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VCE · VCE Units 3 & 4

General Mathematics

Data analysis, recursion, networks and matrices — exam-style practice with fully worked solutions.

20full-length model exams with mark-by-mark answer guides
20detailed note sets — ~120 pages across every topic
64exam-style practice questions with worked solutions
60flashcards for every key term & formula
10official past papers

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Sample revision note

Displaying & Describing Distributions

Classifying Data: The First Decision

Before you can choose a graph or a summary statistic, you must classify the variable. In VCE General Maths every variable is either categorical or numerical, and each of those splits again.

Categorical data places individuals into named groups. It divides into:

  • Nominal — categories with no natural order, e.g. eye colour (blue, brown, green) or method of travel to school (car, train, bike, walk).
  • Ordinal — categories that can be ranked, e.g. T-shirt size (S, M, L, XL) or a satisfaction rating (poor, fair, good, excellent).

Numerical data records quantities you can measure or count. It divides into:

  • Discrete — counted values, usually whole numbers, e.g. number of pets, goals scored, siblings.
  • Continuous — measured values on a continuous scale that can take any value in an interval, e.g. height (174.6 cm), time (12.38 s), mass.

A reliable test for discrete vs continuous: ask "can it ever be a fraction in a meaningful way?" You can have 1.7 metres of height (continuous) but not 1.7 children (discrete).

Why classification matters. The data type controls every later choice:

  • Categorical data is summarised with frequency tables and displayed with bar charts (and you report the mode — the most common category — never a mean).
  • Numerical data is displayed with dot plots, stem plots, histograms or boxplots, and summarised with the mean, median, range, IQR and standard deviation.

Worked classification. A survey records, for each student: (a) postcode, (b) number of text messages sent yesterday, (c) reaction time in seconds, (d) movie rating out of 5 stars. Classify each.

  • (a) Postcode — although it is a number, it is a label; averaging postcodes is meaningless, so it is categorical, nominal.
  • (b) Number of texts — a count, so numerical, discrete.
  • (c) Reaction time — measured, can be any decimal, so numerical, continuous.
  • (d) Star rating — ranked categories, so categorical, ordinal.

Postcode is the classic trap: a numeral printed on the data sheet is not automatically numerical data.

Dot Plots, Stem Plots & Histograms

These three displays are all for numerical data and let you read off shape, centre, spread and outliers.

Dot plots. Each data value is one dot stacked above a number line. Best for small data sets of discrete or rounded values. You can directly count to find the median and spot gaps and outliers.

Stem-and-leaf plots (stem plots). The leading digits form the stem and the final digit forms the leaf. They keep every original value (unlike a histogram) while showing shape. Always include a key, e.g. "6 | 3 means 63". Leaves should be ordered and the plot should look like a sideways bar chart.

Histograms. Data is grouped into equal-width class intervals; bar height is the frequency. Bars touch (unlike a bar chart) because the scale is continuous. Use a histogram for larger data sets.

Describing shape. Report three things in a sentence: shape, centre, spread, plus outliers.

  • Symmetric — a single peak with roughly mirror-image tails.
  • Positively skewed — the long tail points to the right (high values). Most data bunched at the low end.
  • Negatively skewed — the long tail points to the left (low values).

A memory aid: the skew direction is named after where the tail points, not where the hump is.

Worked example — build a stem plot. Resting heart rates (bpm) of 15 students: 58, 62, 71, 65, 49, 73, 68, 60, 77, 55, 63, 81, 66, 70, 59.

Stems are the tens digit (4, 5, 6, 7, 8). Sort the values into stems, then order the leaves:

  • 4 | 9
  • 5 | 5 8 9
  • 6 | 0 2 3 5 6 8
  • 7 | 0 1 3 7
  • 8 | 1

Key: 6 | 2 means 62 bpm. Reading the plot: the bulk of values sit in the 60s, there is a single peak there, and the tails are fairly even with one high value (81) and one low value (49). So the distribution is approximately symmetric, centred near the low-to-mid 60s, with no clear outliers (we confirm outliers formally in a later section). Counting to the 8th value (since n = 15, the median is the 8th) gives 65 bpm as the centre — consistent with our visual read.

Sample exam question

The Brunswick Striders is a running club with 60 members. The club records fitness and training data for every member. (a) The resting heart rates of club members are approximately normally distributed, with a mean of 66 beats per minute (bpm) and a standard deviation of 8 bpm. (i) Using the 68-95-99.7% rule, determine the percentage of members expected to have a resting heart rate between 58 bpm and 82 bpm. (1 mark) (ii) Calculate the standardised score (z-score) for a member with a resting heart rate of 54 bpm. (1 mark) (b) The weekly training distances, in kilometres, of the 60 members have the following five-number summary: minimum = 12, Q1 = 24, median = 32, Q3 = 40, maximum = 68. Use an appropriate calculation to show that the maximum value of 68 km would be shown as an outlier on a boxplot of this data. (2 marks) (c) Eight members completed a 10-km race. Their average weekly training distance, x (km), and race time, y (minutes), are shown below. weekly distance (km): 20 24 28 32 36 40 44 48 race time (minutes): 57.8 56.1 54.6 52.4 51.9 49.2 47.6 47.1 Use your CAS to determine the equation of the least squares regression line that enables race time to be predicted from weekly training distance. Round the values of the intercept and the slope to three significant figures. (2 marks) (d) Interpret the slope of this regression line in terms of the variables race time and weekly training distance. (1 mark) (e) (i) Determine the value of the correlation coefficient, r, rounded to three decimal places. (1 mark) (ii) Determine the value of the coefficient of determination, as a percentage rounded to one decimal place, and interpret it in terms of the variables. (1 mark) (f) (i) Use the least squares regression line to predict the 10-km race time of a member whose average weekly training distance is 38 km. Round your answer to one decimal place. (1 mark) (ii) The member who trained 36 km per week ran the race in 51.9 minutes. Determine the residual value for this member, rounded to two decimal places. (1 mark) (g) The club also runs group fitness classes, and quarterly attendance is seasonal. The seasonal indices are: Q1 (Jan-Mar) 1.15 | Q2 (Apr-Jun) 0.95 | Q3 (Jul-Sep) 0.78 | Q4 (Oct-Dec) unknown (i) Determine the seasonal index for Quarter 4. (1 mark) (ii) The actual attendance in Q1 2026 was 2415. Determine the deseasonalised attendance for this quarter. (1 mark) (iii) A least squares trend line fitted to the deseasonalised attendance data is: deseasonalised attendance = 1980 + 28 x quarter number, where quarter number 1 is Q1 2025. Use this trend line to forecast the ACTUAL attendance in Q3 2027. Round to the nearest whole number. (2 marks)

Show the worked answer

Answer: Worked solution

(a)(i) 1. 58 = 66 - 8 (1 SD below mean); 82 = 66 + 2x8 (2 SD above mean). 2. Percentage = 34 + 34 + 13.5 = 81.5%. (a)(ii) z = (54 - 66)/8 = -1.5. (b) 1. IQR = 40 - 24 = 16. 2. Upper fence = Q3 + 1.5 x IQR = 40 + 1.5 x 16 = 64. 3. Since 68 > 64, the maximum of 68 km is an outlier. (c) CAS linear regression gives intercept 65.657, slope -0.39911. To 3 significant figures: race time = 65.7 - 0.399 x weekly distance. (d) On average, 10-km race time decreases by 0.399 minutes (about 24 seconds) for each additional kilometre of weekly training distance. (e)(i) r = -0.993 (CAS: -0.99312; negative, matching the negative slope). (ii) r^2 = (-0.99312)^2 = 0.98629, so 98.6% of the variation in race time is explained by the variation in weekly training distance. (f)(i) time = 65.657 - 0.39911 x 38 = 50.49 -> 50.5 minutes (interpolation, since 38 is within 20-48). (ii) 1. Predicted = 65.657 - 0.39911 x 36 = 51.29. 2. Residual = actual - predicted = 51.9 - 51.29 = +0.61 minutes (the member was slower than predicted). [Accept 0.56 if the 3-sig-fig equation is used.] (g)(i) Seasonal indices sum to 4: SI(Q4) = 4 - (1.15 + 0.95 + 0.78) = 1.12. (ii) Deseasonalised = actual / SI = 2415 / 1.15 = 2100. (iii) 1. Q3 2027 is quarter number 11 (2025 = 1-4, 2026 = 5-8, 2027 = 9-12). Trend value = 1980 + 28 x 11 = 2288. 2. Actual forecast = 2288 x 0.78 = 1784.64, so approximately 1785 attendances.

All 20 practice exams

  1. Exam 1 — Data analysis: normal distribution and z-scores, boxplot outliers, least squares regression with CAS (slope/intercept interpretation, r, r^2, prediction, residual), and seasonal time series forecasting; Recursion and financial modelling: flat-rate, unit-cost and reducing-balance depreciation, reducing-balance loan via Finance Solver (repayment, recurrence, total interest), annuity drawdown and perpetuity; Matrices: matrix product and element interpretation, determinant and inverse for simultaneous equations, transition matrices including S(n+1) = T.S(n) + B and steady state, dominance ranking with D + D^2
  2. Exam 2 — Original Exam-2 written paper for VCE General Maths U3&4, retail/small-business flavour, exactly 4 questions x 15 marks = 60.; Every numerical answer independently verified in Python (z-scores, finance, matrices, project network, MST, shortest path).; Full worked sample solutions for every labelled part, with mark-by-mark criteria summing to 15 per question.
  3. Exam 3 — Q1 Data analysis: 68-95-99.7 rule with z-scores, boxplot/outlier (IQR fence), least-squares interpretation/prediction/residual, and a 4-quarter time series (seasonal index, deseasonalise, trend-line forecast, reseasonalise) - all using clean verified numbers.; Q2 Recursion & finance: arithmetic recurrence (Myki balance), reducing-balance depreciation as a geometric sequence, compound interest, a reducing-balance loan via Finance Solver (payment, balance, total interest) and a perpetuity scholarship fund.; Q3 Matrices: revenue by matrix product, 2x2 determinant/inverse to solve simultaneous equations, regular transition matrix with steady state, and the S(n+1)=T*S(n)+B recurrence with an added constant matrix.
  4. Exam 4 — Q1 Data analysis: z-scores, 68-95-99.7 rule, least-squares regression with slope/intercept interpretation, residuals, and quarterly seasonal indices (deseasonalising); Q2 Recursion & finance: flat-rate vs reducing-balance depreciation recurrence relations, reducing-balance loan via Finance Solver, perpetuity and annuity; Q3 Matrices: 2x2 determinant/inverse to solve simultaneous equations, transition matrix with state iteration and steady state
  5. Exam 5 — Data analysis: summary statistics, outliers, z-scores and the 68-95-99.7 rule, least-squares regression with coefficient of determination, and deseasonalising time series; Recursion and financial modelling: recurrence relations, compound interest, flat-rate depreciation, reducing-balance loans and perpetuities; Matrices: operations, 2x2 determinant and inverse, solving simultaneous equations, and transition matrices with steady-state behaviour
  6. Exam 6 — Data analysis: five-number summary and boxplot outlier test, 68-95-99.7 rule and z-scores, least-squares regression from summary statistics, slope interpretation, residuals, coefficient of determination, and seasonal indices with trend forecasting - set in a community nutrition study; Recursion and financial modelling: compound-interest recurrence and explicit rule, flat-rate vs reducing-balance vs unit-cost depreciation, reducing-balance loan amortisation with Finance Solver, and a retirement annuity including perpetuity - set around a health-food cafe; Matrices: order and product of matrices for revenue, determinant and inverse of a 2x2 to solve simultaneous equations, a three-state transition matrix with steady state and the S(n+1) = T.S(n) + B growth model, and dominance ranking - set in a wellness centre
  7. Exam 7 — Q1 Data analysis: normal distribution and the 68-95-99.7% rule, z-scores, five-number summary and outlier testing, least-squares regression (slope interpretation, prediction, residuals, coefficient of determination), seasonal indices and deseasonalisation; Q2 Recursion and financial modelling: unit-cost and reducing-balance depreciation, recurrence relations, compound interest investments, effective annual interest rate, reducing-balance loan amortisation with Finance Solver, perpetuities; Q3 Matrices: matrix products and order, determinant and inverse of a 2x2 matrix for simultaneous equations, transition matrices and state matrices, steady state, the rule S(n+1) = T.S(n) + B, communication matrices and two-step connections
  8. Exam 8 — Q1 Data analysis — five-number summary and outlier fences for suburb house prices, normally distributed apartment prices (68-95-99.7 rule, z-scores), least-squares regression of price vs distance from CBD (slope interpretation, prediction, residual, coefficient of determination), and a quarterly house-sales time series (seasonal indices, deseasonalising, trend-line forecasting).; Q2 Recursion and financial modelling — flat-rate depreciation of rental-property appliances, compound-interest deposit savings, a $480,000 reducing-balance home loan (recurrence relation, step-by-step amortisation, interest vs principal split, Finance Solver repayment count and adjusted final repayment), and a perpetuity funded by a property sale.; Q3 Matrices — order and matrix product for office sales x commissions, determinant/inverse of a 2x2 to solve marketing-cost simultaneous equations, a three-region rental transition matrix (interpretation, state matrices, steady state, and the S(n+1) = T.S(n) + B extension), and a dominance-matrix ranking using D + D^2.
  9. Exam 9 — Data analysis: five-number summary and 1.5 x IQR outlier fences, normal distribution (z-scores, 68-95-99.7% rule), least squares regression (slope/intercept interpretation, prediction, residuals, coefficient of determination), seasonal indices, deseasonalising and trend forecasting; Recursion and financial modelling: flat-rate and unit-cost depreciation recurrences, compound interest and effective annual rate, reducing-balance loan amortisation (balance, interest split, term via Finance Solver), perpetuities and annuity lifetimes; Matrices: matrix product modelling and order, determinant and inverse of a 2x2 for simultaneous equations, transition matrices S(n+1) = T.S(n), steady state, and the rule S(n+1) = T.S(n) + B
  10. Exam 10 — Data analysis: rainfall five-number summary and the 1.5xIQR outlier rule, normally distributed January temperatures (68-95-99.7 rule and z-scores), least-squares regression of reservoir evaporation on maximum temperature (slope, r^2, prediction, residual), and seasonal indices with deseasonalisation of quarterly rainfall; Recursion and financial modelling: flat-rate vs unit-cost depreciation of a survey drone, a monthly-compounding investment recurrence, a reducing-balance loan for a Doppler radar (Finance Solver repayment, amortisation split, mid-loan balance, total interest), and a perpetuity research fund; Matrices: order and element interpretation, a cost matrix product, determinant and inverse of a 2x2 to solve simultaneous equations, a fine/wet weather transition matrix (two-step probability and steady state), and the rule S(n+1) = T.S(n) + B for volunteer numbers
  11. Exam 11 — Q1 Data analysis: normal-distribution z-scores and the 68-95-99.7 rule on practice-exam scores (mean 62, SD 9), IQR outlier test on a five-number summary, least-squares line built from summary statistics (slope 1.89, intercept 39.32), r^2 interpretation, prediction and residual, then a termly library-visits time series (missing seasonal index, deseasonalising, trend-line forecast with re-seasonalising).; Q2 Recursion & financial modelling: flat-rate depreciation recurrence vs unit-cost vs reducing-balance on an $8000 photocopier, compound-interest recurrence and effective annual rate for a building fund, a $400,000 reducing-balance gym loan (first-row amortisation, balance after 60 repayments, 114 repayments with adjusted final repayment $3475.90) and a $150,000 scholarship perpetuity vs annuity drawdown.; Q3 Matrices: canteen revenue by matrix product with order/definedness, 2x2 determinant and inverse to find production ticket prices, a 3-state weekly lunch-choice transition matrix on 850 students with integer steady state [310, 260, 280] and the S(n+1) = T.S(n) + B rule, plus a four-house debating dominance matrix ranked by one-step + two-step scores.
  12. Exam 12 — Data analysis: normal distribution (68-95-99.7 rule), 1.5 x IQR outlier testing, least squares regression and interpretation, seasonal time series and trend forecasting; Recursion and financial modelling: flat-rate, unit-cost and reducing-balance depreciation, reducing-balance loan amortisation with Finance Solver, perpetuities and annuities; Matrices: matrix product and element interpretation, inverse-matrix solution of simultaneous equations, transition matrices with steady state and S(n+1) = T.S(n) + B, dominance ranking
  13. Exam 13 — Data analysis: normal distribution (68-95-99.7), z-scores, boxplot outlier test, least squares regression with r^2 and residuals, seasonal indices, deseasonalising and trend forecasting - applied to parcel delivery data; Recursion and financial modelling: flat-rate and unit-cost depreciation, compound interest with effective annual rate, reducing-balance loan amortisation via Finance Solver, annuity withdrawals and perpetuities - delivery fleet and warehouse finance; Matrices: order and elements, cost matrix products, 2x2 determinant/inverse solution of simultaneous equations, transition matrices with steady state and S(n+1) = T.S(n) + B, one-step and two-step dominance - dispatch operations
  14. Exam 14 — Data analysis: five-number summaries and outlier fences, the 68-95-99.7% rule and z-scores, least squares regression from summary statistics with residuals, and seasonal indices with trend-line forecasting.; Recursion and financial modelling: flat-rate vs reducing-balance depreciation, reducing-balance loan recurrences and amortisation components, Finance Solver balances and total interest, and perpetuities vs annuities.; Matrices: matrix products for revenue, determinants and inverses for solving simultaneous equations, transition matrices with state predictions and steady state, and Leslie matrix population modelling.
  15. Exam 15 — Q1 Data analysis: normal distribution and z-scores, outlier fences, least-squares regression (slope, prediction, residual, r^2), seasonal indices, deseasonalising and trend forecasting; Q2 Recursion and financial modelling: flat-rate, reducing-balance and unit-cost depreciation, reducing-balance loan recurrence and amortisation, annuity exhaustion with Finance Solver, perpetuities; Q3 Matrices: matrix equations, determinant and inverse of a 2x2, matrix products and order, transition matrices with steady state and S(n+1) = T.S(n) + B, dominance ranking with M + M^2
  16. Exam 16 — Data analysis: normal distribution and z-scores (68-95-99.7), boxplot outlier test, least-squares regression with CAS (slope/intercept interpretation, prediction, residual, r and coefficient of determination), seasonal indices, deseasonalising and trend forecasting; Recursion and financial modelling: flat-rate, unit-cost and reducing-balance depreciation as recurrence relations, reducing-balance loan amortisation (interest/principal split), Finance Solver for repayments and total interest, annuity duration and perpetuity condition; Matrices: order and matrix product with interpretation, determinant and inverse of a 2x2 to solve simultaneous equations, transition matrices (interpretation, state matrices, steady state, two-step transitions), the rule S(n+1) = T.S(n) + B, dominance matrices and ranking
  17. Exam 17 — Data analysis: normal distribution and z-scores, 1.5xIQR outlier testing, least-squares regression with slope/r^2 interpretation and residuals, seasonal indices and trend-line forecasting; Recursion and financial modelling: flat-rate vs unit-cost depreciation recurrences, compound interest, reducing-balance loan amortisation on the Finance Solver, perpetuities and annuities; Matrices: order and matrix products in context, determinant/inverse solution of simultaneous equations, transition matrices with steady state and the rule S(n+1) = T.S(n) + B, Leslie matrix population projection
  18. Exam 18 — Original Exam-2 written paper (Exam 18 of 20) for VCE General Maths U3&4, hospitality flavour: exactly 4 questions x 15 marks = 60, CAS assumed, extended-response with labelled parts.; Q1 waterfront-bistro data analysis (normal model, 1.5xIQR outlier, least-squares regression with residuals, seasonal indices); Q2 espresso-cafe finance (flat-rate and unit-cost depreciation, savings annuity, reducing-balance loan with final-payment adjustment, perpetuity).; Q3 hotel matrices (order/product revenue, determinant-inverse solve of simultaneous equations, transition matrix with steady state and S(n+1)=T.S(n)+B); Q4 resort networks (Euler trail, minimum spanning tree, shortest path, max-flow min-cut, project scheduling with float and crashing).
  19. Exam 19 — Data analysis: median, IQR and the 1.5 x IQR outlier rule, normal distribution (68-95-99.7 rule and z-scores), least squares regression with slope/intercept interpretation, prediction, residuals and coefficient of determination, plus seasonal indices, deseasonalising and trend-line forecasting for quarterly construction data.; Recursion and financial modelling: flat-rate, unit-cost and reducing-balance depreciation of construction plant, recurrence relations, reducing-balance loan amortisation with Finance Solver (number of repayments, final adjusted repayment, total interest), and an annuity drawdown including the perpetuity condition.; Matrices: matrix order and multiplication for site materials costing, determinant and inverse of a 2 x 2 matrix to solve equipment-hire equations, transition matrices with state matrices, steady state, the rule S(n+1) = T x S(n) + R for a workforce with recruitment, and one- and two-step dominance ranking of tendering firms.
  20. Exam 20 — Data analysis: normal distribution and z-scores, boxplot outliers, least-squares regression (interpretation, prediction, residuals, r^2), seasonal indices and trend forecasting; Recursion and financial modelling: flat-rate vs reducing-balance depreciation, compound interest recurrence, effective interest rate, reducing-balance loan amortisation with final payment, perpetuities; Matrices: matrix products in context, determinant and inverse for simultaneous equations, transition matrices with steady state and the rule S(n+1) = T.S(n) + B, dominance ranking

All 20 revision notes

  • Displaying & Describing Distributions
  • Investigating Associations & Regression
  • Data Transformation & Time Series
  • Data Analysis: CAS Skills & Exam Technique
  • Interpreting & Comparing Distributions in Context
  • Recurrence Relations & Sequences
  • Interest & Depreciation
  • Loans, Annuities & Perpetuities
  • Finance Solver Mastery & Common Errors
  • Modelling Growth & Decay: Choosing the Right Model
  • Matrix Basics & Operations
  • Transition & Leslie Matrices
  • Communication & Dominance Matrices
  • Matrix Applications: Worked Exam-Style Problems
  • Matrices: CAS Skills & Exam Technique
  • Graphs, Networks & Trees
  • Shortest Path, Flow & Matching
  • Project Scheduling & Critical Path
  • Networks: Worked Exam-Style Problems
  • Scheduling & Critical Path: Step-by-Step Method