WACE Mathematics Applications bivariate data and time series deep dive: the 2026 exam

WACEMathematics ApplicationsStudy guide16 min readUpdated 23 Sep 2026

Revision deep dive for the data content of WACE Mathematics Applications: scatterplots, correlation and r squared, least-squares lines, residuals, association versus causation, time series components, moving averages, seasonal indices and forecasting, with worked exam-style examples and links to every related dot point.

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Why this content matters
1. Scatterplots and describing association
2. Correlation and the coefficient of determination
3. The least-squares line
4. Residuals and residual plots
5. Association is not causation
6. Time series: plots and components
7. Smoothing
8. Seasonal indices
9. Trend lines and forecasting
Common mistakes
Check your knowledge

Why this content matters

Bivariate data and time series together make up roughly a third of the WACE Mathematics Applications exam, according to the content ranges in the SCSA design brief. It is also the part of the paper where students most often lose "easy" marks, because the calculation is done by the calculator and the marks are for interpretation. This deep dive links every related dot point on the site. For exam format and timing see the exam strategy guide.

1. Scatterplots and describing association

Dot points: scatterplots and bivariate association, bivariate data and regression.

The explanatory variable (independent) goes on the horizontal axis; the response variable goes on the vertical axis. If a question says "predict the time from the distance", distance is explanatory.
Describe an association with direction, form, strength and outliers.

Describing an association

"There is a strong, negative, linear association between the age of a car and its resale value, with one possible outlier (a 12-year-old car resold for an unusually high price)."

2. Correlation and the coefficient of determination

Dot point: correlation coefficient and coefficient of determination.

Pearson's $r$ measures the strength and direction of a linear association, from $-1$ to $1$ . A common scale: $|r| \ge 0.75$ strong, $0.5 \le |r| < 0.75$ moderate, $0.25 \le |r| < 0.5$ weak. Use the scale your teacher has given you and quote the value.

Interpreting

r^2

For data on daily maximum temperature ( $x$ , in degrees) and cold drink sales ( $y$ ), $r = 0.82$ .

$r^2 = 0.82^2 \approx 0.672$ . So about 67 percent of the variation in cold drink sales can be explained by the variation in daily maximum temperature. The other 33 percent is due to other factors.

Notice the wording: the variation in the response variable is explained by the explanatory variable.

$r$ only describes linear association and is sensitive to outliers. A strong curved relationship can have a small $r$ .

3. The least-squares line

Dot point: least-squares line interpretation.

Interpret, predict and judge reliability

For 20 students, the least-squares line relating weekly practice hours ( $h$ , from 2 to 20 hours) and test mark ( $M$ , out of 100) is $M = 38.5 + 2.4h$ , with $r = 0.86$ .

Slope: Each additional hour of practice per week is associated with an increase of 2.4 marks, on average.
Intercept: A student who does no practice would be predicted to score 38.5. This is an extrapolation (0 is outside 2 to 20), so treat it with caution.
Interpolation: For $h = 12$ : $M = 38.5 + 2.4(12) = 67.3$ . This is within the data range and $r$ is strong, so the prediction is reasonably reliable.
Extrapolation: For $h = 35$ : $M = 38.5 + 84 = 122.5$ , which is impossible for a test out of 100. The model should not be used this far outside the data.

4. Residuals and residual plots

Dot point: residuals and residual plots.

\text{residual} = \text{actual value} - \text{predicted value}

A student who practised 12 hours and scored 60 has residual $60 - 67.3 = -7.3$ : the line over-predicted their mark by 7.3.

A residual plot graphs residuals against the explanatory variable. Random scatter around zero supports a linear model. A clear curve (such as a U shape) means the relationship is non-linear, so the linear model is not appropriate even if $r$ is high.

5. Association is not causation

Dot point: association, causation and the statistical investigation process.

A strong association does not prove that changes in one variable cause changes in the other. Possible non-causal explanations:

Confounding variable: ice cream sales and drownings both rise in hot weather. Temperature drives both.
Coincidence: two unrelated quantities that both happen to trend upward over the same years.
Reverse direction: the association is real, but the cause runs the other way.

In an exam, name a plausible confounding variable in context, and explain how it could affect both variables.

6. Time series: plots and components

Dot points: time series plots and components, time series and forecasting.

A time series plot joins consecutive points in time order. Describe it using its components:

Trend: long-term increase or decrease.
Seasonality: a pattern that repeats over a fixed period of one year or less (quarters, months).
Cycles: longer, irregular rises and falls (such as economic cycles).
Irregular (random) variation: unpredictable fluctuation, including one-off outliers.

7. Smoothing

Dot point: smoothing with moving averages and medians.

Smoothing removes some irregular and seasonal variation so the trend is easier to see. With an even number of points, the average sits between two time periods, so it must be centred.

Centred four-point moving average

Quarterly visitor numbers (thousands) are: Q1 20, Q2 28, Q3 35, Q4 17, next Q1 24.

The first four-point average is $\dfrac{20 + 28 + 35 + 17}{4} = 25$ , which sits between Q2 and Q3.

The next is $\dfrac{28 + 35 + 17 + 24}{4} = 26$ , which sits between Q3 and Q4.

The centred value for Q3 is $\dfrac{25 + 26}{2} = 25.5$ thousand.

8. Seasonal indices

Dot point: seasonal indices and deseasonalising.

Seasonal indices

\text{seasonal index} = \frac{\text{value for the season}}{\text{average of the seasons in that cycle}} \qquad \text{deseasonalised} = \frac{\text{actual}}{\text{seasonal index}}

The seasonal indices across one cycle sum to the number of seasons (4 for quarters, 12 for months).

Seasonal indices from one year

Quarterly sales: Q1 40, Q2 60, Q3 80, Q4 20. The mean is 50, so the indices are $0.8$ , $1.2$ , $1.6$ and $0.4$ (sum 4).

Interpretation of Q3: Q3 sales are typically 60 percent above the quarterly average.

Deseasonalise a later Q3 value of 96: $\dfrac{96}{1.6} = 60$ .

9. Trend lines and forecasting

Dot point: trend lines and forecasting.

The standard sequence for seasonal data is: deseasonalise, fit a trend line to the deseasonalised data, forecast, then reseasonalise.

Forecast and reseasonalise

A least-squares line fitted to deseasonalised quarterly sales is $y = 120 + 5t$ , where $t = 1$ is Q1 of the first year. Forecast actual sales for $t = 13$ (a Q1), given the Q1 seasonal index is $0.8$ .

Deseasonalised forecast: $120 + 5(13) = 185$ . Reseasonalise: $185 \times 0.8 = 148$ .

Comment on reliability: the forecast extrapolates beyond the data, so it assumes the trend and seasonal pattern continue.

Common mistakes

Where data marks go missing

Swapping the explanatory and response variables, which changes the least-squares line.
Writing " $r^2$ shows 67 percent of the data fits the line". It is the percentage of variation in the response explained.
Dividing when you should multiply: deseasonalise by dividing by the index; reseasonalise by multiplying.
Forgetting to centre an even-order moving average.
Claiming causation from correlation.

Check your knowledge

$r = -0.6$ . Describe the strength and direction, and find $r^2$ as a percentage. (Answer: moderate negative; 36 percent.)
A monthly seasonal index is 0.9 and the deseasonalised value is 500. What is the actual value? (Answer: 450.)
A residual plot shows a clear U shape. What does this tell you? (Answer: the relationship is non-linear, so a linear model is not appropriate.)

Then try the data and time series quiz.

Sources & how we know this

Mathematics Applications ATAR course Year 12 syllabus — School Curriculum and Standards Authority (SCSA)
Mathematics Applications past ATAR course examinations — School Curriculum and Standards Authority (SCSA)
Mathematics Applications ATAR course: syllabus and support materials — SCSA

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