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Mean, Median and Mode

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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.

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:

  1. Mean (xÌ… 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.
  2. 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.
  3. 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 Formula

Formula of Mean

Symbol of Mean

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 “xÌ„” (x-bar) when referring to the sample mean.

  • Population Mean: μ (mu)
  • Sample Mean: xÌ„ (x-bar)

These symbols are commonly used in statistical notation to represent the average value of a set of data points.

Properties of Mean

The mean, or arithmetic average, possesses several key properties that make it a fundamental measure in statistics:

  1. Additivity: The mean is additive, meaning that if you have two sets of data and calculate the mean for each separately, the mean of the combined dataset is the sum of the individual means.
  2. Sensitive to Change: The mean is sensitive to changes in the dataset. Even a small change in one of the values can lead to a noticeable change in the mean.
  3. Affected by Outliers: The mean is influenced by outliers or extreme values in the dataset. A single outlier can significantly skew the mean, pulling it towards the extreme value.
  4. Unique Solution: In many cases, the mean is a unique solution that minimizes the sum of the squared deviations of each value from the mean. This property is utilized in various statistical calculations and methods.
  5. Applicability: The mean is applicable to interval and ratio scale data, which includes most quantitative data types.

Mean Formula

The formula to calculate the mean is:

Mean (x̅)  = Sum of Values / Number of Values

If x1, x2, x3,……, xn are the values of a data set then the mean is calculated as:

xÌ… =  (x1 + x2 + x3 + …… + xn) / 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

Mean

x̅ = ∑ fixi / ∑ fi

where,
∑fi is the sum of all frequencies

Mean

x̅ = a + ∑ fixi / ∑ fi

where,
a is Assumed mean
di is equal to xi – a
∑fi the sum of all frequencies

Mean

x̅ = a + h∑ fixi / ∑ fi

where,
a is Assumed mean
ui = (xi – a)/h
h is Class size
∑fi the sum of all frequencies

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. 

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

Symbol of Median

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.

Properties of Median

The median, a measure of central tendency, possesses several key properties:

  1. Robustness to Outliers: Unlike the mean, the median is not significantly affected by outliers or extreme values in the dataset. It provides a more robust representation of the central tendency, making it suitable for skewed distributions or datasets with outliers.
  2. Positional Measure: The median is a positional measure, representing the middle value of a dataset when arranged in ascending or descending order. It divides the dataset into two equal halves, with half of the values falling below the median and half above.
  3. Unique Solution: Like the mean, the median can be a unique solution for a dataset. In cases where the data is odd, the median is the middle value. If the dataset contains an even number of values, the median is the average of the two middle values.
  4. Insensitive to Extreme Values: Since the median is determined based on the position of values rather than their magnitude, it remains insensitive to extreme values, making it a robust measure of central tendency.
  5. Applicability: The median is particularly useful for ordinal, interval, and ratio scale data, especially when the distribution is skewed or contains outliers.

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}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

            = 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 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.

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 “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 utilised in statistical discourse to signify the mode, enhancing clarity and precision in statistical discussions and analyses.

Properties of Mode

The mode, a measure of central tendency, possesses several important properties:

  1. Most Frequent Value: The mode represents the value or values that occur most frequently in a dataset. It identifies the peak(s) or highest frequency of occurrence in the distribution.
  2. Applicability: The mode is applicable to both numerical and categorical data. It can be used to describe the most common value or category in a dataset.
  3. Not Unique: Unlike the mean and median, the mode may not be unique. A dataset can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal), or it may have no mode if all values occur with equal frequency.
  4. Insensitive to Extreme Values: The mode is not influenced by extreme values or outliers in the dataset. It remains the same regardless of the presence of extreme values.
  5. May Not Exist: In some datasets, especially those with continuous distributions, a clear mode may not exist, or it may be difficult to determine.

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 + [(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
l is the lower limit of modal class

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 – 2 Mean

Relation between Mean Median Mode

Mode = 3 Median – 2 Mean

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.

Range of Data

It 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

Difference Between Mean and Average

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.

The key difference lies in their definitions and applications. While the mean pertains specifically to the arithmetic average, the term “average” can encompass different measures of central tendency, though it is commonly understood to refer to the mean in many contexts.

While “mean” and “average” are frequently used interchangeably, especially in casual discourse, they have nuanced distinctions in their definitions and interpretations within the realms of mathematics and statistics.

Difference Between Mean and Median

The mathematical average is known as the mean of the data set, whereas the positional average is considered the Median. 

The 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 the 5th observation.

Median = 20000

Therefore, the median is ₹20,000.

After comparing the mean and median for the above data set. It is evident that the mean salary is Rs 21666.67, and the median is Rs 20,000

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.

  • The mean is the average of all values and is sensitive to outliers.
  • The median is the middle value when data is sorted and is not affected by outliers.
  • The mode is the most frequently occurring value in the dataset.

Each measure offers unique insights into the dataset’s central tendencies, with the mean being influenced by outliers, the median remaining robust, and the mode identifying the most common value. The choice of measure depends on the dataset’s characteristics and the nature of the data.

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

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 = 53

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 given data 15, 42, 65, 65, 95.

Solution:

Given data set 15, 42, 65, 65, 95

The number with highest frequency = 65

Mode = 65

Related Articles

Statistics Formulas

Shortcut method for Arithmetic Mean

Calculation of Median of Discrete Series

Calculation of Mode in Discrete Series

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 – 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)/2th 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?

The 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.



Last Updated : 04 Mar, 2024
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