Course content

• Comparing two data sets using these statistics:
  • a measure of average: mean or median
  • a measure of spread: interquartile range or standard deviation

Textbook page references

• TeeJay Maths Book N5 pp.104-115
• Zeta National 5+ Maths pp.266-272
• Leckie National 5 Maths pp.346-361

What is a data set?

A data set is just a group of numbers. Examples of data sets:

• the goals scored by a team in each game through a league season
• the number of pets each pupil in your Maths class has
• the heights of a group of plants four weeks after their seeds were sown.

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Average and spread

Average is a number that gives us an idea of what value is in the middle of a data set: about halfway between the maximum and minimum. The two measures of average used at National 5 are mean and median.

Spread, or dispersion, is a number that gives us an idea of how spread out (varied) or close together (consistent) the data items are. The two measures of spread used in Nat 5 Maths are interquartile range and standard deviation.

Mean

Mean \((\overline x)\) is the sum of the data items \((\Sigma x)\) divided by how many data items there are \( (n) \).

$$ \overline x = \frac{\Sigma x}{n} $$

\(\Sigma\) (sigma) is the Greek capital S. It just means "add up all the..." so \(\Sigma x\) is the sum of all the \(x\)-values.

Example:  The mean of 9, 4, 11 and 6 is

\( \overline x =\large \frac{\Sigma x}{n} \normalsize = \large \frac{9+4+11+6}{4} \normalsize = \large \frac{30}{4} \normalsize =7.5 \)

Median

Median is the middle number in an ordered list.

Example: 15 19 12 10 16.

Put them in order: 10 12 15 16 19.
This example is easy because \(n\) is odd.
The median is 15 because it has two data items on either side.

Example:  34 26 41 29.

Put them in order: 26 29 34 41.
\(n\) is even so we need to calculate the median. It's the mean of 29 and 34.
Median = \( \large \frac{29+34}{2} \normalsize \) = 31.5

Quartiles

Example:  3 5 6 9 12 15 20.

The median is 9. It splits the data set in half:
3 5 6 9 12 15 20.

Each "half" contains three data items.

We can do the same again in each half:

• 3 5 6 in the lower half
• 12 15 20 in the upper half.

We have now "quartered" the data set. The values that separate each quarter are called quartiles.

• The lower quartile \(Q_1\) = 5
• The median \(Q_2\) = 9
• The upper quartile \(Q_3\) = 15

Sidenote: This data set has 7 data items, so they aren't really "halves" and "quarters", but it's useful to call them that so that we can understand how to find the quartiles. I'll drop the quote marks now!

Example:  1 2 4 5 6 7 7 8 9.

The median is 6, but this time there are four data items in each half. So:

• Lower quartile \(Q_1\) = \(\frac{2+4}{2}\) = 3
• Median \(Q_2\) = 6
• Upper quartile \(Q_3\) = \(\frac{7+8}{2}\) = 7.5

Example:  1 3 4 7 9 10.

Here the median needs to be calculated, but the lower and upper quartiles are easy because they are just the middle numbers in groups of three.

• Lower quartile \(Q_1\) = 3
• Median \(Q_2\) = \(\frac{4+7}{2}\) = 5.5
• Upper quartile \(Q_3\) = 9

Example:  1 1 4 5 8 8 10 11.

This is the worst type. All three quartiles need to be calculated.

• Lower quartile \(\frac{1+4}{2}\) = 2.5
• Median \(Q_2\) = \(\frac{5+8}{2}\) = 6.5
• Upper quartile \(Q_3\) = \(\frac{8+10}{2}\) = 9

Interquartile range

Interquartile range (IQR) is a measure of spread, defined as \( Q_3-Q_1 \).

Semi-interquartile range (SIQR) is another measure of spread, defined as half of the IQR:

$$ SIQR = \frac{Q_3-Q_1}{2} $$

SIQR was part of the National 5 Maths course until 2021, so it appears in some past papers. The course now only includes IQR.

Example:  20 24 26 27 30 31.

\(Q_1=24\) and \(Q_3=30\).
So \( IQR = 30-24 = 6 \).

Standard deviation

Standard deviation (\(s\)) is probably the best measure of spread, but it is trickier to calculate.

Two different ways of calculating \(s\) are given on the N5 formula list:

$$ \begin{eqnarray} s &=& \sqrt{\frac{\Sigma{\left( x-\overline x \right)^2 }}{n-1}} \\[6pt] s &=& \sqrt{\frac{\Sigma{x^2}-\frac{\left(\Sigma x \right)^2}{n}}{n-1}} \end{eqnarray} $$

If the standard deviation was zero, all the data items would be the same. The bigger the SD is, the more varied (spread out) the data items are.

Here are two examples of how to calculate standard deviation, using each of the formulae:

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Example 1 (1st formula)

Midday temperatures in °C were recorded in Glasgow over the period from 1^st to 9^th March, inclusive:

6  8  11  12  8  5  9  7  6

Calculate the standard deviation of these temperatures, correct to 2 significant figures.

Show answer

Example 2 (2nd formula)

Midday temperatures in °C were recorded in Edinburgh over the period from 1^st to 9^th March, inclusive:

7  6  10  10  9  6  8  7  7

Calculate the standard deviation of these temperatures, correct to 2 significant figures.

Show answer

Example 3 (exam-style question)

Using the answers to the previous examples, make two valid comments comparing the temperatures at midday in Glasgow and Edinburgh over the period from 1^st to 9^th March, inclusive.

Show answer

Recommended student books

Zeta Maths: National 5+ practice book 
Leckie: National 5 Maths textbook 

Example 4 (non-calculator)

SQA National 5 Maths 2015 P1 Q5

The standard deviation of 1, 2, 2, 2, 8 is equal to \(\sqrt{a}.\) Find the value of \(a.\)

Show answer

Example 5 (calculator)

SQA National 5 Maths 2021 P2 Q6

A company operates a bus route from the city centre to the airport. The number of passengers on six of its buses on a Monday was

32 27 34 29 31 33

(a) Calculate the mean and standard deviation of the number of passengers.

(b) The mean number of passengers the following Saturday was 28 and the standard deviation was 3.2. Make two valid comments comparing the number of passengers on each bus on Monday and Saturday.

Show answer

Example 6 (non-calculator)

SQA National 5 Maths 2023 P1 Q9

A magazine company conducted a survey of the ages of its readers. A sample of ten readers' ages, in years, are shown below.

33 55 38 47 36 41 42 41 35 31

(a) Calculate the median and interquartile range of the ages of readers for this sample.

A newspaper company also conducted a survey of the ages of its readers. The median age of a sample of its readers was 41 years and the interquartile range was 9 years.

(b) Make two valid comments comparing the ages of the readers of the magazine and the ages of the readers of the newspaper.

Show answer

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Maths.scot worksheet

• Data sets worksheet • Answer sheet • See all National 5 Maths worksheets

Past paper questions

• All past paper questions by topic

Median and IQR (or SIQR): • 2015 Paper 1 Q10 (SIQR) • 2017 Paper 1 Q2 (SIQR) • 2019 Paper 1 Q5 (SIQR) • 2021 Paper 1 Q5 (SIQR) • 2023 Paper 1 Q9 (IQR)

Mean and standard deviation: • 2013 Specimen Paper 2 Q8 • 2014 Paper 2 Q4 • 2015 Paper 1 Q5 (with surds) • 2016 Paper 2 Q6 • 2017 Paper 1 Q12 (with surds) • 2018 Paper 2 Q5 • 2021 Paper 2 Q6 • 2022 Paper 2 Q5

Other great resources

Videos - Maths180.com

Video - Mr Graham Maths

Video - DLB Maths SQA 2014 Paper 2 Q4 solution

Video - Mr Hamilton Online SQA 2015 Paper 1 Q5 solution

Videos - S Tarbard 1. Standard deviation: 1st formula 2. Standard deviation: 2nd formula

Notes and videos - Mistercorzi 1. Quartiles and averages 2. Standard deviation

PowerPoint - MathsRevision.com

1. Notes - BBC Bitesize 2. Test yourself - BBC Bitesize

Notes - Maths4Scotland

Notes - National5.com

Lesson notes - Maths 777 1. Median, IQR and SIQR 2. Standard deviation

Notes and examples - Maths Mutt

Quartiles notes - Maths is Fun

Practice questions - Maths Hunter

Worksheet - Calderglen HS

Essential Skills worksheet Standard deviation (Answers)

Worksheet - Starting Points Maths Word problems

Exercises - Larkhall Academy Pages 34-44 Ex 1-3

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