Deciding which average to use Revision Notes for Edexcel GCSE Statistics

Deciding which average to use

When working with data in statistics, you have three main types of average to choose from: the mode, median, and mean. Each has its own strengths and weaknesses, making them suitable for different situations. Understanding when to use each average is crucial for accurately representing your data.

The three types of average

Mode

The mode is the most frequently occurring value in a dataset. It's the simplest average to identify once you've organised your data.

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Advantages of the mode:

Very straightforward to find and will always be an actual data value from your set
Works with any type of data, including non-numerical categories (like favourite colours or car models)
Extreme values or outliers won't affect it at all

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Disadvantages of the mode:

Sometimes there isn't a mode (when all values appear equally often)
You can't use it to work out measures of spread like range or standard deviation

Median

The median is the middle value when all data points are arranged in order from smallest to largest.

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Advantages of the median:

Extreme values don't throw it off, making it reliable when you have outliers in your data
Very useful for calculating quartiles, which help you understand the interquartile range and skew of your data

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Disadvantages of the median:

The median might not be one of your actual data values, especially with an even number of data points

Mean

The mean is what most people think of as the "average" - you add up all the values and divide by how many values you have.

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Advantages of the mean:

It uses every single piece of data in your calculation, so nothing gets ignored
Essential for calculating standard deviation and skew, which are important statistical measures

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Disadvantages of the mean:

Gets heavily influenced by extreme values or outliers, which can make it misleading
Usually isn't one of your actual data values

Working with the median

To find the median position in a dataset, use this formula:

$\text{Median position} = \frac{n + 1}{2}$

Where $n$ is the total number of data values.

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Worked Example: Finding Median Position

Let's see this in action with a stem-and-leaf diagram example:

If you have 15 data points, the median position would be: $\frac{15 + 1}{2} = \frac{16}{2} = \text{8th value}$

So you'd count to the 8th value in your ordered list to find the median.

When you add more data points, the median position changes. For example, if two more data points (38 and 40) are added to make 17 total values: $\frac{17 + 1}{2} = \frac{18}{2} = \text{9th value}$

The median position shifts to the 9th value, and the actual median value increases accordingly.

Working with the mean

The mean has a very useful property: you can multiply it by the number of data values to find the total of all the data.

$\text{Total of all data values} = \text{Mean} \times \text{Number of data values}$

This relationship works in reverse too - if you know the total and the number of values, you can find the mean.

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Worked Example: Using Mean to Calculate Totals

Here's a practical example: If Jack has taken 10 tests with a mean score of 17.5 out of 20, what's his total score so far?

Step 1: Calculate current total $\text{Total so far} = 10 \times 17.5 = 175 \text{ marks}$

Step 2: Calculate required total for desired mean Now, if Jack needs a mean of 18 after his 11th test, what total will he need? $\text{Desired total} = 11 \times 18 = 198 \text{ marks}$

Step 3: Find the required score Since he currently has 175 marks, he needs: $198 - 175 = 23 \text{ marks on his next test}$

Conclusion: Since each test is only marked out of 20, this is impossible - Jack can't achieve his target mean of 18.

Choosing the right average

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The key to choosing the right average depends on your data and what you want to show:

Use the mode when you want to know the most common value, especially with categorical data or when extreme values shouldn't influence your answer.
Use the median when your data has outliers or extreme values that would skew the mean, or when you need to calculate quartiles.
Use the mean when you want to use all your data and when extreme values are genuinely part of the pattern you're studying, or when you need to calculate measures like standard deviation.

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Key Points to Remember:

The mode is the most frequently occurring value and works with any type of data
The median is the middle value and isn't affected by extreme values
The mean uses all the data but can be misleading when there are outliers
You can multiply the mean by the number of values to find the total
Choose your average based on your data type and what you want to demonstrate

Deciding which average to use (Edexcel GCSE Statistics): Revision Notes