Mean, Median & Mode | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Mean, Median & Mode

Why do we have different types of average?

• You'll often hear the phrase "on average" used frequently.
• For instance:
  • Politicians discussing the economy.
  • Sports analysts making assessments.
• However, not all data is numerical.
• For example:
  • The political party people voted for in the last election.
• Even when the data is numerical, some of it can produce misleading outcomes.
• This is why we have three types of averages.

What are the three types of average?

Mean:

• This is typically what is meant by "average."
• It's like an ideal world where everything is shared equally among everyone.
• It is calculated as the total of all values divided by the number of values.
• Problems with the mean arise when there are outliers (unusually high or low values) in the data, which can skew the mean to be higher or lower than the actual pattern in the data.
• A formula for the mean could be represented as: Mean = (Total Amount) / (Number of Data).

Median:

• The median is similar to the word "medium," meaning in the middle.
• It represents the middle value, but the data must be sorted in numerical order first.
• The median is used instead of the mean to avoid the influence of outliers.
• If there is an odd number of values, there is one middle value.
• If there is an even number of values, there are two middle values; the median is the halfway point between these two.
• To find this halfway point, add the two middle values and divide by 2, which is the same as finding the mean of the middle two values.

Mode:

• Mode is used when data is not numerical.
• MOde means the Most Often.
• It is often used for categories like "favorite...," "sold the most," or "most popular."
• Mode is sometimes called the modal value.
• It refers to the value that appears most frequently in the data.
• The mode can also be applied to numerical data.
• If no value occurs more often than the others, there is no mode.
• If two values occur most frequently, the data is bi-modal; whether this is appropriate depends on the context of the data.

Calculations with the Mean

What does calculations with the mean involve?

Because the mean has a formula, you could be asked to use this formula in reverse or in other ways:

• Mean = Total of values ÷ Number of values.
• This formula involves three quantities: the mean, the total of values, and the number of values.
• If you know any two of these quantities, you can find the third one.

What calculations with the mean might I have to do?

Typical questions may ask you to:

• Work backwards from a known mean.
• Combine means from two data sets.

As this involves problem-solving, there may be unique questions you haven't encountered before. It's essential to understand what the mean is, how it works, and what it represents.

How do I solve problems involving calculations with the mean?

• Known mean, unknown data value:
  • This involves working backwards from the mean to find an unknown data value.
  • Let the unknown data value be x.
  • Using the mean formula, set up an equation in terms of x.
  • Rearrange and solve the equation to find x.
• Combined means for two data sets:
  • If you know the mean for two different data sets and want to find the overall mean:
    • Find the total values for both data sets.
    • Divide by the total number of values across both data sets.
  • Alternatively, if you know the overall mean and need to work back to find the mean of one or both data sets, or an unknown data value.
• Others:
  • Due to the problem-solving nature of these questions, there will be some variations in question styles.
  • The above two methods should cover the majority of questions you will encounter.
  • The best way to approach questions involving the mean is to:
    • Write down the quantities you know.
    • Write down the quantities you don't know.
    • Use the mean formula to link the unknown and known values.