Back to Statistics and Probability

Higher Applications of Mathematics

Probability, tree diagrams and Venn diagrams

Representing outcomes and combined events.

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 representing combined events, overlapping groups and expected frequencies. 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

  • Probability = favourable outcomes / total outcomes.
  • For independent events, multiply along branches of a tree diagram.
  • For overlapping sets, use the intersection once in the centre of a Venn diagram.

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

Summary statistics output

A class compares journey times to a sports venue, measured in minutes.

Simulated output

> summary(travel$minutes)
Min.   1st Qu.   Median   Mean   3rd Qu.   Max.
18.0     26.0      30.5    31.4     36.0    48.0

> sd(travel$minutes)
[1] 6.8
Mean31.4
Median30.5
Standarddeviation 6.8

Mean

31.4 min

The average journey time in the sample.

Median

30.5 min

Half the journeys were shorter than this and half were longer.

Standard deviation

6.8 min

A typical spread from the mean; smaller would mean more consistent times.

What it means

The typical journey took just over 30 minutes. The standard deviation shows there was some variation, so one journey time should not be treated as exact for everyone.

What to write

The mean journey time was 31.4 minutes and the median was 30.5 minutes, so a typical journey was about 31 minutes. The standard deviation of 6.8 minutes shows the journey times varied by several minutes.

Weak answer: The standard deviation is 6.8, so the average is 6.8.

Watch out

Remember that standard deviation is not the average. It describes spread, not centre.

Which value would you quote to describe consistency?

Choose an option, then check the feedback.

Worked examples

Worked example 1

Choose the method

A school canteen survey asks pupils whether they buy soup, a sandwich or both.

  1. Place the number who chose both in the overlap first.
  2. Fill the soup-only and sandwich-only regions.
  3. Add the regions to check the total number surveyed.

The diagram shows each pupil once, so totals and probabilities can be checked.

Worked example 2

Carry out and interpret

A school canteen survey asks pupils whether they buy soup, a sandwich or both.

  1. Convert each region to a probability by dividing by the total.
  2. Use the Venn diagram to find the required event.
  3. State the answer as a fraction, decimal or percentage.

The probability statement must match the region or route being used.

Worked example 3

Check the conclusion

A school canteen survey asks pupils whether they buy soup, a sandwich or both.

  1. For a two-stage choice, draw the first set of branches.
  2. Add the second-stage probabilities to each branch.
  3. Multiply along the route for the combined event.

Tree diagrams make combined-event probabilities clearer than a list of outcomes.

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.

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