. There are various measures of central tendency and the most important three measures of central tendency are:
 Mean
 Median
 Mode
What are Mean, Median, and Mode?
Mean, median, and mode are measures of central tendency used in statistics to summarize a set of data.
Mean (xÌ… or Î¼): The mean, or arithmetic average, is calculated by summing all the values in a dataset and dividing by the total number of values. It’s sensitive to outliers and is commonly used when the data is symmetrically distributed.
Median (M): The median is the middle value when the dataset is arranged in ascending or descending order. If there’s an even number of values, it’s the average of the two middle values. The median is robust to outliers and is often used when the data is skewed.
Mode (Z): The mode is the value that occurs most frequently in the dataset. Unlike the mean and median, the mode can be applied to both numerical and categorical data. It’s useful for identifying the most common value in a dataset.
What is Mean?
Mean is the sum of all the values in the data set divided by the number of values in the data set. It is also called the Arithmetic Average. Mean is denoted as xÌ… and is read as x bar.
The formula to calculate the mean is:
Mean Symbol
The symbol used to represent the mean, or arithmetic average, of a dataset is typically the Greek letter “Î¼” (mu) when referring to the population mean, and “xÌ„” (xbar) when referring to the sample mean.
 Population Mean: Î¼ (mu)
 Sample Mean: xÌ„ (xbar)
These symbols are commonly used in statistical notation to represent the average value of a set of data points.
Mean Formula
The formula to calculate the mean is:
Mean (xÌ…) Â = Sum of Values / Number of Values
If x_{1, }x_{2,} x_{3},……, x_{n} are the values of a data set then the mean is calculated as:
xÌ… = Â (x_{1 }+ x_{2} + x_{3 }+ . . . + x_{n}) / n
Example: Find the mean of data sets 10, 30, 40, 20, and 50.
Solution:
Mean of the data 10, 30, 40, 20, 50 is
Mean = (sum of all values) / (number of values)
Mean = (10 + 30 + 40 + 20+ 50) / 5 = 30
Mean of Grouped Data
Mean for the grouped data can be calculated by using various methods. The most common methods used are discussed in the table below:
Direct Method  Assumed Mean Method  Step Deviation Method 

xÌ… = âˆ‘ f_{i}x_{i} / âˆ‘ f_{i} Where, 
xÌ… = a + âˆ‘ f_{i}x_{i} / âˆ‘ f_{i} Where, 
xÌ… = a + hâˆ‘ f_{i}x_{i} / âˆ‘ f_{i} Where, 
Read More about Mean, Median and Mode ofÂ Grouped Data.
What is Median?
A Median is a middle value for sorted data. The sorting of the data can be done either in ascending order or descending order. A median divides the data into two equal halves.Â
The formula to calculate the median of the number of terms if the number of terms is even is shown in the image below:
The formula to calculate the median of the number of terms if the number of terms is odd is shown in the image below:
Median Symbol
The letter “M” is commonly used to represent the median of a dataset, whether it’s for a population or a sample. This notation simplifies the representation of statistical concepts and calculations, making it easier to understand and apply in various contexts. Therefore, in Indian statistical practice, “M” is widely accepted and understood as the symbol for the median.
Median Formula
The formula for the median is:
If the number of values (n value) in the data set is odd then the formula to calculate the median is:
Median = [(n + 1)/2]^{th} term
If the number of values (n value) in the data set is even then the formula to calculate the median is:
Median Â = [(n/2)^{th} term + {(n/2) + 1}^{th }term] / 2
Example: Find the median of given data set 30, 40, 10, 20, and 50.
Solution:
Median of the data 30, 40, 10, 20, 50 is,
Step 1: Order the given data in ascending order as:
10, 20, 30, 40, 50
Step 2: Check n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.
Step 3: Here, n = 5 (odd)
Median = [(n + 1)/2]^{th} term
Median = [(5 + 1)/2]^{th} term
Â Â Â Â Â Â = 30
Median of Grouped Data
The median of the grouped data median is calculated using the formula,
Median = l + [(n/2 – cf) / f]Ã—h
where
 l is lower limit of median class
 n is number of observations
 f is frequency of median class
 h is class size
 cf is cumulative frequency of class preceding the median class.
Read More about MedianÂ of Grouped Data.
What is Mode?
A mode is the most frequent value or item of the data set. A data set can generally have one or more than one mode value. If the data set has one mode then it is called “Unimodal”. Similarly, If the data set contains 2 modes then it is called “Bimodal” and if the data set contains 3 modes then it is known as “Trimodal”. If the data set consists of more than one mode then it is known as “multimodal”(can be bimodal or trimodal). There is no mode for a data set if every number appears only once.
The formula to calculate the mode is shown in the image below:
Symbol of Mode
In statistical notation, the symbol “Z” is commonly used to represent the mode of a dataset. It indicates the value or values that occur most frequently within the dataset. This symbol is widely utilised in statistical discourse to signify the mode, enhancing clarity and precision in statistical discussions and analyses.
Mode Formula
Mode = Highest Frequency Term
Example: Find the mode of the given data set 1, 2, 2, 2, 3, 3, 4, 5.
Solution:
Given set is {1, 2, 2, 2, 3, 3, 4, 5}
As the above data set is arranged in ascending order.
By observing the above data set we can say that,
Mode = 2
As, it has highest frequency (3)
Mode of Grouped Data
The mode of the grouped data is calculated using the formula:
Mode = l + [(f_{1} + f_{0}) / (2f_{1} – f_{0} – f_{2})] Ã— h
where,
 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,
 h is the size of class intervals, and
 l is the lower limit of modal class.
Read More about Mode of Grouped Data.
Relation between Mean Median Mode
For any group of data, the relation between the three central tendencies mean, median, and mode is shown in the image below:
Mode = 3 Median â€“ 2 Mean
Mean, Median and Mode: Another name for this relationship is an empirical relationship. When we know the other two measures for a given set of data, this is used to find one of the measures. The LHS and RHS can be switched to rewrite this relationship in various ways.
What is Range?
In a given data set the difference between the largest value and the smallest value of the data set is called the range of data set. For example, if height(in cm) of 10 students in a class are given in ascending order, 160, 161, 167, 169, 170, 172, 174, 175, 177, and 181 respectively. Then range of data set is (181 – 160) = 21 cm.
Range of Data
Range is the difference between the highest value and the lowest value. It is a way to understand how the numbers are spread in a data set. The range of any data set is easily calculated by using the formula given in the image below:
Range Formula
The formula to find the Range is:
Range = Highest value – Lowest Value
Example: Find the range of the given data set 12, 19, 6, 2, 15, 4.
Solution:
Given set is {12, 19, 6, 2, 15, 4}Â
Here,Â
Lowest Value = 2
Highest Value = 19
Range = 19 âˆ’ 2Â = 17
Difference Between Mean and Median
The key differences between mean and median are listed in the following table:
Aspect  Mean  Median 

Definition  The sum of all values divided by the count  The middle value of a sorted dataset 
Calculation  Mean = SumÂ ofÂ allÂ values/Count  Median is the middle value when the data is arranged in ascending or descending order 
Sensitivity to Outliers  Can be highly influenced by extreme values in the dataset  Less sensitive to extreme values, outliers have minimal impact 
Use Cases  Commonly used in statistical analysis and mathematics  Useful when extreme values skew the data or when the distribution is not symmetric 
Let’s see the following example to understand the difference.
Difference between Mean and Median is understood by the following example. In a school, there are 8 teachers whose salaries are 20000 rupees, a principal with a salary of 35000, find their mean salary and median salary.
Mean = (20000+20000+20000+20000+20000+20000+20000+20000+35000)/9 = 195000/9 = 21666.67
Therefore, the mean salary is â‚¹21,666.67.
For median, in ascending order: 20000, 20000, 20000, 20000, 20000, 20000, 20000, 20000, 35000.
n = 9,
Thus, (9 + 1)/2 = 5
Thus, the median is 5^{th} observation.
Median = 20000
Therefore, the median is â‚¹20,000.
Note: Mean gets easily affected by extreme values.
Differences between Mean, Median and Mode
Mean, median, and mode are measures of central tendency in statistics.
Feature 
Mean 
Median 
Mode 

Definition 
Mean is the average of all values. 
Median is the middle value when data is sorted. 
Mode is the most frequently occurring value in the dataset. 
Sensitivity 
Mean is sensitive to outliers. 
Median is not sensitive to outliers . 
Mode is not sensitive to outliers. 
Calculation 
Calculated by adding up all values of a dataset and dividing them by the total number of values in dataset. 
Calculated by finding the middle value in a list of data. 
Calculated by finding which value occurs more number of times in a dataset. 
Representation 
Value of mean may or may not be in dataset. 
Value of median is always a value from the dataset. 
Value of mode is also always a value from the dataset. 
Difference Between Mean and Average
Aspect  Mean  Average 

Definition  The sum of all values divided by the count  The sum of all values divided by the count 
Formula  xÌ„=âˆ‘ x/nâ€‹â€‹  Same as the mean formula 
Importance  Commonly used in statistics and mathematics  Often used interchangeably with “mean.” 
Sensitivity  Affected by outliers  Can be less sensitive to outliers. 
Application  Used for analyzing data sets  Commonly used in everyday language and contexts. 
Representation  Usually represented symbolically as Î¼  Often referred to simply as “average” or “avg.” 
Context  Often used in research and analysis  Informally used in everyday conversation. 
The terms “mean” and “average” are frequently used in mathematics and statistics, often interchangeably. However, they possess subtle distinctions in their meanings and applications.
Mean, in statistical terms, represents the arithmetic average of a dataset. It is calculated by summing up all the values in the dataset and dividing the sum by the total number of values. For instance, if you have the numbers 2, 4, 6, 8, and 10, the mean would be (2 + 4 + 6 + 8 + 10) / 5 = 6.
On the other hand, “Average” is a broader term that can refer to various measures of central tendency, including mean, median, and mode. In common usage, however, “average” often specifically denotes the mean. Like the mean, it involves summing up a set of values and dividing by the number of values to obtain a representative value.
Read More: Difference between Mean and Average.
How does Mean Median Mode link to Real Life?
In our daily life we came across various instances where we have to use the concept of mean, median and mode. There are various application of mean, median and mode, here’s how they link to real life:
 Mean: Mean, or average, is used in everyday situations to understand typical values. For example, if you want to know the average income of people in a city, you would calculate the mean income.
 Median: Median is in household income data, the median income provides a better representation of the typical income than the mean when there are extreme values. In real estate, the median house price is often used to gauge the affordability of homes in a particular area.
 Mode: Mode represents the most frequently occurring value in a dataset and is used in scenarios where identifying the most common value is important. For example, in manufacturing, the mode may be used to identify the most common defect in a production line to prioritize quality control efforts
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Conclusion – Mean, Median and Mode
Mean, Median and Mode are the measure of central tendency which helps us analyze and interpret data across various fields. Mean, often used as the arithmetic average, is sensitive to extreme values. On the other hand, the median, representing the middle value of any dataset. Meanwhile, the mode, indicating the most frequently occurring value.
Solved Questions on Mean, Median, and Mode
Question 1: Study the bar graph given below and find the mean, median, and mode of the given data set.
Solution:
Mean = (sum of all data values) / (number of values)
Mean = (5 + 7 + 9 + 6) / 4 Â
Â Â Â Â Â = Â 27 / 2Â
Â Â Â Â Â = 6.75Order the given data in ascending order as: 5, 6, 7, 9
Here, n = 4 (which is even)
Median Â = Â [(n/2)^{th} term + {(n/2) + 1}^{th} term] / 2
Median Â = (6 + 7) / 2 Â
Â Â Â Â Â Â Â = Â 6.5Mode = Most frequent valueÂ
Â Â Â Â Â = 9 Â (highest value)Range = Highest value – Lowest valueÂ
Range = 9 – 5Â
Â Â Â Â Â Â = 4
Question 2: Find the mean, median, mode, and range for the given data
190, 153, 168, 179, 194, 153, 165, 187, 190, 170, 165, 189, 185, 153, 147, 161, 127, 180
Solution:
For Mean:
190, 153, 168, 179, 194, 153, 165, 187, 190, 170, 165, 189, 185, 153, 147, 161, 127, 180
Number of observations = 18
Mean = (Sum of observations) / (Number of observations)
Â Â Â Â Â = (190+153+168+179+194+153+165+187+190+170+165+189+185+153+147 +161+127+180) / 18
Â Â Â Â Â = 2871/18
Â Â Â Â Â = 159.5
Therefore, the mean is 159.5
For Median:
The ascending order of given observations is,
127, 147, 153, 153, 153, 161, 165, 165, 168, 170, 179, 180, 185, 187, 189, 190, 190, 194
Here, n = 18
Median = 1/2 [(n/2) + (n/2 + 1)]^{th} observation
Â Â Â Â Â Â Â = 1/2 [9 + 10]^{th} observation
Â Â Â Â Â Â Â = 1/2 (168 + 170)
Â Â Â Â Â Â Â = 338/2
Â Â Â Â Â Â Â = 169Thus, the median is 169
For Mode:
The number with the highest frequency = 153
Thus, mode = 53
For Range:
Range = Highest value â€“ Lowest value
Â Â Â Â Â Â = 194 â€“ 127
Â Â Â Â Â Â = 67
Question 3: Find the Median of the data 25, 12, 5, 24, 15, 22, 23, 25
Solution:
25, 12, 5, 24, 15, 22, 23, 25
Step 1: Order the given data in ascending order as:Â
5, 12, 15, 22, 23, 24, 25, 25Â
Step 2: Check n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.
Step 3: Here, n = 8 (even) then,
Median = [(n/2)^{th} term + {(n/2) + 1)^{th} term] / 2
Median = [(8/2)^{th} term + {(8/2) + 1}^{th} term] / 2Â
Â Â Â Â Â Â = (22+23) / 2Â
Â Â Â Â Â Â = 22.5
Question 4: Find the mode of given data 15, 42, 65, 65, 95.
Solution:
Given data set 15, 42, 65, 65, 95
The number with highest frequency = 65
Mode = 65
FAQs on Mean, Median and Mode
What are the mean, median, and mode?
Mean, Median and Mode are the measures of central tendency. These three measures of central tendency are used to get an overview of the data. They represent the true essence of the given data set.
What is the relation between mean, median, and mode?
The relationship between mean median and mode is:
Mode = 3 Median â€“ 2 Mean
How to find mean, median, and mode?
Mean, Median, and Mode of any given data set is calculated using the suitable formulas which are discussed above in the articles.
How to find the mean?
Mean is also called the average, it is calculated for ungrouped data using the formula:
 Mean = (Sum of observations)/(Number of observations)
In case of Grouped Data, the mean is calculated by the three methods
 Direct method
 Assumed mean method
 Step deviation method
How to find the median?
Median is the middle term of the data when it is arranged in either ascending or descending order. It is calculated using the formula:
 Median = (n + 1)/2^{th }observation {when n is odd}
 Median = Average of (n/2)^{th} and [(n/2) + 1]^{th} observations {when n is even}
How to find the mode?
Value with the highest frequency is called the mode. Mode is calculated by observation first the given set of values is arranged in either ascending or descending order then the value with the highest frequency is noted as Mode.