Statistics
What the Statistics domain tests
Level 3 Statistics assesses the full investigative cycle: posing a question, planning data collection, choosing displays and models, analysing with appropriate evidence, and drawing conclusions that acknowledge uncertainty. The external standards cluster around time series (AS91580), bivariate measurement data (AS91581), formal inference (AS91582), probability concepts (AS91585), and probability distributions (AS91586). Expect to interpret context, justify method choice, and communicate uncertainty rather than just compute.
Core rules and concepts
- Bivariate data: describe association with a scatterplot, the correlation coefficient (r), and the least-squares regression line. Check linearity, scatter, outliers, and influential points using residuals. (r) measures the strength and direction of linear association only; it is not the slope and does not prove causation.
- Time series: decompose into trend, seasonal, and irregular components. Use moving means to smooth trend; seasonal indices compare actual values with trend. Forecast only within the observed pattern and state assumptions.
- Formal inference: use sample data to make a claim about a population parameter. Bootstrap confidence intervals estimate sampling variability; the interval is about the parameter, not individual observations.
- Probability: conditional probability (P(A|B)=P(A \text{ and } B)/P(B)). Check independence by comparing (P(A|B)) with (P(A)); do not assume it from appearance.
Worked example
A random sample of 120 students finds 78 favour a later start. The sample proportion is (78/120=0.65). A bootstrap procedure gives a 95% confidence interval of [0.57, 0.73]. Reasoning: the bootstrap re-samples with replacement from the observed data to mimic sampling variability, so the middle 95% of re-sampled proportions gives an approximate interval. Interpret: we are 95% confident the population proportion is between 0.57 and 0.73. It does not mean 95% of samples fall there, nor that the true proportion has a 95% probability of being in this fixed interval.
Traps that cost marks
Saying 'proves' from correlation; extrapolating regression beyond the data range; ignoring residual plots; interpreting a confidence interval as a probability statement about the parameter; confusing (P(A|B)) with (P(B|A)); forecasting a time series without checking trend and seasonal stability.