It’s useful for different situations that you come across in real life, as well as for academic purposes. There are different types of average and we can help you find the mean, median, mode as well as the range, and give you the confidence to overcome any challenge with averages. Let’s go for it!

How To Find The Mean, Median, Mode & Range
How To Find The Mean, Median, Mode & Range

A Quick Look At Different Types of Averages

The mean, median, mode, and range each provide different insights into the nature of a set of numbers, also known as a dataset. Why do you need to learn these? Because whatever context you’re using them for, knowing the difference between averages and when you use them will get you well on the way to being an expert in this topic! Let’s see how:

  • The mean takes into account an entire set of numbers and gives us an overall average - for instance, you might use it if you were looking at the average set of scores in a class.
  • The median gives us the middle value of a data set when ordered from lowest to highest, which is useful when you have outliers that might affect the average too much.
  • The mode reveals the number that appears most often in a set of numbers. This could be useful when trying to find what’s most popular in a set of numbers.
  • The range tells us the difference between the largest and smallest numbers in a dataset, which can help you understand how spread out the numbers are.

An outlier is a value in a set of numbers that is significantly higher or lower than the other numbers, and it can skew or distort the overall picture of the dataset's average. It’s basically a number that decided to go a bit rogue from the rest!

How To Find The Mean

The mean is a way to give you an overall average of an entire set of numbers, and is used extensively in everyday life and academically. When you hear people talk about an average, more often than not, they’re talking about the mean average.

To find the mean, you add up all the numbers in a set and then divide by how many numbers there are in the set. The mean is sometimes seen as a bit mean compared to other averages, especially because of the division; but don’t worry, we’ll break it down and you’ll start to like it!

Step-by-Step Guide to Calculating the Mean

Calculating the mean is straightforward, but following these steps carefully will ensure you get it right every time:

  1. 1. List all the numbers: Make sure you have all the numbers written down in front of you.
  2. 2. Add up all the numbers: Add up your set of numbers to get a total sum.
  3. 3. Count how many numbers you have: Find out how many numbers you have in your dataset.
  4. 4. Divide the total sum by the count of numbers: This will give you the mean.

Example: Calculate the Mean

Problem: Let's say you want to find the mean score from a test a class took. The scores were: 4, 8, 6, 5, and 7.

First, we’ll add up the scores: 4 + 8 + 6 + 5 + 7 = 30

Then, we count how many scores there were in the class: There are 5 scores in total.

Finally, we divide the total sum by the count of scores. 30 / 5 = 6.

The mean of the scores is 6.

Not too bad, right?

Challenge Yourself!

Problem: Find the mean of this set of numbers: 10, 15, 20, 25, 30, 35.

Solution:

  • Add up the numbers: 10 + 15 + 20 + 25 + 40 = 110
  • Count how many numbers you have: There are 6 numbers
  • Divide the total sum by amount of numbers: 110 / 5 = 22
  • The mean score for this dataset is 22

Remember: Because we’re dividing, sometimes the answer might have a decimal! Take a look at our decimals worksheets to get a refresher.

 

Want to try your hand at some more problems on how to find the mean? Take a look at our online Find the Mean Worksheet.

 

How To Find The Median

The median is the middle value in a dataset when it is arranged in order. It is a very useful measure, especially when the data includes outliers or extremely high or low values, because it provides a clear picture of the centre of the dataset without being messed up by those extremes. Here, we’ll take you through step-by-step on how to find the median, and soon you’ll be a master!

Step-by-Step Guide to Finding the Median

Here's how you can find the median in any set of numbers:

  1. List all the numbers: Make sure you have the entire set of numbers listed in front of you, especially if there is more than 1 of the same number. So if you have two 4’s in your set, you still list them both!
  2. Order the numbers: Arrange the data points from smallest to largest. If you have an odd amount of numbers, this helps you find the exact middle number - if you have an even amount of numbers, this helps you identify the two numbers in the middle.
  3. Find the middle point: Check if the number of data points is odd or even:
    • Odd: The median is the number right in the middle of the list.
    • Even: You need to find the middle two numbers, add them together, and then divide by two to get the median.

 

Example 1: Finding the Median with an Uneven Set of Numbers

Problem: Find the median of the following test scores: 3, 9, 5, 8, and 2.

First, we’ll order the scores: 2, 3, 5, 8, 9

The middle score (median) is 5 because it is the third score in a list of five numbers.

Sometimes a helpful way to do this is to cross through one number at each end until you reach the middle. So in this example you would cross out 2, then 9, then 3, then 8 which would leave you with 5.

Example 2: Finding the Median with an Even Set of Numbers

Problem: Find the median of these ages: 7, 12, 5, and 10.

First, we order the ages: 5, 7, 10, 12

Since there are four numbers, the second and third numbers are in the middle. These are 7 and 10.

Add the two middle numbers: 7 + 10 = 17

Divide the sum by 2: 17 / 2 = 8.5

The median of the ages is 8.5.

Challenge Yourself!

Problem: Calculate the median of this set of numbers: 14, 17, 9, 8, 22, and 15.

Solution:

  • Order the numbers: 8, 9, 14, 15, 17, 22
  • Find the middle number(s): You could cross off from each end - 8, 22, 9, 17 which leaves you with 2 numbers: 14 and 15.
  • Another way to look at it is, there are six numbers, so the median will be the average of the third and fourth numbers, which are 14 and 15.
  • Add the numbers together: 14 + 15 = 29
  • Divide by 2: 29 / 2 = 14.5.
  • The median is 14.5.

Looking for an extra challenge? Try out our Finding the Median Worksheet to apply the knowledge you’ve learnt so far!

How To Find The Mode

The mode is the number that appears most frequently in a set of numbers. It's particularly useful for finding the most common or popular elements in any set of data. This could range from finding the most popular product in a survey to identifying the most frequent number in a series of measurements. Let’s discover how to find the mode in any set of numbers!

Step-by-Step Guide to Finding the Mode

Finding the mode requires you to carefully look at all the numbers and find out which one appears most frequently. Here’s an easy way to accurately determine the mode:

  • List all the numbers: Arrange your data in ascending order (smallest to largest). This helps in easily spotting the most frequent numbers.
  • Count the occurrences of each number: As you scan through your ordered list, keep a count of how often each number appears.
  • Identify the highest frequency: The mode is the number that occurs most often. There might be more than one mode if multiple numbers have the same highest frequency.

Example 1: Finding the mode in a dataset

That set of numbers is pretty messy! We’ll order them from lowest to highest first: 2, 3, 3, 5, 5, 5, 5, 8

Now we’ll count the occurrences: The number 5 appears four times in total, which is more frequently than any other number.

Therefore, the mode is 5 as it is the number that comes up the most. It’s the most common number. 

Easy as that!

Example 2: Finding the mode in a dataset with more than one mode

Problem: Determine the mode of these numbers: 12, 17, 12, 9, 17, 9, 9, 17.

First we order the numbers: 9, 9, 9, 12, 12, 17, 17, 17

Then we count the occurrences: We see that the numbers 9 and 17 both appear three times.

No problem! That just means the modes are 9 and 17 - there is more than one mode!

Challenge Yourself!

Problem: Find the mode in this list of test scores: 88, 74, 88, 90, 74, 88, 90, 74.

Solution:

  • Order the numbers: 74, 74, 74, 88, 88, 88, 90, 90
  • Count the occurrences: The numbers 74 and 88 each appear three times, making them both the modes of this dataset.

What if all the numbers appear the same amount of times?

If you have a set of numbers where every value appears the same number of times (for instance, 2, 4, 6, 8, 10), then your dataset has no mode.

Now you know how to find the mode, you’ll easily be able to tackle any problem and also help you with certain situations in real life! Try out our Find the Mode worksheet if you want an extra challenge.

How To Find The Range

The range is a simple statistical measure that shows the difference between the highest and lowest values in a dataset. It can be really handy because it gives you a quick sense of how spread out the numbers in a set are, which is crucial in fields such as weather forecasting, sports, and many other areas. Let’s take a look at the steps and some examples, and you’ll pick it up in no time!

Step-by-Step Guide to Calculating the Range

Although not a type of average, the range is useful to know along with an average and is nice and simple to work out. Let’s learn how to do it together:

  1. Order the numbers: Arrange the set of numbers in ascending order.
  2. Identify the highest and lowest numbers in your dataset.
  3. Subtract the lowest number from the highest number: This difference is the range. Simple!

Example: Calculate the Range for Temperatures Over a Week

Problem: Find the range of temperatures for a week that were recorded as 15°C, 20°C, 18°C, 21°C, 16°C, 19°C, and 17°C.

First, we’ll order the temperatures to make it easier for ourselves: 15°C, 16°C, 17°C, 18°C, 19°C, 20°C, 21°C Take a look at the first number in your list to find the lowest: 15°C Then find the last number to find the highest: 21°C All that’s left is to work out the difference by calculating the highest minus the lowest: 21°C - 15°C = 6°C

So, the range of the temperatures over the week is 6°C.

Challenge Yourself!

Problem: Calculate the range of these test scores: 65, 82, 90, 70, 58, 84.

Solution:

  • Order the scores: 58, 65, 70, 82, 84, 90
  • The lowest score is 58
  • The highest score is 90
  • Calculate the range: 90 - 58 = 32

The range of the scores is 32 points.

Extra Knowledge: Understanding the Impact of an Outlier

Problem: Consider a dataset with an outlier: 10, 12, 15, 20, 22, 100.

Steps to calculate the range:

Highest number = 100

Lowest number = 10

Range = 100 - 10 = 90

The presence of the outlier (100) significantly increases the range. Without the outlier (if the highest number was 22), the range would be 12 (22 - 10). This shows how one extreme value can dramatically affect what the range tells you about your set of numbers.

Everything making sense so far? Try out our Finding the Range Worksheet to have a go at testing your knowledge!

Practice Problems and Solutions

  1. Find the mean of these test scores: 55, 60, 65, 70 and 75.
  2. Determine the median of this set of numbers: 12, 15, 10, 8, 20.
  3. Identify the mode in this list of numbers: 4, 2, 4, 7, 7, 5, 4, 7, 7, 5.
  4. Calculate the range of the following ages: 7, 13, 10, 18, 21, 10.

Answers:

  1. Add all the scores together (55+60+65+70+75 = 325) and then divide by the number of scores (5), resulting in a mean of 65.
  2. First, order the numbers (8, 10, 12, 15, 20). The median is the middle number, which is 12 in this ordered list.
  3. First, order the numbers to make it easier (2, 4, 4, 4, 5, 5, 7, 7, 7, 7). The number 7 appears most frequently (four times), which makes it the mode of this dataset.
  4. The highest age is 21 and the lowest is 7. Subtracting these (21 - 7) gives you a range of 14 years.

Well done on working through this blog on working out the mean, median, mode and range! Remember, mastering these statistical skills not only boosts your maths understanding but also improves your ability to make sense of the world around you. Keep practising, and soon you'll be able to tackle any data challenges that come your way with confidence and ease!

Use the below worksheets to keep practising the skills you’ve learnt along the way, and soon enough working out the mean, median, mode and range will be a breeze.