Averages

The term ‘average’ is one which you will have most probably come across before, but this term is not exact. This is because there are many different averages that can be worked out for data and each has its own use. Therefore, when someone says to get the average of something, really they need to be more specific about which average. The main three averages which we will be looking at are:

• Mean
• Mode
• Median

Each of these averages have their own use and some are more useful than others depending on the situation. As well as this, they each have their own method for being calculated and will give differing answers to one another for the same set of data.

Mean

The mean of a set of data is probably the most common of the three different averages. It is also probably the most useful when we are looking at numerical data and is also rather easy to calculate.

To get the mean we must:

1) Add up all the values that we are given.

2) Divide this ‘total’ by the amount of separate statistics we were given.

This can also be shown as  which is said as ‘the sum of the data divided by n‘ where n is the amount of individual statistics.

Example

Find the mean of the following data.

5, 9, 17, 6, 3, 12, 14, 18, 9, 7, 2, 11, 10, 4, 6, 8

Following the steps outlined above we must first add all of these numbers up. This gives us 141. Now we need to divide this amount by how many individual numbers we have. There are 16 different numbers that are in the list.

So we calculate  (2dp)

So the data we were given has a mean of 8.81

Clearly the answer that we have above (8.81) is not even in the list that we were given! This is completely fine when we are looking for the mean as the value does not have to actually be a given value and will be a new number most of the time. This is because the mean is simply an average of the numbers we have and does not have to be part of the data that we started out with.

Mode

The mode of a set of data is quite simply the ‘most frequent’. This clearly means the one which crops up the most in our data! So if we were to flip a coin multiple times to get 5 heads and 7 tails, tails would be the mode of the data as this result happened more often than any other outcome.

When finding the mode of data in a frequency table or when the data is split into segments, the mode is exactly the same and very easy to calculate when the data is already in a frequency table – it is simply the row with the highest frequency.

Example

Find the mode of the following data that is split into categories.

Here the highest frequency is 27, therefore we can say that the mode of the data is a score of 3.

You may be wondering what happens if there are two or more frequencies which are the highest. In this case the mode of the data is simply all of the most frequent scores. So if the score of 4 also had a frequency of 27 in the above table then the mode would be both the scores 3 and 4.

Median

The median is again fairly simple to calculate but this time the data must be numerical. This is due to the fact that we need to have the data ordered from lowest to highest to find the median.

How to find the median:

1) Order the data from lowest to highest.

2) Count the amount of individual statistics that we have, call this n.

3) The median is the middle value of the ordered list, so we need to count from the first value through the list until we find the value exactly in the middle of the n different values.

Another way to think of finding the middle number is to count along to the number that is  numbers into the ordered list. This will then ensure that we have the middle value.

Suppose that when we find the value of  we were to get an answer that wasn’t whole (the only possible outcomes are a number that is whole or one which is a half). In this case we must look for the number which lies between the two values.

So, if we needed the number that is 7.5 in the list, we find the 7th and the 8th and then get the number between these two values.

Example

Find the median of the already ordered list of data below:

1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 8, 9, 10

Counting how many individual numbers we have we see there are 13. Therefore, we need to find the value that is the  value in the list.

1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 8, 9, 10

The 7th value is 4, so the median of the data is 4.

Example

Using the same list as the previous example, we find that one value was missed. Adding the missed value of 5, find the new median.

The previous list was

1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 8, 9, 10

When adding the new value of 5 into this list we must make sure that the values are still in the correct order, lowest to highest. Doing this we get:

1, 1, 1, 2, 3, 3, 4, 5, 5, 7, 8, 8, 9, 10

We now have 14 pieces of data which means we need to look for the  value. This means we need to find the 7th and 8th value and then find the value halfway between these. The 7th and 8th values are 4 and 5, and then finding the middle of these two values gives us a median of 4.5.

Comparing the mean, median and mode

There are many differences between these three statistics and for the same set of data each value could be very different. The values of the means and median do not necessarily have to be a part of the original data list, however, the mode being the most frequent value must be the same as one of the values originally given.

All of these different averages have their own uses and, depending on the situation, one may be better than the others. Also, all of the different methods can be used for finding averages of data that is grouped and in a frequency table. This is what we will move on to look at next.