National 5 Maths
Statistics: Data Sets

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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 numbers

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

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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.

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.

To use the first formula, we need to calculate the mean.

$$ \begin{eqnarray} \overline x &=& \frac{\Sigma x}{n} \\[6pt] &=& \frac{6+8+11+12+8+5+9+7+6}{9} \\[6pt] &=& \frac{72}{9} \\[6pt] &=& 8 \end{eqnarray} $$

Then we make a table with these headings:

$$ \begin{array} {|c|c|} \hline x & x-\overline x & (x-\overline x)^2 \\ \hline \sf\small 6 & \sf\small 6-8=-2 & \sf\small (-2)^2=4 \\ \hline \sf\small 8 & \sf\small 8-8=0 & \sf\small 0^2=0 \\ \hline \sf\small 11 & \sf\small 11-8=3 & \sf\small 3^2=9 \\ \hline \sf\small 12 & \sf\small 12-8=4 & \sf\small 4^2=16 \\ \hline \sf\small 8 & \sf\small 8-8=0 & \sf\small 0^2=0 \\ \hline \sf\small 5 & \sf\small 5-8=-3 & \sf\small (-3)^2=9 \\ \hline \sf\small 9 & \sf\small 9-8=1 & \sf\small 1^2=1 \\ \hline \sf\small 7 & \sf\small 7-8=-1 & \sf\small (-1)^2=1 \\ \hline \sf\small 6 & \sf\small 6-8=-2 & \sf\small (-2)^2=4 \\ \hline & \Sigma{\left( x-\overline x \right) }=0 & \Sigma{\left( x-\overline x \right)^2 }= 44 \\ \hline \end{array} $$

$ \Sigma{\left( x-\overline x \right)} $ isn't used in the formula but it should always be zero, so it's a good idea to check before squaring each entry in the middle column. If it doesn't add to zero, go and find your mistake!

We are finally ready to finish this off:

$$ \begin{eqnarray} s &=& \sqrt{\frac{\Sigma{\left( x-\overline x \right)^2 }}{n-1}} \\[6pt] &=& \sqrt{\frac{44}{9-1}} \\[6pt] &=& \sqrt{\frac{44}{8}} \\[6pt] &=& 2.34521... \\[6pt] &\approx& 2.3^\circ C \end{eqnarray} $$

Video solution by Clelland Maths

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.

Example 3 (exam-style)

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.

First we'll need to quickly calculate the mean for Edinburgh:

$ \overline x = \large \frac{\Sigma x}{n} \normalsize = \large \frac{70}{9} \normalsize \approx 7.8^\circ C $

It can be a good idea to put the means and standard deviations into a comparison table:

	Mean	S.D.
Glasgow	8.0°C	2.3°C
Edinburgh	7.8°C	1.6°C

Comparing the means, we can write any of the following:

On average, Glasgow was warmer.
In general, Glasgow was warmer.
On average, Glasgow had higher temperatures.
In general, Glasgow had higher temperatures.
On average, Edinburgh was colder.
In general, Edinburgh was colder.
On average, Edinburgh had lower temperatures.
In general, Edinburgh had lower temperatures.

Comparing the standard deviations, we can write any of the following:

Glasgow's temperatures were more varied.
Edinburgh's temperatures were less varied.
Glasgow's temperatures were less consistent.
Edinburgh's temperatures were more consistent.
The spread of temperatures was more in Glasgow.
The spread of temperatures was less in Edinburgh.

Video solution by Clelland Maths

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 $a.$

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.

(a) First, let's find the mean:

$$ \begin{eqnarray} \overline x &=& \frac{\Sigma x}{n} \\[6pt] &=& \frac{32+27+34+29+31+33}{6} \\[6pt] &=& \frac{186}{6} \\[6pt] &=& 31 \end{eqnarray} $$

We could use either formula for standard deviation here, but as we've already found the mean, let's choose the first version.

$$ \begin{array} {|c|c|} \hline x & x-\overline x & (x-\overline x)^2 \\ \hline \sf\small 32 & \sf\small 32-31=1 & \sf\small 1^2=1 \\ \hline \sf\small 27 & \sf\small 27-31=-4 & \sf\small (-4)^2=16 \\ \hline \sf\small 34 & \sf\small 34-31=3 & \sf\small 3^2=9 \\ \hline \sf\small 29 & \sf\small 29-31=-2 & \sf\small (-2)^2=4 \\ \hline \sf\small 31 & \sf\small 31-31=0 & \sf\small 0^2=0 \\ \hline \sf\small 33 & \sf\small 33-31=2 & \sf\small 2^2=4 \\ \hline & \Sigma{\left( x-\overline x \right) }=0 & \Sigma{\left( x-\overline x \right)^2 }= 34 \\ \hline \end{array} $$

$$ \begin{eqnarray} s &=& \sqrt{\frac{\Sigma{\left( x-\overline x \right)^2 }}{n-1}} \\[6pt] &=& \sqrt{\frac{34}{6-1}} \\[6pt] &=& \sqrt{\frac{34}{5}} \\[6pt] &=& 2.6076... \\[6pt] &\approx& 2.6 \end{eqnarray} $$

(b) It can be a good idea to put the means and standard deviations into a comparison table:

	Mean	S.D.
Monday	31	2.6
Saturday	28	3.2

On average, the Monday buses had more passengers, as 31 > 28.
The number of passengers on each bus varied more on Saturday, as 3.2 > 2.6.

Video solution by Clelland Maths

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.

Example 7 (non-calculator)

SQA National 5 Maths 2024 P1 Q5

The prices, in pounds (£), of the camera on display in a shop are listed below.

155 160 190 210 230 240

(a) Calculate the median and interquartile range of these prices.

On a website, a sample of camera prices have a median of £195 and an interquartile range of £73.

(b) Make two valid comments comparing the prices of the cameras in the shop and on the website.

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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)
• 2024 Paper 1 Q5 (IQR)
• 2025 Paper 1 Q3 (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
• 2025 Paper 2 Q4

Standard Grade: Credit (1986–2013)
• Exam questions and answers

Buy our favourite N5 textbook

Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press

Statistics worksheets

Maths.scot worksheet
• Comparing data sets (Answers)

Maths Hunter worksheet
• Practice questions (no answers)

Essential Skills worksheet
• Standard deviation (Answers)

CJ Maths worksheets
1. Quartiles & IQR (no answers)
2. Standard deviation (no answers)
3. Comparing statistics (no answers)

MyMathsGuy.com worksheet
• Comparing data (no answers)

National5Maths.co.uk worksheets
1. Standard deviation (no answers)
2. SD exam-style (no solutions)

Starting Points Maths worksheet
• SD word problems (no answers)

Larkhall Academy worksheet
• Pages 34-44 Ex 1-3 (no answers)

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Notes and videos

Videos - Maths180.com

Video - Mr Graham Maths

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

Notes and examples - Maths Mutt

Quartiles notes - Maths is Fun

Click here to study the standard deviation notes on National5.com.

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⇦ Nat 5 topic list ⇧ Top of this page

National 5 MathsStatistics: Data Sets

Page sections

Topic content

Textbook page numbers

Buy our favourite textbook

What is a data set?

Average and spread

Mean

Median

Quartiles

Interquartile range

Standard deviation

Looking for an N5 Maths tutor?

Example 1 (1st formula)

Example 2 (2nd formula)

Example 3 (exam-style)

Example 4 (non-calculator)

Example 5 (calculator)

Example 6 (non-calculator)

Example 7 (non-calculator)

Buy N5 Maths practice papers

Past paper questions

Buy our favourite N5 textbook

Statistics worksheets

Buy N5 Maths revision guides

Notes and videos

National 5 Maths
Statistics: Data Sets