Unit 3: Bivariate data and time series analysis, sequences and Earth geometry

12 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

Topic 1: Bivariate data analysis 1 - how do we describe and measure the relationship between two numerical variables?

Identify the explanatory and response variables, construct and use a scatterplot to describe an association between two numerical variables in terms of direction, form and strength, calculate Pearson's correlation coefficient r and the coefficient of determination R^2 from raw data using technology and interpret them, and use r to determine R^2 and vice versa

The QCE General Mathematics 2025 syllabus sub-topic on associations between two numerical variables. Explanatory and response variables, scatterplots, describing direction, form and strength, Pearson's r and the coefficient of determination R^2 from a calculator, converting between r and R^2, outliers and non-linear form, with fully worked examples.

14 min read →

Topic 2: Bivariate data analysis 2 - how do we model a linear relationship and use it to make predictions?

Fit a least-squares line y = mx + c to bivariate numerical data using technology and using m = r(s_y / s_x) and c = y-bar - m x-bar, interpret the slope and y-intercept in context, distinguish between interpolation and extrapolation, use the line to make predictions, and recognise and explain the potential dangers of extrapolation

The QCE General Mathematics 2025 syllabus sub-topic on fitting a linear model. The least-squares line y = mx + c from a calculator's statistics mode and from m = r(s_y/s_x) and c = y-bar - m x-bar, interpreting slope and intercept in context, interpolation versus extrapolation, predictions and the dangers of extrapolation, with fully worked examples.

15 min read →

Topic 2: Bivariate data analysis 2 - how do we use residuals and a residual plot to judge whether a straight line is the right model?

Calculate residuals for a least-squares line, construct a residual plot, and use it to assess the appropriateness of fitting a linear model to the data

The QCE General Mathematics 2025 syllabus skill of residual analysis. Residual equals actual minus predicted, what its sign means, constructing a residual plot against the explanatory variable, reading random scatter versus a curved pattern, why a high R squared is not enough, and working backwards from residuals, with a blood pressure question modelled on QCAA's style.

15 min read →

Topic 3: Time series analysis - how do we describe patterns over time and smooth out short-term fluctuation?

Construct and describe time series plots (trend, seasonality and irregular fluctuations), smooth a series with a simple moving mean or median for an odd number of data, deseasonalise using seasonal indices found by the average percentage method, and fit a least-squares line to model the long-term trend

QCE General Mathematics Unit 3 Topic 3 (Time series analysis): constructing and describing time series plots by trend, seasonality and irregular fluctuations, smoothing with odd-numbered moving means and medians, choosing the window, and an overview of seasonal indices, deseasonalising and least-squares trend forecasts, with script-checked worked examples.

15 min read →

Topic 4: Growth and decay in sequences - how do we model the loss in value of an asset over time with arithmetic and geometric sequences?

Use arithmetic sequences to model straight-line depreciation and geometric sequences to model diminishing-value depreciation, with recursion, tables, graphs and the nth-term rules, and solve practical problems involving the value of an asset over time

The QCE General Mathematics 2025 syllabus applications of sequences to depreciation. Straight-line depreciation as an arithmetic sequence, diminishing-value depreciation as a geometric sequence, recursion, tables and graphs, the rules t(n) = t(1) + (n - 1)d and t(n) = t(1) r^(n - 1), the term-number trap, finding when a value falls below a threshold, and comparing the two methods.

15 min read →

Topic 4: Growth and decay in sequences - how do we model quantities that change by a constant amount or a constant ratio?

Use recursion to generate arithmetic and geometric sequences, display their terms in tabular and graphical form, use the rules t(n) = t(1) + (n - 1)d and t(n) = t(1)r^(n - 1), and model and analyse practical situations involving linear and geometric growth and decay, such as simple interest, taxi fares and population growth (use of logarithms not required)

The QCE General Mathematics 2025 syllabus topic on growth and decay in sequences. Recursion, tables and graphs, the arithmetic rule t(n) = t(1) + (n - 1)d and the geometric rule t(n) = t(1)r^(n - 1), simple interest, taxi fares, population and bacterial growth, decay, trial and improvement without logarithms, and comparing linear with geometric growth.

15 min read →

Topic 5: Earth geometry and time zones - how do we locate points on the Earth and measure distances along its surface?

Understand great circles and angles of latitude and longitude, locate and state positions on Earth's surface, and calculate angular distance and distance between two places on the same meridian (D = 111.2 x angular distance) or the same parallel of latitude (D = 111.2 cos θ x angular distance)

QCE General Mathematics Unit 3 Topic 5 (Earth geometry and time zones): great and small circles, latitude and longitude in decimal degrees and degrees and minutes, and the syllabus formulas D = 111.2 x angular distance (same meridian or the equator) and D = 111.2 cos θ x angular distance (same parallel), with script-checked worked examples.

15 min read →

Topic 5: Earth geometry and time zones - how do we relate longitude to local time and work out time differences between places?

Understand the meaning of Greenwich Mean Time (GMT), the International Date Line and Coordinated Universal Time (UTC), understand the link between longitude and time, determine the number of degrees of longitude for a given time difference, calculate time differences between two places on Earth, and solve practical problems involving time zones, making allowances for daylight saving where necessary

The QCE General Mathematics 2025 syllabus sub-topic on time zones. GMT, UTC and the International Date Line, 15 degrees of longitude per hour and 4 minutes per degree, time differences from longitudes and from UTC offsets, daylight saving, flights, phone calls and broadcasts across zones and the Date Line, with fully worked examples.

14 min read →

Why does an observed association between two variables not necessarily mean that one causes the other, and how do we identify and communicate non-causal explanations when solving practical problems?

Recognise and explain that an observed association between two variables does not necessarily mean there is a causal relationship, identify and communicate non-causal explanations including coincidence or the influence of another variable, and solve practical problems involving associations

The QCE General Mathematics 2025 syllabus sub-topic on association and causation. Why association does not imply causation for categorical and numerical data, the influence of another variable, coincidence, reversed direction, how to communicate these in exam answers, and a full practical investigation using two-way tables, r, R squared and the least-squares line.

14 min read →

How do we use a two-way frequency table and percentages to decide whether two categorical variables are associated?

Understand bivariate data, construct two-way frequency tables with row and column sums and percentages, and use an appropriately percentaged table to identify and describe an association between two categorical variables in context

The QCE General Mathematics 2025 syllabus sub-topic on two categorical variables. What bivariate data is, building a two-way frequency table with row and column sums, choosing row or column percentages from the explanatory variable, deciding whether an association exists, and describing it systematically in context, with original exam-style questions.

14 min read →

How do we calculate seasonal indices with the average percentage method, deseasonalise a time series, fit a least-squares trend line and use it to forecast?

Deseasonalise a time series by calculating seasonal indices using the average percentage method, fit a least-squares line to model the long-term trend, and solve practical problems that involve forecasting with time series data

The QCE General Mathematics 2025 syllabus skills for analysing seasonal time series. Seasonal indices by the average percentage method, what an index means, deseasonalising and reseasonalising, fitting a least-squares trend line to deseasonalised data with time as the explanatory variable, forecasting, and the limits of forecasts, with full worked examples.

16 min read →

Extension (not in the 2025 syllabus) - how do we straighten a curved relationship so a least-squares line can be fitted?

Apply a square, logarithmic or reciprocal transformation to one variable to linearise a non-linear association, fit a least-squares line to the transformed data, use the transformed equation to predict, and choose the transformation that best straightens the scatter

Extension material beyond the QCAA General Mathematics 2025 syllabus: when a residual plot shows a curve, straighten the data with a square, log or reciprocal transformation, fit a least-squares line to the transformed data, compare residual plots and r squared, and predict by substituting and back-transforming, with script-checked worked examples.

15 min read →