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Mean, Median and Mode

Mean, Median and Mode

Last Updated : 08 Apr, 2025

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, their formulas, and examples.

Table of Content

Symbol of Mode

Measures of Central Tendency

The 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?

The mean, median, and mode are measures of central tendency used in statistics to summarize a set of data.

Mean (x̅ or μ): 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. The Mean is denoted as x̅ and is read as x bar.

The formula to calculate the mean is:

Mean Formula — Formula for Mean

Mean Symbol

The symbol used to represent the mean, or arithmetic average, of a dataset is typically the Greek letter "μ" (mu) when referring to the population mean, and "x̄" (x-bar) when referring to the sample mean.

Population Mean: μ (mu)
Sample Mean: x̄ (x-bar)
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 (x̅) = Σxi/ n

If x_1,x_2, x₃,......, x_n are the values of a data set then the mean is calculated as:

x̅ = (x₁+ x₂ + x₃+ . . . + 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

The 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

x̅ = ∑ f_ix_i / ∑ f_i

Where,
∑f_i is the sum of all frequencies

x̅ = a + ∑ f_ix_i / ∑ f_i

Where,

a is the Assumed mean
d_i is equal to x_i – a
∑f_i the sum of all frequencies

x̅ = a + h∑ f_ix_i / ∑ f_i

Where,

a is the Assumed mean
u_i = (x_i – a)/h
h is Class size
∑f_i the sum of all frequencies

Read More about Mean, Median and Mode of Grouped Data.

What is the 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 halves.

The formula to calculate the median of the number of terms if the number of terms is even is shown in the image below:

Median of Even Terms — Median Formula for Even Terms

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 of Odd Terms — Median Formula for Odd Terms

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 = [(n/2)^th term + {(n/2) + 1}^thterm] / 2

Example: Find the median of the the 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
= 30

Median of Grouped Data

The median of the grouped data median is calculated using the formula,

Median = l + [(n/2 - cf) / f]×h

where

l is the lower limit of the median class
n is the number of observations
f is the frequency of the median class
h is class size
cf is the cumulative frequency of the class preceding the median class.

Read More about Median 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 values. If the data set has one mode then it is called "Uni-modal". 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 "multi-modal"(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:

Mode Formula — Formula of Median

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 utilized 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,
Using the formula
Mode = Highest Frequency Term
Mode = 2
As, it has highest frequency (3)

Mode of Grouped Data

The mode of the grouped data is calculated using the following formula:

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

where,

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

Read More about Mode of Grouped Data.

Relation between Mean, Median, And 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 – 2 Mean

Relation between Mean Median Mode — Mode = 3 Median – 2 Mean

Mean, Median,the 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 the 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, the range of the 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 — Formula to Find Range

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}
Here,
Lowest Value = 2
Highest Value = 19
Range = 19 − 2 = 17

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.	The median is the middle value when data is sorted.	Mode is the most frequently occurring value in the dataset.
Sensitivity	The mean is sensitive to outliers.	The median is not sensitive to outliers.	The 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 the 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	The value of the mean may or may not be in the dataset.	Value of the median is always a value from the dataset.	The value of the mode is also always a value from the dataset.

Note: Mean gets easily affected by extreme values.

Let's look at 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 ₹21,666.67.
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 ₹20,000.
Mode is the data with maximum frequency
Mode = 20,000.

Read More: Difference between Mean and Average.

How does Mean Median Mode link to Real Life?

In our daily lives, we come across various instances where we have to use the concepts of mean, median and mode. There are various applications 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

People Also Read:
Statistics Formulas	Shortcut method for Arithmetic Mean
Calculation of Median of Discrete Series	Calculation of Mode in Discrete Series

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.

Example 1

Solution:

Mean = (sum of all data values) / (number of values)
Mean = (5 + 7 + 9 + 6) / 4
= 27 / 2
= 6.75
Median = Order the given data in ascending order as: 5, 6, 7, 9
Here, n = 4 (which is even)
Median = [(n/2)^th term + {(n/2) + 1}^th term] / 2
Median = (6 + 7) / 2
= 6.5
Mode = Most frequent value
= 9 (highest value)
Range = Highest value - Lowest value
Range = 9 - 5
= 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)
= (190+153+168+179+194+153+165+187+190+170+165+189+185+153+147 +161+127+180) / 18
= 2871/18
= 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
= 1/2 [9 + 10]^th observation
= 1/2 (168 + 170)
= 338/2
= 169
Thus, the median is 169
For Mode:
The number with the highest frequency = 153
Thus, mode = 153
For Range:
Range = Highest value – Lowest value
= 194 – 127
= 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:
5, 12, 15, 22, 23, 24, 25, 25
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] / 2
= (22+23) / 2
= 22.5

Question 4: Find the mode of the given data 15, 42, 65, 65, 95.

Solution:

Given data set 15, 42, 65, 65, 95
The number with highest frequency = 65
Mode = 65

Unsolved Practice Questions on Mean, Median, and Mode

Question 1: A company recorded the weekly sales (in dollars) of five salespersons as follows: $450, $520, $480, $510, and $490, Find the mean sales value for this group?

Question 2: Find the median of the following data set: 12, 15, 20, 9, 17, 25, 10.

Question 3: A survey collected the number of books read by a group of 10 people last year: 5, 7, 6, 5, 9, 7, 8, 5, 10, 6. What is the mode of the data set?

Question 4: In a classroom, the scores (out of 100) for a test are: 56, 78, 67, 45, 56, 90, 56, 67, 78, and,82. Find the mean, median, and mode of the scores.

Question 5: In a skewed distribution the mean of the data is 40 and median of the data is 35. Calculate the mode of the data set.

Answers to Practice Questions
Ans 1: Mean = $490	Ans 2: Median = 15.	Ans 3: Mode = 5.
Ans 4: Mean = 67.5, Median = 67, Mode = 56.	Ans 5: Mode = 25

Conclusion

Mean, Median and Mode are essential statistical measures of central tendency that provide different perspectives on data sets. The mean provides a general average, making it useful for evenly distributed data. The median gives a middle value, providing a better view of central tendency when dealing with skewed distributions or extreme values and, the mode highlights the most frequent value, making it valuable in categorical data analysis.

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.

Mean, Median and Mode

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