Back to Statistics and Probability

Higher Applications of Mathematics

Scatter graphs and correlation

Describing relationships between variables.

Statistics referenceStatistics mixed quiz

Statistics reference

Check probability, diagrams, averages, regression, intervals and tests.

Mixed revision quiz

Review Statistics and Probability with adaptive feedback.

Before you start

  • Be confident reading values from tables and graphs.
  • Check units, sample size and what each variable represents.
  • Use context in written answers, especially when interpreting results.

Method chooser

Which statistics method do I use?

Are there stages or repeated chance events?

Use a tree diagram.

Do categories overlap?

Use a Venn diagram.

Do you need a typical value?

Calculate an average.

Do you need to compare consistency?

Compare spread using standard deviation or IQR.

Are you comparing distributions?

Use box plots or histograms.

Are you checking a relationship?

Use scatter graph and correlation.

Do you need a prediction line?

Use linear regression.

Are you estimating a population value?

Use a confidence interval.

Are you judging evidence against a claim?

Use a hypothesis test.

Are the same people measured twice?

Use a paired t-test.

Are you comparing two proportions?

Use a z-test for two proportions.

Statistics lesson

Key idea

  • This topic focuses on describing relationships between two numerical variables without overclaiming causation. In Higher Applications, the aim is to use statistical methods to make careful decisions from real data.
  • Good statistical work has three parts: choose a suitable method, carry it out accurately, then explain what the result means in the situation.
  • When writing conclusions, use cautious language such as 'this suggests' or 'there is evidence to suggest'. Data can support a conclusion, but it rarely proves it completely.

Key formulae, definitions and methods

  • Positive correlation means both variables tend to increase together.
  • Negative correlation means one variable tends to decrease as the other increases.
  • Correlation does not prove causation.

Technology output practice

Interpreting statistical output

Read the simulated output, pick out the key value, then turn it into a written conclusion. This is a learning preview, not a real RStudio environment.

Context

Correlation output

A pupil investigates the relationship between weekly revision hours and assessment score.

Simulated output

> cor(study$hours, study$score)
[1] 0.82

r

0.82

A strong positive correlation.

Direction

Positive

Higher revision hours tended to go with higher scores.

Limit

Association

Correlation alone does not prove that one variable caused the other.

What it means

The value is close to 1, so the association is strong and positive. The context still matters: other factors may also affect score.

What to write

There is a strong positive relationship between weekly revision hours and assessment score. Pupils who revised for longer tended to score higher, but this does not prove revision time was the only cause.

Weak answer: The correlation is 0.82.

Watch out

Many pupils stop after copying r. Add direction, strength and context, and avoid claiming proof of cause.

Which conclusion is best?

Choose an option, then check the feedback.

Worked examples

Worked example 1

Choose the method

A sports club records training hours and 5 km run times for members.

  1. Plot each pair of values accurately.
  2. Look for direction, strength and unusual points.
  3. Describe the pattern in context.

A scatter graph shows whether a relationship is visible before using calculations.

Worked example 2

Carry out and interpret

A sports club records training hours and 5 km run times for members.

  1. Estimate whether the relationship is positive, negative or none.
  2. Comment on strength using the closeness of points.
  3. Avoid saying one variable definitely causes the other.

The interpretation should say what the relationship means in context.

Worked example 3

Check the conclusion

A sports club records training hours and 5 km run times for members.

  1. Use the graph to make a cautious prediction inside the data range.
  2. State that extrapolation outside the range is less reliable.
  3. Include units in the prediction.

Predictions from scatter graphs should be cautious and within the data range.

Watch out

  • Choosing a method because it is familiar rather than because it matches the data.
  • Giving a numerical answer without explaining what it means in context.
  • Mixing up sample evidence with certainty about the whole population.
  • Ignoring outliers, skewness, units or the scale on a graph.
  • Using causal language when the data only shows association.

Technology connection

Related RStudio and Spreadsheet topics

RStudio: Scatter plots and correlation in RStudioSpreadsheet: Scatter graphs and trendlines

Next step

Move into practice

Use the learning notes to choose suitable summaries and conclusions, then try varied data sets, tables, p-values and interpretation prompts.

Statistics mixed quiz