# Average and Spread

## Topic Overview

In order to draw conclusions from any sets of mathematical data, you must compare the different values within these sets. Within the topic of '**Average and Spread**', you will be introduced to various different measures which can be used to compare and evaluate sets of mathematical data:

In the field of mathematics, the term '**average**' can have multiple meanings. The average of a set of data is the value which is an accurate overall representative of that set of data. The term average can refer either to the mean, the median or the mode of a set of data. However, it is most commonly used to refer to the mean of a set of data.
In the field of mathematics, the term '**spread**' refers to the way in which a set of mathematical data lies. It is measured in a variety of ways including the range, the inter-quartile range and the standard deviation of a set of data.

The '**mean**' is the term used to describe the average of a set of data. It provides a central value for a set of numbers. In order to calculate the mean, you must add all of the values included within your data set and then divide by how many values are within your data set.

The '**median**' is the term used to describe the middle number within a data set. In order to calculate the median, place all of the values within your data set in numerical order. The number which lies exactly in the middle of this data set is the median.

The '**mode**' is the term used to describe the value which appears most often within a particular data set.

The '**range**' is the term used to describe the difference between the lowest and highest values within a data set. To calculate the range, subtract the highest value in your data set from the lowest value.

Within a data set, there exist '**quartiles**'. These quartiles are the values which divide your data set values into quarters. As a result, the '**inter-quartile range**' is the term used to describe the difference between the first and third quartile of a set of data.

In order to calculate the inter-quartile range, you must place your data set values in numerical order and divide your data set into four equal parts. The values at these points are your quartiles. Once you have identified these values, simply subtract the third quartile from the first quartile in order to calculate the value of the inter-quartile range.

An '**outlier**' is the term used to describe a value within your data set which is significantly smaller or larger than the other values.

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems related to the concepts of average and spread. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Calculate the median, mean, and mode of various data sets
- Calculate the range, quartiles and inter-quartile range of various data sets

Listed below are a series of summaries and worked examples to help you solidify your knowledge about the concepts of average and spread.

## Worked Examples

**1 - Calculating the mean**

During your GCSE maths exam, you will be required to calculate the mean (common measure of average) from a set of values:

*Example*

(a) - Calculate the mean from the following data set:

52, 60, 53, 52, 51, 54, 55, 56, 57, 58

** Solution**
(a) - In order to calculate the mean from this data set, you must add all of the values and then divide by the total number of values:

52 + 60 + 58 + 52 + 51+ 54+ 55 + 57 + 56 + 53 = 548

There are 10 values in the data set, so you must divide 548 by 10 = 54.8

Therefore, the mean for this set of values = 54.8

**2 - Calculating the median**

During your GCSE maths exam, you will also be required to calculate the median (the middle number) of a set of values:

*Example*

(a) - Calculate the median of the following data set:

7, 14, 9, 8, 6, 12

*Solution*

(a) - In order to calculate the median, you must first place all of your values in numerical order:

6, 7, 8, 9, 12, 14

If you have an odd number of total values, the median is the middle number.

Alternatively, if you have an even number of values, the median will be the mean of the two values in the centre of your data set.

In this example, you have an even number of values, and as a result you must calculate the mean of the two central values:

6, 7, 8, 9, 12, 14

8 + 9 = 17

17/2 = 8.5

Therefore, the median of this set of values is 8.5

*Example*

(b) - Calculate the median of the following data set:

1.5, 2.5, 1.5, 4.5, 3.5, 5, 2, 3, 0.5

*Solution*

(b) - Similarly to the previous example, you must place these values in numerical order:

0.5, 1.5, 1.5, 2, 2.5, 3, 3.5, 4.5, 5

In this example, you have an odd number of values; therefore the median is the middle value:

0.5, 1.5, 1.5, 2, 2.5, 3, 3.5, 4.5, 5

Therefore, the median for this set of values is 2.5

**3 - Calculating the mode**

During your GCSE maths exam, you will also be required to calculate the mode (the most common number) of a set of values:

*Example*

(a) - Identify the mode of the following data set:

5, 7, 3, 9, 6, 8, 5, 8, 6, 7, 5, 8, 6, 4, 8

*Solution*

(a) - In order to identify the mode, it is helpful to list your values in numerical order:

3, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9

By doing so, it is easier to count how many times each number appears. For example, you can see that '8' has appeared 4 times, whereas all of the other values appear 3 times or less:

3, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9

Therefore, you can identify that 8 is the mode for this set of values.

**4 - Calculating the range**

During your GCSE maths exam, you will also be required to calculate the range (the difference between the highest and lowest numbers) of a set of values:

*Example*

(a) - Calculate the range of the following data set:

5, 9, 8, 7, 3, 6, 12, 10

*Solution*

(a) - To calculate the range of these values, you must subtract the highest value from the lowest value. In order to do so, it is helpful if you place your values in numerical order:

3, 5, 6, 7, 8, 9, 10, 12

From this numerical order, it is evident that the lowest value is 3 and the highest value is 12.

Therefore, the range of this set of values = 12 - 3 = 9

**5 - Calculating the inter-quartile range**

As explained in the Topic Overview, a set of values can be divided into quartiles:

The **first (lower) quartile** is the value which divides the lower half of the data into two halves.

The **median** is the **second quartile** because it divides the entire data into two halves.

The **third (upper) quartile** is the value which divides the upper half of the data into two halves.

There are three simple rules which you can follow to identify the median, lower and upper quartiles of a set of values:

**Lower quartile is the [(n + 1) ÷ 4] th value**

**Median is [(n + 1) ÷ 2] th value**

**Upper quartile is the 3 [(n + 1) ÷ 4] th value,**

where 'n' refers to the total number of values within the data set.

In order to calculate the inter-quartile range, you must subtract the upper quartile from the lower quartile.

*Example*

(a) (i) - Identify the median, lower and upper quartiles for the following data set:

10, 12, 4, 8, 3, 6, 11, 4, 8, 7, 9

(ii) - Calculate the inter-quartile range for this data set.

*Solution*

(a) (i) - In order to identify the median, lower and upper quartiles, it is helpful to place your values into numerical order:

3, 4, 4, 6, 7, 8, 8, 9, 10, 11, 12

Once you have done so, you can use the rules mentioned above in order to easily identify the median, lower and upper quartiles:

Lower quartile is the [(n + 1) ÷ 4] th value,

Median is [(n + 1) ÷ 2] th value,

Upper quartile is the 3 [(n + 1) ÷ 4] th value.

There are 11 values in this data set, therefore n = 11.

As such:

**The lower quartile = (11+1) ÷ 4 = 3rd value**

**The median = (11+1) ÷ 2 = 6th value**

**The upper quartile = 3(11 +1) ÷ 4 = 9th value**

By looking at your ordered data set, you can easily identify these values:

3, 4, 4, 6, 7, 8, 8, 9, 10, 11, 12

Therefore, the lower quartile = 4, the median = 8, and the upper quartile = 10

(ii) Now that you know the lower and upper quartiles, you can easily calculate the inter-quartile range:

Inter-quartile range = Upper Quartile - Lower Quartile = 10 - 4 = 6

Therefore, the inter-quartile range for this data set = 6

**6 - Identifying and rationalising outliers**

During your GCSE maths exam, you may be required to identify and evaluate any outliers within a data set. As mentioned in the Topic Overview, 'outliers' are values which lie outside the range of the other values.

For example, out of the following data set:

31, 36, 34, 33, 120, 30, 35, 32

120 is an outlier because it is significantly larger than all of the other values.

These outliers can significantly impact the average and spread of various data sets. For example, if you calculate the mean of this data set including the outlier of 120, you are presented with a value of 43.875.

However, if you omit the outlier, you are presented with a more accurate value of 33.

During your GCSE maths exam, you may be faced with outliers such as this. In these circumstances, you will be required to make a judgement as to whether to include or omit these outliers. As long as you display your working out and provide a logical reason for removing these outliers, you will earn maximum marks during your exam.

*Example*

(a) - Calculate the mean for the following data set:

4, 5, 6, 8, 9, 63, 2, 3, 7

*Solution*

(a) - In this example, the value of 63 is an outlier because it is significantly larger than the other values.

If you calculate the mean including this outlier, you are presented with a value of 11.89

If you calculate the mean omitting this outlier, you are presented with a value of 5.5

Therefore, you can state that the mean for the data set is 5.5. However, you must note that you have omitted the outlier of 63 because it is significantly larger than the other values and will skew the average and spread of the data if it is included in your calculations.

## Exam Tips

- Remember that, in order to calculate the mean, you must add all of the values included within your data set and then divide by how many values are within your data set.
- In order to calculate the median, place all of the values within your data set in numerical order. The number which is exactly in the middle of this data set is the median.
- The mode is the term used to describe the value which appears most often within a particular data set.
- In order to calculate the range of a data set, subtract the highest value in your data set from the lowest value.
- Memorise the following three rules in order to identify the median, lower and upper quartiles of a set of values: Lower quartile is the [(n + 1) ÷ 4] th value, Median is [(n + 1) ÷ 2] th value, Upper quartile is the 3 [(n + 1) ÷ 4] th value.
- In order to calculate the inter-quartile range, you must subtract the upper quartile from the lower quartile.
- When faced with outlier values, make sure that you provide a clear and logical reason for omitting or including these extreme values.
- Write down ALL of your working out when calculating the average and spread of various data sets.

## Topic Summary

If you have a comprehensive understanding of the average and spread mathematical terms which are mentioned throughout this revision guide, you can easily identify and evaluate the properties of various data sets. It is important to memorise the difference between the following terms and the different ways in which you can calculate their respective values:

- Mean
- Mode
- Median
- Range
- Lower and Upper Quartiles
- Inter-quartile range
- Outliers

Before your GCSE maths exam, attempt as many past paper questions as possible to familiarise yourself with the different properties of these average and spread terms. With practice, you will be able to confidently recognise, identify and calculate both basic and complex average and spread examination questions.

## Related Topics

- Probability Diagrams
- Sampling
- Charts and Tables
- Conditional Probability