What does the Statistics area cover in SQA National 5 Applications of Maths?

The Statistics area covers reading and drawing statistical diagrams such as tables, bar charts, pie charts, line graphs and stem-and-leaf diagrams, calculating averages (mean, median and mode) and measures of spread (range, the five-figure summary, the semi-interquartile range and standard deviation), comparing two data sets using both an average and a spread, drawing a line of best fit on a scattergraph and using its equation to estimate values, and calculating probability and expected frequency to make and justify decisions about risk.

What is the difference between the semi-interquartile range and the standard deviation?

Both measure spread. The semi-interquartile range is half the difference between the upper and lower quartiles, so it describes the spread of the middle half of the data and ignores extreme values. The standard deviation measures, on average, how far every value lies from the mean, so it uses all the data and is more sensitive. The semi-interquartile range pairs with the median, and the standard deviation pairs with the mean. A smaller value of either means more consistent data.

How do you compare two data sets in National 5 Applications of Maths?

Always quote both an average and a measure of spread, using matching measures for both sets, then state a conclusion in context. Compare the medians (or means) to say which set is typically higher, then compare the semi-interquartile ranges (or standard deviations) to say which is more consistent. A set with the same average but a smaller spread is the more reliable. Never compare a median from one set with a mean from the other.

How do you calculate probability and expected frequency?

When all outcomes are equally likely, probability is the number of favourable outcomes divided by the total number of outcomes, giving a value between 0 and 1. The probability of an event not happening is 1 minus the probability it does. Expected frequency is the probability multiplied by the number of trials, so a probability of 0.2 over 50 trials gives an expected 10 occurrences. Probability and expected frequency are then used to weigh up risk and justify a decision.

ScotlandApplications of Mathematics

SQA National 5 Applications of Mathematics Statistics: diagrams, averages, spread, standard deviation, scattergraphs and probability

A deep-dive SQA National 5 Applications of Mathematics guide to the Statistics area. Covers extracting and interpreting data from tables and statistical diagrams, the mean, median, mode and range, the five-figure summary and semi-interquartile range, standard deviation, comparing data sets, scattergraphs with a line of best fit, and probability with risk and expected frequency.

Generated by Claude Opus 4.815 min readNational 5Updated 2026-06-15

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What the Statistics area actually demands
Statistical diagrams and reading data
Averages, spread and standard deviation
Comparing data sets
Scattergraphs and probability
How the Statistics area is examined
Check your knowledge

What the Statistics area actually demands

Statistics is a major strand of Applications of Mathematics, and it rewards careful reading, accurate calculation and clear interpretation. The examiners want you to read data from diagrams, summarise it with averages and spread, compare two sets fairly, model a relationship with a line of best fit, and reason about probability and risk. This guide walks through every topic of the area, then sets out the patterns the SQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Statistical diagrams and reading data

The area opens with extracting and interpreting data from tables, bar charts, pie charts, line graphs and stem-and-leaf diagrams. The skill is checking the scale or key first, then reading values accurately. Pie charts use $360^\circ$ for the whole, so each sector angle is its fraction of the total times $360^\circ$ . Stem-and-leaf diagrams keep data ordered, ready for the median and quartiles.

Averages, spread and standard deviation

Averages are the mean, median and mode; spread is the range, the five-figure summary and the semi-interquartile range $\tfrac{Q_3 - Q_1}{2}$ . The standard deviation $s = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n - 1}}$ uses every value and is the most sensitive measure of spread. The median pairs with the semi-interquartile range, the mean with the standard deviation.

Comparing data sets

To compare two data sets, quote both an average and a measure of spread, using matching measures for each set, and conclude in context. The average shows which set is typically higher; the spread shows which is more consistent.

Scattergraphs and probability

Scattergraphs show correlation: positive (rising), negative (falling) or none. A line of best fit follows the trend, and its equation $y = mx + c$ estimates values. Probability is favourable outcomes over total outcomes, from $0$ to $1$ ; the complement is $1 - P(A)$ ; and expected frequency is probability times the number of trials, used to weigh risk and justify decisions.

How the Statistics area is examined

A typical SQA profile for Statistics:

Accurate reading. Diagrams must be read against the correct scale or key.
Full comparison. Comparing data sets needs both an average and a spread, stated in context.
Justified decisions. Probability and risk questions reward a conclusion supported by the figures.

Check your knowledge

A mix of recall and method questions covering the Statistics area. Attempt them, then check against the solutions.

In a pie chart, a category is $25\%$ of the total. Find its angle. (2 marks)
Find the median of $4, 6, 9, 11, 15$ . (1 mark)
For $Q_1 = 7$ and $Q_3 = 19$ , find the semi-interquartile range. (2 marks)
A line of best fit is $y = 3x + 2$ . Estimate $y$ when $x = 5$ . (2 marks)
A bag has $4$ red and $6$ blue counters. Find the probability of red. (2 marks)

Solutions to check-your-knowledge questions

Step 1: Find the sector angle for a category that is 25% of a pie chart

A pie chart represents the whole as $360^\circ$ , so each category's angle is its proportion of $360^\circ$ . Multiplying $0.25$ by $360^\circ$ gives the angle directly:

0.25 \times 360 = 90^\circ.

Step 2: Find the median of five ordered values

The median is the middle value once the data are arranged in order. With five values the third position is the middle, so no calculation beyond locating the centre is needed:

4,\ 6,\ \mathbf{9},\ 11,\ 15 \implies \text{median} = 9.

Step 3: Calculate the semi-interquartile range from the quartiles

The semi-interquartile range is half the difference between the upper and lower quartiles, measuring the spread of the middle half of the data:

\text{SIQR} = \dfrac{Q_3 - Q_1}{2} = \dfrac{19 - 7}{2} = 6.

Step 4: Estimate a value from the equation of a line of best fit

Substitute the given $x$ -value into the equation of the line of best fit to estimate the corresponding $y$ -value. Multiplying the coefficient by the input and adding the intercept gives the estimate:

y = 3 \times 5 + 2 = 17.

Step 5: Find the probability of drawing a red counter

Probability is the number of favourable outcomes divided by the total number of outcomes. With 4 red counters out of 10 counters in total, the probability simplifies to:

P(\text{red}) = \dfrac{4}{10} = \dfrac{2}{5}.

Sources & how we know this

SQA National 5 Applications of Mathematics Course Specification — SQA (2023)
SQA National 5 Applications of Mathematics - Course overview and resources — SQA (2023)