Difference Between Mean, Median, and Mode

Introduction

If you truly think about it, you will realize that we all are constantly surrounded by numbers, aggregates, and statistics. For example, have you recently watched a cricket match and seen the run rate project on the screen? Or have you recently got the result of your examination and told your parents about your aggregate percentage?

 

As you might have guessed, in both of these cases, we are dealing with a collection of data that is highlighted by a single value. This is also known as statistics. And in this article, readers will be able to learn more about statistics, measures of central tendencies, and the difference between median and mode. We will look at the difference between mean and mode.

 

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Before we talk about the mean and mode difference, we need to have a grasp on the basics of this concept. 

 

As we mentioned above, statistics is a branch of mathematics that deals with the collection of information and data. The collection of information and data is done for a particular purpose. For example, if there is a game of cricket and one wants to create a written record of each run for each ball, then that would give us the statistics of the game. A representation like this can be done in several ways, including graphs, tables, bar graphs, pie charts, and pictorial representations.

 

Going in the same direction, it should be noted that statistics also consists of various important topics. The measure of central tendency is one such topic. To understand what measures of central tendencies mean, you need to know that in statistics we often tend to represent a set of data through a representation value.

 

The representation value would provide an approximate definition for the entire collection of data,and this value is known as the measure of central tendency. 

 

We are very familiar with data and also we come across data every day as it is very common these days and can be found in articles, bank statements, newspapers, bills etc. To study the various features or characteristics of any given set of data, we use mean, median and mode which are known as the measures of central tendencies. One should also remember that the measures of central tendency refer to values around which the entire collection of data is centred.  

 

There are various parameters that are used for measures of central tendency. However, the most common parameters are:

  • Mean

  • Median

  • Mode

 

To choose the best measure of central tendency always depends on the type of data given.  We will learn more about these parameters in the next section of the article.

 

Understanding Mean, Median and Mode

Let’s start our discussion by first understanding the meaning of mean. According to experts, the mean can be defined as the value of central tendency that represents the average of any given collection of data. In other words, we can say it is the sum of all the observations divided by the number of observations. Mean is also the most commonly used measure of central tendency. 

 

The formula for calculating the arithmetic mean is given below : 

 

Mean = Sum of all the observations/ Total number of observations

 

Mean is always denoted by x̄ which is pronounced by x bar. 

 

It should be noted by students that the mean is applicable for both discrete data and continuous data. Let’s take an example to further be familiar with this topic. Assume that there are n values in a set of data. These values are x1, x2, x3, x4, …, xn. This means that the mean of data can be given by:

 

\[\overline{x} = \frac{x_1 + x_2 + x_3 + … + x_n}{n}\]

 

This can also be represented by:

 

\[\overline{x} = \frac{1}{n} \sum_{i=1}^n x_i\]

 

In contrast to the mean, the median can be defined as the mid-value of any given data set. The median value is only selected when the data is arranged in a particular order. 

 

The steps for finding the median value of a data set are mentioned below. 

  • Arrange the data in ascending or descending order.

  • If the number of observations or values that are present in the data is odd, then the median value can be calculated by:

\[(\frac{n+1}{2})^{th}\] observation

  • If the number of observations or values that are present in the data is even, then the median value can be calculated by:

\[(\frac{n}{2})^{th}\] and \[(\frac{n}{2} + 1)^{th}\] observation

 

For example, let's take a set of data - 2,4,3,6,2. First, we have to arrange this data in ascending order: 2,2,3,4,6. The total number of observations is 5. Thus, the median will be the middle value of the data set. Here, in this data set, the middle value is 3. So, the median will be 3. 

 

Now, let’s talk about the mode. The mode can be described as the most frequently occurring number of a data set. In other words, the most common number in any data set given is the number which is called mode. For example, let’s consider a data set of marks that were obtained by various students in science. This data set is given below in the form of a table. 

Name

Preet

Sharon

Mansingh

Geet

Pulkit

Happy

Marks Obtained (Out of 100)

73

80

73

70

73

65

 

From this table, you must be able to observe that the maximum frequency observation is 73. This is because there are three students out of six students who have scored 73 marks. Hence, the mode for this data collection is 73.

 

Difference between Mean and Median

  • Mean is considered as an arithmetic average whereas the median is considered as a positional average. 

  • It is considered to be highly sensitive to outlier data in comparison to the median. Whereas the median is not as sensitive to outlier data as is the mean. 

  • Mean explains the central value of any data set, whereas median explains the mid-value of any given data set. 

  • Mean cannot be used for measuring the uneven data as there are a lot of external factors which limit the use of mean. While the median is considered to be reliable for measuring the uneven data. 

 

Till now, we have looked at what are mean, median, and mode. Now, in the next section, we will try to understand the differences that exist between mean, median, and mode. To make it easier for students to learn this topic, we have mentioned all the differences in the form of a table below. 

S.No

Mean

Median

Mode

1

To calculate the mean value, the average is taken for all the numbers in a data set.

To calculate the median, the middle value of a data set is taken after arranging it in a particular order.

Mode refers to the number that occurs the most frequently in a given data set or collection.

2

When it comes to mean, all the numbers are added together and that sum is divided by the total number of data or values that are present in the data collection.

Median places all the given numbers in an ascending or descending order.

This central tendency shows the frequency of occurrence.

3

The result of calculating the mean is the average score.

In the median, after arranging the values, the next step is to find the middle number from the list.

It is possible for an individual to get one value of mode or no value of mode from a data set or collection.

4

For example, if an individual wants to find the average of all the four numbers, including 2, 4, 6, and 8, we will need to add all the numbers first.

For example, if the given data set is 4, 2, 8, 10, 19, then we will first arrange the number in ascending order, which will be 2, 4, 8, 10, 19. After that, we can select the middle number. This is easy because there are five numbers and we know that the middle number is 8 from this data set. 

For example if there is a series of data like:

3, 3, 5, 6, 7, 7, 1, 1, 1, 4, 5, 6, and we have to find the frequency of every number, then we can do that by calculating the times that every number occurs. In this data set, 3 occurs for two times, 5 occurs for two times, 6 occurs for two times, 7 occurs for two times, 8 occurs for a single time, 1 occurs for three times, and four occurs a single time too. We can use this information to find the value of mode, which is the number or numbers with the highest frequency.

 

Fun Facts About Mean, Median and Mode

You know about the difference between mean, median, and mode. But did you know that all the three measures of central tendencies are related to one another? Mean, median, and mode are related to one another by the formula that is mentioned below.

 

Mode = 3 Median - 2 Mean

 

Also, according to this formula, the three measures of central tendencies are arranged in a moderately skewed distribution. This relation is also known as the empirical relationship. It is used for finding one of the measures when the value of the other two measures is already known for a data set.

 

This relationship can also be written in different forms by just interchanging the RHS and LHS. Also, did you know that in statistics, the range is explained as the difference between the highest and lowest data value in a set? The formula that one can use for calculating the range of a data set is mentioned below.

 

Range = Highest value - Lowest value

FAQs on Difference Between Mean, Median, and Mode

1. Mention the Three Measures of Central Tendencies.

The three measures of central tendencies are:

  • Mean

  • Median

  • Mode

2. What do you Understand by the Mean in Statistics? Provide an Example with Your Answer.

Mean can be defined as the average of any given data set. The formula for calculating the value of the mean is mentioned below.


\[\text{Mean} = \frac{\text{Total of Observations}}{\text{Number of Observations}}\]


For example, if a data set is 2, 5, 6, 7, 8, and we have to find the value of the mean, then we will follow the steps that are mentioned below.


\[Mean = \frac{(2 + 5 + 6 + 7 + 8)}{5}\]


\[Mean = \frac{28}{5} = 5.6\]


Hence, the value of the mean is 5.6

3. What is the main difference between median and mode? 

Median is the value that is in the middle of any given set of data. For finding the median of any data set, firstly we have to arrange the data set in ascending order. Then if the data set is small, we can find the middle value just by observing the set, otherwise by using the formula we can find the median of that data set. Mode is the number whose frequency is most in the given set of data. It is being used to determine the value which is being used frequently. Modes are of three types namely, bimodal, trimodal and multimodal. 

4. What is the main difference between mean and median ? 

The main difference between mean and median is that mean is used to find out the average of any given data set while the median is used to find out the mid-value of the data set while the data is being arranged in an ascending or descending order. The mean is always used for normal distributions while the median is mostly used for skewed distributions. 

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