Mean, Median & Mode (CIE IGCSE Maths: Core)

Why do we have different types of average?

• You’ll hear the phrase “on average” used a lot
  • For example
    • by politicians talking about the economy
    • by sports analysts
• However not all data is numerical
  • For example
    • the party people voted for in the last election
  • Even when data is numerical, some of the data may lead to misleading results
• This is why we have 3 types of average

What are the three types of average?

1. Mean

• This is what is usually meant by “average”
  • it’s like an ideal world where everybody has the same
  • everything is shared out equally
• It is the TOTAL of all the values DIVIDED by the NUMBER OF VALUES
  • Add all the numbers together and divide by how many there are
• Problems with the mean occur when there are one or two unusually high (or low) values in the data (outliers)
  • these can make the mean too high (or too low) to reflect any patterns in the data

2. Median

• This is similar to the word medium, which can mean in the middle
• So the median is the middle value – but beware, the data has to be arranged into numerical order first
• We would use the median instead of the mean if we did not want extreme values (outliers) affecting our data
• If there are an odd number of values, there will only be one middle value
• If there are an even number of values we would get two values in the middle
  • In this case we take the half-way point between these two values
  • This is usually obvious but, if not, add the two middle values and divide by 2
    • this is the same as finding the mean of the middle two values

3. Mode

• Not all data is numerical and that is where we use mode
• MOde means the Most Often
• It is often used for things like “favourite …” or “… sold the most” or “… were the most popular”
• Mode is sometimes referred to as modal
  • you may see phrases like “modal value”
  • they all mean the same thing, the value occurring most often
• Be aware that the mode can be applied to numerical data
• Sometimes if no value/data occurs more often than the others we say there is no mode
  • If two values occur the most we may say there are two modes (bi-modal)
    • whether it is appropriate to do this will depend on what the data is about

What calculations with the mean might I have to do?

• Mean = Total of values ÷ Number of values
  • it is a formula involving 3 quantities
  • if you know any 2, you can find the other one
• Typical questions ask you to either
  • work backwards from a known mean or
  • combine means for two data sets
• As this is in the area of problem solving there may be something unusual that you haven’t seen before
  • you will need to make sure you understand what the mean is, how it works and what it shows

How do I solve problems involving calculations with the mean?

• You can rearrange the formula for the mean or use a formula triangle
  • Total of values = Mean × Number of values
  • Number of values = Total of values ÷ Mean
• If you know the mean and you have to find a missing value then:
  • Find the total of the values
  • Subtract the values you do know
• If you are working with two sets of data then make it clear what the mean or total is referring to