Averages and range Revision Notes for Edexcel GCSE Maths

Averages and range

What are averages and range?

There are three different types of average: the mean, median, and mode. Each average tells us something different about a data set. The range tells us how spread out the data is by showing the difference between the highest and lowest values.

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Understanding when to use each type of average is crucial for data analysis. The mean gives the mathematical centre, the median shows the positional centre, and the mode reveals the most common value.

Mean

Definition

The mean is what most people call the "average". It is calculated by adding all values together and dividing by the number of values.

How to calculate the mean

Add all the values together
Divide by the number of values
Don't round your answer unless the question asks you to

chatImportant

Always check if the question asks for rounding. Many students lose marks by rounding when they shouldn't, or by not rounding when they should.

Formula

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

Alternative formula: $\text{Sum of values} = \text{Mean} \times \text{Number of values}$

Example

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Worked Example: Calculating the Mean

For the numbers: 4, 5, 9, 7, 4, 4

Step 1: Add them up $4 + 5 + 9 + 7 + 4 + 4 = 33$

Step 2: Divide by the count $\text{Mean} = \frac{33}{6} = 5.5$

Therefore, the mean is 5.5

Median

Definition

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

How to find the median

Write the values in order from smallest to largest
Find the middle value
If there are two middle values, the median is halfway between them (add them and divide by 2)

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The median is particularly useful when you have extreme values (outliers) in your data, as it isn't affected by these extreme values like the mean is.

Example

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Worked Example: Finding the Median

For the numbers: 4, 5, 9, 7, 4, 4

Step 1: Put in order 4, 4, 4, 5, 7, 9

Step 2: Find the middle There are 6 numbers, so the middle is between the 3rd and 4th values

Step 3: Calculate the median The middle values are 4 and 5 $\text{Median} = \frac{4 + 5}{2} = 4.5$

Mode

Definition

The mode is the value that appears most often in a data set.

How to find the mode

Count how many times each value appears
The value that appears most frequently is the mode
There can be more than one mode if several values appear equally often
There might be no mode if all values appear the same number of times

chatImportant

A common mistake is thinking there's always a mode. If all values appear the same number of times, there is no mode. Also remember that you need at least two identical values to have a mode.

Example

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Worked Example: Finding the Mode

For the numbers: 4, 5, 9, 7, 4, 4

Step 1: Count the frequencies

4 appears 3 times
5 appears 1 time
9 appears 1 time
7 appears 1 time

Step 2: Identify the most frequent The value 4 appears most often (3 times)

Therefore, the mode is 4

Range

Definition

The range shows how spread out the data is. It is the difference between the largest and smallest values.

How to calculate the range

$\text{Range} = \text{Largest value} - \text{Smallest value}$

Example

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Worked Example: Calculating the Range

For the numbers: 4, 5, 9, 7, 4, 4

Step 1: Identify the extreme values

Largest value = 9
Smallest value = 4

Step 2: Calculate the difference $\text{Range} = 9 - 4 = 5$

Exam tips

Here are essential strategies for success in exams involving averages and range:

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Critical Exam Reminders:

Make sure you know the difference between the mean, median, and mode
For the median, always put values in order first
When calculating the mean, don't round unless asked
The mode is the most common value - if you have cards or data, at least two items must have the same value
Use the formula $\text{Sum of values} = \text{Mean} \times \text{Number of values}$ to find missing values

Problem solving with missing values

If you know the mean and some values, you can find missing values using the relationship between sum, mean, and count.

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Worked Example: Finding Missing Values

Three cards show 5, 5, and an unknown number. The mean is 4.

Step 1: Use the formula $\text{Sum of values} = \text{Mean} \times \text{Number of values}$

Step 2: Calculate the total sum $\text{Sum} = 4 \times 3 = 12$

Step 3: Set up the equation $5 + 5 + ? = 12$

Step 4: Solve for the unknown $10 + ? = 12$ $? = 2$

Therefore, the unknown number is 2.

Summary

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Key Points to Remember:

Mean: Add all values and divide by the count - gives the mathematical average
Median: The middle value when data is arranged in order - not affected by extreme values
Mode: The most frequently occurring value - useful for finding the most common item
Range: Largest value minus smallest value - shows how spread out the data is
Always order your data before finding the median, and check your arithmetic when calculating the mean

Averages and range (Edexcel GCSE Maths): Revision Notes