Difference Between Mean, Median, and Mode

We deal with numbers all the time, whether it's on an individual or corporate level. But rough numbers or data sets don't make a whole lot of sense until we have proper tools to study them. Mean, median, and mode are some of those tools.

If you're looking at a large group of data, these terms can help you identify important benchmarks in your information. We'll explain what they really mean and examine the differences between the terms.

Mean

The mean is the sum of all the values in a dataset, divided by the total number of values in the data.

Typically, there's a range of values in any given set of data. The average is one measure of the center of a set of data. We also call this the mean. It can also be referred to as the arithmetic mean. Imagine you are a teacher and all of your students score differently on their exams. When your supervisor asks about your class's performance, you tell them the average score of the class. This is the class's mean or the mean score of students.

As mean takes into account every value in the data set, it is affected by extreme values or outliers. For example, if two students do poorly on the test, earning F's, this would lower the class mean.

Example 1

The mean of 12, 16, 9, 13, 21

• Mean = (12 + 16 + 9 + 13 + 21) / 5
• Mean = 14.2

Example 2

The mean of 100, 99, 99, 95, 94, 90, 88, 86, 80, 75, 63

• Mean = (100 + 99 + 99 + 95 + 94 + 90 + 88 + 86 + 80 + 75 + 63) / 11
• Mean = 88.09

Median

The median is the middle value or number in a given data set organized sequentially. In order to find the median, we need to arrange the data set in either ascending or descending order. The middle value is the median.

Unlike mean, we do not take into account every value in the dataset to calculate the median. So, the median is not affected by outliers or extreme values. The median is considered as a positional average while the mean is considered as an arithmetic average

Every now and then, we'll have two numbers in the middle (when the number of values is even). Find the average of those two numbers (a+b/2) to find the median. You can also use the median to separate the given data set into two halves, one half has all the higher values and the other half has all the lesser values.

Example 1

Find the median of the data 20 10 10 40 57

• Arrange the given data in ascending order: 10 10 20 40 57
• The middle number of the data set is the median (5/2 = 2.5 -> 3rd value in the data set)
• Median = 20

Example 2

Find the median of the data 20 10 10 25 40 57

• Arrange the given data in ascending order: 10 10 10 15 40 57
• As the total number of values is even (here 6), the average of two middle numbers is our median
• Median = (10+15)/2 = 25/2
• Median = 12.5

Mode

The mode is the most common number or the most repeated number in a given data set. While the mean represents the entire dataset, the mode doesn't. Mean takes into consideration each and every value while mode doesn't. It is simply the value that appears most frequently.

There is always a mean and a median for a given dataset—but this is not the case with mode. There may be no mode if no value appears more than any other.

Example 1

Find the mode of 15, 15, 19, 23, 24, 19, 16, 19, 23, 30, 19, 22

• We can start by arranging the given data in ascending order: 15, 15, 16, 19, 19, 19, 19, 23, 23, 24, 30
• We can easily find the most repeated number from the above data (only one number repeats 4 times)
• Mode = 19

Example 2

Find the mode of 16, 19, 93, 45, 63, 24, 87, 33, 52, 23, 11, 01, 100

• There is no mode for the above data since there is no value that appears or repeats more than any other value

The Relationship Between Mean, Median, and Mode

Most of the time, our aim is to calculate the central tendency of a dataset. When working on a dataset, to measure central tendency, the mean is preferred over the other two entities because of the fact that it takes into consideration every single value in the dataset.

However, if the dataset contains outliers or extreme values, the median is preferred over the mean since it isn't affected by those extreme values or outliers. In any case, mean and median are preferred over mode for calculating the central tendency of the data. There is a formula that shows the mathematical connection between the three terms:

Mode = 3 x Median - 2 x Mean

The above formula is derived from the observation that the difference between the mean and mode is almost equal to three times the difference between the mean and median.

Mean - Mode = 3(Mean - Median)

Depending on the situation, you can choose the term that will best represent the sections of data that you're identifying.

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