Selecting and interpreting statistical diagrams, comparing data sets using measures of centre and spread, identifying outliers and misleading graphs, and choosing an appropriate sampling method.

What this dot point is asking

The SQA wants you to choose the right diagram for a data set, read and compare distributions using centre and spread, spot outliers and misleading graphs, recognise the type of data, and pick a sampling method that avoids bias. This is the descriptive-statistics foundation that the project and the inferential topics build on.

Choosing a statistical diagram

Different diagrams suit different data. A bar chart compares categories with gaps between bars. A histogram shows the distribution of grouped continuous data, with bars touching and area representing frequency. A box plot summarises a distribution with the five-number summary (minimum, lower quartile, median, upper quartile, maximum) and is ideal for comparing two or more groups side by side. A scatter plot shows the relationship between two variables.

Comparing data sets

When comparing two distributions, the SQA expects one statement about centre and one about spread, both in context.

• Centre: the median (middle value, resistant to outliers) or the mean (average, affected by outliers).
• Spread: the interquartile range $\text{IQR} = Q_3 - Q_1$ (the middle $50\%$, resistant to outliers) or the standard deviation (typical distance from the mean).

A smaller spread means more consistent data. A higher centre means larger values on average. Phrasing matters: say "class A scored higher on average" and "class A's marks were more consistent", not just the bare numbers.

Outliers

An outlier is a value far from the rest of the data. The standard rule uses the interquartile range.

Compute the two boundaries and check whether the value lies outside them. Outliers may be genuine extreme cases or data-entry errors, and they pull the mean and standard deviation but barely move the median and IQR, which is why those resistant measures are often preferred.

Data types, misleading graphs and sampling

Data types

Data is quantitative (numerical, either discrete counts or continuous measurements) or qualitative (categories). The type guides which diagram and which average suit it.

Misleading graphs

Be alert to a vertical axis that does not start at zero (exaggerating differences), unequal class widths in a histogram, or a truncated or distorted scale. Recognising these is examinable.

Sampling

When you cannot survey a whole population you take a sample, and the method must avoid bias so the sample represents the population.

Try this

Q1. A data set has $Q_1 = 22$ and $Q_3 = 34$. Find the IQR and the upper outlier boundary. [2 marks]

• Cue. $\text{IQR} = 12$; upper boundary $= 34 + 1.5 \times 12 = 52$.

Q2. Two teams' scores have medians $18$ and $24$ and IQRs $9$ and $4$. Compare them in context. [2 marks]

• Cue. The second team scores higher on average (median $24$) and is more consistent (IQR $4$).

Q3. A school wants a sample reflecting its year groups, which differ in size. Name a suitable sampling method and why. [2 marks]

• Cue. Stratified sampling, because it samples each year group in proportion to its size, representing all groups fairly.