How to Find the Average of a Group of Numbers
What Do We Mean by 'Find the Average'?
When somebody tells you the average of a group of numbers, they would generally be talking about one of three distinct things: the mean, the median or the mode.
Each has a different way of being calculated, and each has its own advantages and disadvantages when being used to compare groups of data.
In this article, we're going to have a look at how each of the three averages is calculated and when each of the different averages is most useful.
Each has a different way of being calculated, and each has its own advantages and disadvantages when being used to compare groups of data.
In this article, we're going to have a look at how each of the three averages is calculated and when each of the different averages is most useful.
How to Find the Mean of a Group of Numbers
The mean is the most commonly used average in daily life and the one most people are referring to when they say 'the average.' Finding it is fairly simple; you just add all of your values together and then divide by how many values there are.
Example
Find the mean of 9, 5, 1, 9, 6, 2 and 3.
Add the numbers together: 9 + 5 + 1 + 9 + 6 + 2 + 3 = 35
Divide by how many numbers there are: 35 ÷ 7 = 5
The mean average of these numbers is 5.
One way of thinking about the mean is if you had a group of people, each with a different number of sweets, how many would they each have if they pooled the sweets together and shared them equally?
Example
Find the mean of 9, 5, 1, 9, 6, 2 and 3.
Add the numbers together: 9 + 5 + 1 + 9 + 6 + 2 + 3 = 35
Divide by how many numbers there are: 35 ÷ 7 = 5
The mean average of these numbers is 5.
One way of thinking about the mean is if you had a group of people, each with a different number of sweets, how many would they each have if they pooled the sweets together and shared them equally?
Finding the median
How to Find the Median of a Group of Numbers
The median of a group of numbers is the number that would be in the middle of the group if they were written out in order.
Taking the same group of numbers as before, 9, 5, 1, 9, 6, 2 and 3, we first need to rewrite the list in size order (it doesn't matter if you pick ascending or descending, the middle will still be in the same place).
In size order: 1, 2, 3, 5, 6, 9, 9
The middle number in the list is 5; hence, this is the median.
Taking the same group of numbers as before, 9, 5, 1, 9, 6, 2 and 3, we first need to rewrite the list in size order (it doesn't matter if you pick ascending or descending, the middle will still be in the same place).
In size order: 1, 2, 3, 5, 6, 9, 9
The middle number in the list is 5; hence, this is the median.
How to Find the Median When There Are an Even Amount of Numbers
With the list above, we had an odd amount of numbers, so there was only one number in the middle, and the median could be found easily, but what about if we had an even amount of numbers?
In this case, we take the two middle numbers and find the mean of these by adding them together and dividing by 2.
Example
Find the median of the following list of numbers: 10, 32, 15, 43, 45, 8, 18, 24.
Rewrite the list in size order: 8, 10, 15, 18, 24, 32, 43, 45
Find the mean of the two middle numbers: (18 + 24) ÷ 2 = 21
The median is 21.
In this case, we take the two middle numbers and find the mean of these by adding them together and dividing by 2.
Example
Find the median of the following list of numbers: 10, 32, 15, 43, 45, 8, 18, 24.
Rewrite the list in size order: 8, 10, 15, 18, 24, 32, 43, 45
Find the mean of the two middle numbers: (18 + 24) ÷ 2 = 21
The median is 21.
How to Find the Mode of a Group of Numbers
The mode is probably the simplest of the three commonly used averages to find. It is the number that appears most often in the group and is easily remembered, as mode and most begin with the same two letters.
Example
Find the mode of 9, 5, 1, 9, 6, 2 and 3.
When dealing with a long list of numbers, it can be helpful to put them in order as we did with the median, as this can help you avoid missing any numbers and make repeated numbers easier to count.
However, this is not necessary, especially in a small group such as this. We can see with a quick glance that the most common number here is 9. Hence, this is the mode (also known as the modal value).
A group of numbers can be bimodal if two numbers appear equally and more often than any other. If there are no numbers that appear most frequently, there is no mode.
Example
Find the mode of 9, 5, 1, 9, 6, 2 and 3.
When dealing with a long list of numbers, it can be helpful to put them in order as we did with the median, as this can help you avoid missing any numbers and make repeated numbers easier to count.
However, this is not necessary, especially in a small group such as this. We can see with a quick glance that the most common number here is 9. Hence, this is the mode (also known as the modal value).
A group of numbers can be bimodal if two numbers appear equally and more often than any other. If there are no numbers that appear most frequently, there is no mode.
How to Find the Range of a Group of Numbers
While not an average, the range is also a useful value to find alongside the averages when interpreting data.
The range is a measure of how spread out a group of numbers is and is calculated by subtracting the smallest value from the largest value.
Example
Find the range of our group of numbers: 9, 5, 1, 9, 6, 2 and 3.
First we see that the smallest number is 1 and the largest number is 9.
Therefore the range = 9 − 1 = 8
The larger the range, the more spread out a group of numbers is, while a small range means they are closer together and more consistent.
The range is a measure of how spread out a group of numbers is and is calculated by subtracting the smallest value from the largest value.
Example
Find the range of our group of numbers: 9, 5, 1, 9, 6, 2 and 3.
First we see that the smallest number is 1 and the largest number is 9.
Therefore the range = 9 − 1 = 8
The larger the range, the more spread out a group of numbers is, while a small range means they are closer together and more consistent.
How to Find Averages From a Frequency Table
Learn how to find the mean, median, mode and range from a frequency table in this quick maths article.
So Which Average Is the Best to Use?
We've seen that are three common averages which can be used to compare groups of data: the mean, median and mode. But which one is the best? The answer to this depends upon what sort of data you are using and what sort of result you are after.
The Mean: Advantages and Disadvantages
As mentioned earlier, the mean is generally the average most commonly used and is a very powerful tool for comparing data. Its strongest advantage over the other averages is that it uses every value, not just the middle one or the most common one.
However, there is one particularly large flaw with the mean: extreme values can distort it.
One example of this would be if you took the mean average of salaries in a small company. Suppose five employees each earning £20 000 a year and one director earning £50 000. The mean salary in the office would be (5 × 20 000 + 50 000) ÷ 6 = £25 000. This isn't particularly representative of the data as it has been skewed by the outlier of the director's earnings. In this case, the median and mode would have been far more appropriate.
However, there is one particularly large flaw with the mean: extreme values can distort it.
One example of this would be if you took the mean average of salaries in a small company. Suppose five employees each earning £20 000 a year and one director earning £50 000. The mean salary in the office would be (5 × 20 000 + 50 000) ÷ 6 = £25 000. This isn't particularly representative of the data as it has been skewed by the outlier of the director's earnings. In this case, the median and mode would have been far more appropriate.
The Median: Advantages and Disadvantages
The median has the advantage of not being skewed by outliers. In the salaries example above, it doesn't matter how large the director's salary becomes; it will not affect the middle value. That will remain at £20 000. Likewise, if an inexperienced worker joins the company on a much lower salary than everybody else, this new lower outlier won't affect the median either.
The median is also relatively easy to find, with no calculations required.
The main disadvantage of the median is that it doesn't use all of the numbers in the data set.
The median is generally the best average to use when your data has some extreme outliers, which would distort the mean, such as in the salary example above.
The median is also relatively easy to find, with no calculations required.
The main disadvantage of the median is that it doesn't use all of the numbers in the data set.
The median is generally the best average to use when your data has some extreme outliers, which would distort the mean, such as in the salary example above.
The Mode: Advantages and Disadvantages
Just as with the median, one advantage of the mode is that it is not affected by outliers. It is also generally the easiest to calculate of the three averages and is the only one of the three that can be used for non-numerical data (e.g., if you were tallying the colour of cars driving past, the mode might end up being silver).
Again just like the median, the mode has the disadvantage of not using every value in the set. It also has the problem of sometimes not existing. If there is no most common number, then there is no mode.
Generally, the mode is most useful when you want to find the most likely number. For example, if you were stocking a shoe shop, the modal shoe size sold would be extremely useful as this would be the shoe size you would then reorder most of.
Again just like the median, the mode has the disadvantage of not using every value in the set. It also has the problem of sometimes not existing. If there is no most common number, then there is no mode.
Generally, the mode is most useful when you want to find the most likely number. For example, if you were stocking a shoe shop, the modal shoe size sold would be extremely useful as this would be the shoe size you would then reorder most of.
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