What is the mean?
The mean is a type of average. In a set of data, the mean is the total sum of all the values divided by the number of values in the set.
There must be at least 2 numbers in the data set for you to be able to find the mean. They should be connected to each other or have some sort of relationship for the mean to be important. For example, the mean is useful if you are comparing the average temperatures of each day across a month or the different marks that students get in a class.
Other types of average are the mode, median, and mean. One of the main instances where we could calculate the mean would be when we want to figure out an average or norm of a set of values.
How to find the mean in maths
To do this, we find the total of all the numbers in a data set, before dividing by the number of values that we have. Simply put, add them all together and divide by how many there are!
Let’s do an EXAMPLE step by step:
Step 1. First is add all the numbers together. 2 + 5 + 8 + 9 + 11 = 35
Step 2. Now, we have six numbers in total so, 35 ÷ 6 = 5.
Step 3. Our final answer, and the mean of these numbers, is 5.
However, it is important to remember that the mean is not always a whole number, sometimes, it is a decimal. Usually, a question will tell you how they want the answer to be given to one, two or three decimal places or to the nearest whole number.
How to work out the mean with negative numbers
Even if our data set contained negative numbers, the process to calculate the men would be the same. Let’s try an example with negative numbers:
Find the mean of 2,6,6,9, -1 and -4.
Step 1. First, we need to add them together. Now because we have two negative numbers included here it would be simpler to add the positive numbers together first. And then combine the two negative numbers and subtract that from our positive total:
23-5=18.
Step 2. Next, we need to divide by the number of values that we have. That gives us 18÷6=3.
Step 3. The mean of these numbers is 3.
What is the median?
In an organised list of numbers, the median is the number that’s in the middle, literally speaking. However, we don’t mean it in terms of the numerical value of the number, but the literal number in the middle of the list.
For example:
If you have a list of numbers like this one 1, 5, 7, 8, 10, 15, 22, the median number is 8. This is because it’s in the middle of the list. TOP TIP: make sure the list has been organised into numerical order BEFORE trying to find the median.
When there is an odd number of numbers in the data set finding the median is easy. However, if there is an even number within the data set then you won’t find a middle number. In these instances, we should add the two middle numbers together and then divide by two (because there are two numbers to work out the median from)
Example:
Here is a data set:
9, 15, 24, 56, 71, 88.
We can see here that there are two middle numbers: 24 and 56. So, to find the average we add these together and divide by 2. 24 + 56 = 80 now we divide by 2 to get our median number of 40.
Why is it important to learn about the median?
Mean, median and mode are all very useful for analysing data. Whether this is small amounts of data or large, they’re good indicators of what the data looks like.
Calculating the median is a great way to quickly identify a centre point in a data set. This is helpful when the data is scattered with no common trend or pattern, this is also called disproportionate. However, the median isn’t a good representation of the average in a data set, which is why we also calculate the mean.
A step-by-step guide
Now let’s look at a step by step break down to help us get the hang of finding the median.
Step one: put the data set into ascending numerical order. (Start with the lowest number and put them in order up to the largest).
EXAMPLE:
Here is a data set:
37, 80, 1, 9, 7 3, 22, 10, 34
In order it looks like:
1, 3, 7, 9, 10, 22, 34, 37, 80.
Step 2: Then, you must count how many numbers you have in your sorted list. Are there an odd number or an event number?
EXAMPLE:
In this data set there are 9 numbers. That’s an odd number, this makes finding the median easier.
Step 3. when we have an odd number in a data set all we need to do is make sure that there are an equal amount of numbers on the left and right and then we’ll eventually find the middle
For example, here 4 numbers to the right and left, leaving the 5th number in the middle alone as our median.
However, if there is an even number (let’s say there were 8 numbers) we would have 3 to the right and left leaving us two numbers in the middle to calculate the median from.
EXAMPLE:
1, 3, 7, 9, [10] 22, 34, 37, 80.
As we can see here there are equal amounts of numbers to the right and left of our middle number. If we had a large data set (of maybe 57) we simply divide the number by two and use to the nearest whole number. So, if our data set was 57, 57 divide by 2 would leave us with 28.5, so our median would be the 29th number! It’s that easy, and it’s also the reason why we must have the numbers in ascending order before we start the process!
What is the Mode?
The mode in maths is the value that appears most often in a set of data, the most repeated number.
So, in the list 4, 4, 6, 6, 6, 9, 10, 10, 12 the modal number is 6 as it appears most often.
The mode in maths is one of the keyways to detect the average within a set of data. By finding the average, we can understand the most common value. This might tell us something significant about the data set and hence can be very useful.
Example: if we were to look at the goals scored by a school football team over the course of a season, we would be able to identify an average goal score by finding the mode.
0, 1, 1, 2, 2, 2, 3, 3. Looking at the data of goals scored we can see that the team averaged at 2 goals per match.
Analysing data can be tricky to learn at first, so starting with the simple concepts of the mode, median, mean and range are useful and a great place to start. TOP TIP: you can remember what the mode means by looking at the first two letters, "M" and "O" remembering that the mode is the number that appears Most Often.
More Than One Modal Number
Sometimes, a data set will have more than one mode.
TOP TIP: unlike other methods of data analysis, you don’t need to order the data set yet, the mode can be found whilst the data is in its original form. That is because we are just looking for the number that repeats itself the most times. So long as the child can use a form of marking (maybe by underling a repetitive number or using a dot) to help them identify the most repeated number they do not need to spend the time organising the data into numerical order.
Example:
Example 1: Find the modal number(s) in this set of data...
3, 3, 8, 3, 10, 21, 9, 4, 7, 4, 12, 4
When looking at this data set, you can see that both the number 3 and the number 4 appear three times. Therefore, there must be two modal numbers for this set of data.
The modal numbers are: 3 and 4.
TOP TIP: When a set of data has two modes, it is called ‘bimodal’.
Example 2: Find the modal number(s) in this set of data...
88, 70, 70, 21, 88, 70, 70, 88, 90, 90, 90, 88, 35, 10, 90
By analysing this set of data, you can see that 88, 70, and 90 appear 4 times. This means that there are 3 modes for this set of data.
The modal numbers are: 88, 70, and 90.
When a set of data has more than two modes, it is called ‘multimodal’.
Modal Classes
In some instances, you will find that each number in the data set occurs the same number of times. When you are working with data like this, finding the mode won’t be very helpful. Instead, you can group the data into ‘modal classes’, and you can then identify the mode of each of the individual classes. This will give you a better understanding of the data set as a whole.
For example, find the mode of this set of data: 1, 4, 2, 7, 8, 9, 10, 17, 19, 20, 32, 34, 37, 40, 43
Now, you must separate your data into appropriate groups. It’s important to make sure all your groups are of equal sizes so that you can accurately compare them.
The groups for this data set should be:
1 – 4: 3 values (1, 4, and 2)
7 – 10: 4 values (7, 8, 9, and 10)
17 – 20: 3 values (17, 19, and 20)
31 – 34: 2 values (32 and 34)
36 – 39: 1 value (37)
40 – 43: 2 values (40 and 43)
Now, we can see that one class clearly contains more values than the others: 7-10 contains 4 values. This means that the modal class is 7-10.
This method of dividing data up into different groups and figuring out the modal class is particularly useful when dealing with a set of data that contains anomalies (outliers/irregularities) that could skew the results.
Sound confusing, let’s try another example!
Find the mode in this set of data:
20, 22, 25, 26, 27, 28, 29, 33, 46, 50, 57, 58, 59, 61, 63, 72, 74, 75
The first step is to analyse the data closely and find appropriate groups for your numbers. TOP TIP: keep the groups the same size!
For this data set the groups would look like this. (we can see that there are gaps in the flow of the groups – they do not go in numerical order, there are numbers missing – but this is because we only need to create groups to contain numbers from our data set. So long as the groups are all the same size it does not matter that we do not follow the complete numerical order.
20 – 24
25 – 29
30 – 34
46 – 50
55 – 59
60 – 64
71 – 75
Now that you have your groups, you can figure out how many values there are in each.
20 – 24: 2 values (20 and 22)
25 – 29: 5 values (25, 26, 27, 28, and 29) 30 – 34: 1 value (33)
46 – 50: 2 values (46 and 50)
55 – 59: 3 values (57, 58, and 59)
60 – 64: 2 values (61 and 63)
71 – 75: 3 values (72, 74, and 75)
We can now clearly see that the group ‘25 – 29’ has the most values within it, therefore, the modal class for this set of data is 25 – 29.
Advantages and Disadvantages of the Mode
Let’s have a look at some of the key advantages and disadvantages of finding the mode:
Advantages of finding the mode:
The mode is simple and easy to understand.
The mode is not affected by extremely large or small values in a set of data.
The mode is easy identifiable.
The mode can be located on a graph.
Disadvantages of finding the mode:
To find the mode, there much be repeat values in a data set.
It is hard to find the mode when the data set contains a small number of values.
A data set can have one mode, more than one mode, or a modal class.