Mean, Median & Mode (Cambridge O Level Maths)

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
• 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
• A formula for the mean could be thought of as 

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 does calculations with the mean involve?

• Because the mean has a formula it means you could be asked questions that use this formula backwards and in other ways
• 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

What calculations with the mean might I have to do?

• 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?

• Known mean, unknown data value
  • This is working backwards from the mean, to an unknown data value
  • Call the unknown data value , say
  • Using the 'formula' for the mean, set up an equation in 
  • Rearrange and solve the equation to find , the unknown data value
• Combined means for two data sets
  • This is where we know the mean for two different data sets but would like to know the overall mean
  • We would need to find the overall total of values from both data sets
  • Then divide by the total number of values across both data sets 
  • Alternatively we may know the overall mean and want to work back to the mean of one or both of the data sets, or an unknown data value
• Others
  • Due to the problem solving nature of such questions there will be some variation in question styles
    • The above two should give you a good idea and cover the vast majority of questions
  • The best way to start tackling questions with the mean is to

    • write down the quantities you do know
    • write down those you don't know
    • use the 'formula' for the mean to link the unknown and known values