STAT5002 Introduction to Statistics
The Complete Exam Bible · S2 2026STAT5002 Overview
- Postgraduate unit
- 6 credit points
- Semester 2, 2026
- Taught in R
- Assumed knowledge: HSC Mathematics
STAT5002 Introduction to Statistics is a postgraduate unit at the University of Sydney that introduces the basic concepts and methods of statistical data analysis, with every computation done in R. It assumes HSC Mathematics and has no prerequisites, so it suits students arriving from many disciplines.
- Assessed by A 60% final exam, a 20% Week 8 quiz, a 10% assignment and 10% weekly quizzes
- Organising idea The box model gives the standard error behind every interval and test
- Hardest step Matching the right standard error to a sum, average, proportion or difference
- How to prepare Write each test as hypotheses, assumptions, statistic, p-value, conclusion
How STAT5002 is assessed
| Component | Weight | Format |
|---|---|---|
| Final Exam | 60% | Written exam in the formal exam period: multiple choice questions and extended answer questions, 2 hours, AI prohibited |
| Quiz | 20% | In-person, closed-book supervised multiple choice quiz in the Week 8 workshops, 60 minutes, AI prohibited |
| Assignment | 10% | Individual data analysis, 5 to 10 pages, online submission, due in Week 12, AI allowed |
| Weekly Online quizzes | 10% | Weekly revision quizzes, except Weeks 1, 8 and 13; online, AI allowed |
What STAT5002 covers
The chapters follow the order of the unit's lecture notes and weekly schedule. Weeks 1 to 4 describe data: graphical and numerical summaries, the normal curve, correlation and the regression line. Weeks 5 and 6 build the probability machinery, the box model and the Central Limit Theorem. Weeks 6 to 9 use that machinery for inference on proportions and means and for chi-squared tests.
The outline then moves to multiple regression, model selection and logistic regression in Weeks 10 to 12.
Graphical Summaries and Initial Data Analysis
Variable types, barplots, histograms on the density scale, boxplots and logical operators in R · (Week 1)Before any formal analysis, a statistician takes a first broad look over the data, called initial data analysis.
- Four parts of a first look: background, structure, wrangling and summaries
- Variables, subjects and the dimension of a data frame
- Qualitative variables: ordinal versus nominal categories
- Quantitative variables: discrete counts versus continuous measurements
- Checking storage with str() and converting with as.factor() and as.numeric()
- Barplots from table(), and stacked or side by side bars for two variables
Numerical Summaries of Centre and Spread
Mean, median, standard deviation, standard units, quartiles, IQR and boxplot thresholds · (Week 2)Graphs show the shape of a variable; numerical summaries compress it into a few numbers that are easy to report and compare. Two families of summary are built here and used throughout the unit: a measure of centre and a matching measure of spread.
- What a numerical summary keeps and what it throws away
- Sample mean, deviations and the balancing point
- Sample median for odd and even sample sizes
- Robustness: how one extreme value moves the mean and the median
- Reading skewness from the mean and median
- Why the average deviation is always zero
The Normal Curve and Areas Under It
Standardising, areas and quantiles with pnorm and qnorm, the 68%, 95%, 99.7% rule · (Week 3)This chapter turns a bell-shaped histogram into a model you can calculate with. The normal curve is fixed by two numbers, the mean μ for its centre and the SD σ for its spread, and the course writes it as N(μ, σ²), where the second entry is the variance.
- Smoothing a density-scale histogram into a normal curve
- General normal curve N(mean, variance) and the standard curve N(0, 1)
- The normal curve formula and what mu and sigma control
- Area under the curve as a proportion of the data
- Lower tail, upper tail and interval areas with pnorm
- Standard units and rescaling to the standard curve
Scatter Plots and the Correlation Coefficient
Correlation as an average of products in standard units, invariance and its warnings · (Week 3)This chapter is about pairs of measurements taken on the same individuals, such as height and weight, or hours of practice and a test score.
- Bivariate data and the scatter plot
- Five summaries: two means, two SDs and r
- Correlation as the average product of standard units
- Quadrants at the point of averages and the sign of r
- Interpreting the size and sign of r
- Shift, scale and swap invariance of r
The Regression Line, Residuals and R Squared
Slope and intercept from summaries, prediction, residuals, R squared and residual plots · (Week 4)Once you can measure how tightly a scatter plot clusters with the correlation coefficient, the next step is to summarise the cloud with a line you can actually predict from.
- Five summaries that define a scatter plot
- Slope b = r SDy/SDx and intercept a = ȳ − b x̄
- Why the SD line overshoots at the extremes
- Fitting the line in R with lm(y ~ x)
- Prediction from the line versus the baseline mean
- Predicting x from y means refitting, not rearranging
Probability Rules and Simulation
Complements, mutually exclusive versus independent events, conditional probability, simulation · (Week 5)Probability is the language every later topic in the unit is written in, from the box model to p-values, so this chapter makes the basic rules automatic.
- Frequentist definition of probability
- Sample space, events, union, intersection and complement
- Complement rule for at least one questions
- Conditional probability as a share of the conditioning event
- Multiplication rule, general and independent forms
- Addition rule, general and mutually exclusive forms
The Box Model: Expected Value and Standard Error
Draws from a box, expected value and standard error of sums and averages, the 0-1 box · (Week 5)Think of the box model as the simplest way to reason about randomness in an introductory statistics unit: imagine tickets in a box, each carrying a number, and a sample of tickets drawn at random. That picture turns two familiar summaries into new tools.
- Box model: tickets, draws and random samples
- Distribution of a single draw with unequal chances
- Expected value E(X) as the mean of the box
- Standard error SE(X) as the SD of the box
- Chance error and the error box
- Computing formula: mean square minus squared mean
The Law of Averages and the Central Limit Theorem
Chance error, why the percentage error shrinks, normal approximation for sums and averages · (Week 6)Once you can find the expected value and standard error of a sum or an average, the next question is what happens as the number of draws grows. Two results answer it.
- Kerrich's coin experiment as a box model
- The 0 1 box with mean and SD of one half
- Chance error of a count grows like the square root of n
- Chance error of a proportion shrinks like one over root n
- Law of averages and the law of large numbers
- Shape of the box of all possible averages
Unknown Proportions: Intervals and Z-tests
Prediction and confidence intervals for a proportion, the HATPC framework, one-sided tests · (Weeks 6 and 7)Most yes/no questions in statistics come down to an unknown proportion: the share of voters who back a policy, of parcels that arrive damaged, of days with rain.
- The 0-1 box: mean p and SD √(p(1−p))
- Standard error of the sample proportion
- Prediction intervals when p is known
- Why p(1−p) makes intervals widest near 0.5
- Wilson confidence interval by inverting prediction intervals
- The binom.confint() function and its conf.level argument
Unknown Means: T-statistic, T-test and Bootstrap
Z-test for a mean, estimating the standard error, Student's t, t.test() and bootstrap intervals · (Week 8)Once the question is about an average rather than a proportion, a new problem appears: the SD of the box is usually unknown as well. This chapter walks through three ways to test a hypothesis about an unknown mean μ and to build an interval for it.
- Z-test for a mean when the box SD is known
- Confidence intervals for a mean at 95%, 98% and 99%
- Population SD versus sample SD in R
- The T-statistic and its fatter tails
- Student's t distribution and n − 1 degrees of freedom
- The R functions dt, pt and qt as analogues of dnorm, pnorm and qnorm
Two-Sample and Paired T-tests
Two-box model, pooled and Welch tests, simulation, paired differences · (Week 8)Comparing two means is where most real studies end up: a new method against an old one, one group against another, before against after.
- Two-box model for two independent samples
- Expected value and SE of a difference of sample means
- Z-statistic when both box SDs are known
- Equal variances assumption and the pooled SD estimate
- Classical two-sample T-test on m + n - 2 degrees of freedom
- Confidence interval for a difference of means with qt()
Chi-squared Tests of Goodness of Fit and Independence
Pearson's statistic, degrees of freedom, expected counts, simulation and two-way tables · (Week 9)Chi-squared tests answer questions about counts: does a set of categories occur in the proportions someone claims, and are two categorical variables related? The chapter starts with goodness of fit.
- Box model for a categorical variable
- Hypotheses for goodness of fit: H0 p = p0 against not H0
- Expected frequencies E = n p0
- Pearson's chi-squared statistic
- Chi-squared distribution as a sum of squared standard normals
- Upper-tail p-values with pchisq() and chisq.test()
The unit opens with initial data analysis: classifying variables, then summarising them with barplots, histograms, boxplots, means, medians and standard deviations. It then builds the normal curve and uses standard units to find areas and percentiles, before describing relationships between two variables with the correlation coefficient and the least squares regression line, judged with residuals and R squared.
The middle of the semester turns to chance. Probability rules lead to the box model, in which the expected value and standard error of a sum or an average are calculated, and the Central Limit Theorem explains why those sums and averages follow the normal curve. Everything after that is inference.
For an unknown proportion you build prediction and confidence intervals and run Z-tests using the HATPC framework of hypotheses, assumptions, test statistic, p-value and conclusion. For unknown means you move to the T-statistic, Student's t, one-sample, two-sample and paired T-tests, and bootstrap simulation. Chi-squared tests of goodness of fit and independence close the inference block.
The unit outline continues with multiple linear regression, model selection, and logistic and non-parametric regression in Weeks 10 to 12. Assessment is a 60% final exam of multiple choice and extended answer questions, a 20% supervised quiz in Week 8, an individual data-analysis assignment worth 10% and weekly online revision quizzes worth 10%.
This guide works through the first nine weeks of content in twelve chapters, each with original worked examples, figures and practice questions.
Worked example · free
Predicting a commute time from distance with the regression line
- +1Slope: b = r × SD(time) ÷ SD(distance) = 0.8 × 9.0 ÷ 5.0 = 1.44 minutes per km.
- +1Intercept: the line passes through the point of averages, so a = 31.0 − 1.44 × 12.4 = 13.144. The line is time = 13.14 + 1.44 × distance.
- +1Prediction at 20 km: 13.144 + 1.44 × 20 = 41.94 minutes, about 41.9 minutes.
- +1Residual = observed − predicted = 47 − 41.94 = 5.06 minutes. It is positive, so the point sits above the line: this student is slower than the line predicts.
- +1Coefficient of determination: r² = 0.8² = 0.64, so the line explains 64% of the variation in commute time; the other 36% is left in the residuals.
Key terms
- Standard units
- A value's distance from the mean measured in standard deviations, found by subtracting the mean and dividing by the SD. Standard units let you compare values on different scales and read areas from the standard normal curve.
- Correlation coefficient
- A number between minus one and one measuring how tightly a scatter plot clusters around a straight line. It is unchanged when either variable is shifted by a constant or rescaled by a positive constant, and it says nothing about curved patterns or causation.
- Regression line
- The least squares line for predicting one variable from another, with slope r times the ratio of SDs and passing through the point of averages.
- Residual
- The observed value minus the value predicted by the regression line. Residuals average to zero, and their pattern in a residual plot shows whether a straight line is appropriate.
- Box model
- A model of a chance process as random draws, with replacement, of tickets from a box. The mean and SD of the box give the expected value and standard error of any sum or average of the draws.
- Expected value
- The long-run average result of a chance quantity. For the sum of n draws from a box it equals n times the mean of the box.
- Standard error
- The typical size of the chance error in a chance quantity, playing the role that an SD plays for a list. For the sum of n draws it equals the square root of n times the SD of the box.
- Central Limit Theorem
- The result that the sum or average of a large number of draws from a box follows the normal curve approximately, whatever the shape of the box.
- P-value
- The chance, computed assuming the null hypothesis is true, of a test statistic at least as extreme as the one observed. Small values are evidence against the null hypothesis.
STAT5002 FAQ
Is this introductory statistics unit hard if my maths is rusty?
The assumed knowledge is HSC Mathematics and there are no prerequisites, so the algebra is light: averages, square roots and reading a formula. The real load is conceptual. You must decide which standard error belongs to which quantity and write a conclusion in context.
Students who rework each week's exercises in R before the next lecture usually keep pace; the inference weeks build directly on the box model, so falling behind in Weeks 5 and 6 is costly.
How is the overall mark made up?
The Semester 2, 2026 unit outline lists four components: a Final Exam worth 60%, an in-person Quiz in Week 8 worth 20%, an individual data-analysis Assignment worth 10% and Weekly Online quizzes worth 10%. The two supervised tasks prohibit AI and together carry 80 of the 100 marks, so exam-style practice without notes matters most.
What does the final exam look like?
The outline describes a written exam in the formal exam period with multiple choice questions and extended answer questions, two hours long, with AI prohibited. It is compulsory: not attempting it results in an AF grade. The exam date is set in the University's exam timetable, so confirm it there and check Canvas for any permitted materials.
How do the weekly online revision quizzes count towards the grade?
Together they are worth 10%. The outline says there are ten equally weighted quizzes, with the best eight counting, and that special consideration should not be requested for them. The Week 5 quiz page on Canvas calls that quiz worth 1% of the total. Because the two descriptions differ, check Canvas for how your 10% is calculated.
How much R do I need to know for tests?
The lecture notes run every analysis in R, and the weekly exercise sheets show how functions such as pnorm and qnorm can appear in tests and exams, often as a choice between candidate lines of code. Learn what each function returns, an area or a quantile, and practise reading t.test and chisq.test output rather than memorising syntax.
Does this guide cover every week of the unit?
It covers the content of Weeks 1 to 9: graphical and numerical summaries, the normal curve, correlation and regression, probability and box models, the Central Limit Theorem, Z-tests, T-tests and chi-squared tests. The unit outline schedules multiple linear regression, model selection, and logistic and non-parametric regression for Weeks 10 to 12; study those from the Canvas lecture materials.
Can I use AI tools while studying for this unit?
For study, yes, and the outline allows AI for the individual assignment and the weekly online quizzes, provided you acknowledge it. It prohibits AI in the Week 8 Quiz and the Final Exam. Sia can explain a standard error step by step or check your HATPC write-up, but practise the supervised tasks without help.
Questions students ask in each STAT5002 chapter
- Why is the tallest bar of a histogram not always the interval with the most data?
- Height on the density scale is the proportion divided by the interval width. A narrow interval with a moderate share can stand taller than a wide interval holding more subjects, so the count lives in the area, not the height. Chapter 01 · Graphical Summaries and Initial Data Analysis
- Why does R's sd() give a different answer from my calculator?
- R's sd() divides the sum of squared deviations by n minus 1, the sample version. A calculator set to the population version divides by n. Multiply the R value by the square root of (n minus 1)/n to convert. Chapter 02 · Numerical Summaries of Centre and Spread
- What is the difference between pnorm and qnorm in R?
- pnorm takes a value on the data scale and returns the proportion of the curve below it. qnorm goes the other way: you give it a proportion between 0 and 1 and it returns the value with that share below. If the question gives a percentage and asks for a measurement, you need qnorm. Chapter 03 · The Normal Curve and Areas Under It
- Does a correlation of 0 mean there is no relationship between x and y?
- No. A value of 0 only rules out a linear association. Points that follow a symmetric arch or U shape can have r equal to zero while y is completely determined by x, which is why the scatter plot must always come first. Chapter 04 · Scatter Plots and the Correlation Coefficient
- How do I get the regression slope from the correlation and the two standard deviations?
- Multiply r by the SD of the response and divide by the SD of the predictor: b = r × SDy/SDx. Then the intercept is the mean of y minus b times the mean of x. Use the same SD version, sample or population, for both, since only their ratio matters. Chapter 05 · The Regression Line, Residuals and R Squared
- What is the difference between mutually exclusive and independent events?
- Mutually exclusive events cannot both happen, so their joint probability is zero. Independent events can happen together, and knowing one occurred leaves the chance of the other unchanged. Two exclusive events that each have positive probability are therefore always dependent. Chapter 06 · Probability Rules and Simulation
- What is the difference between standard deviation and standard error?
- The SD describes the spread of the tickets in a box or of a list of data. The standard error describes the likely size of the chance error in a random quantity such as a draw, a sum or an average. For one draw they are numerically equal, but they answer different questions. Chapter 07 · The Box Model: Expected Value and Standard Error
- Does the law of averages mean the number of heads catches up with half the tosses?
- Not at all. The standard error of the number of heads is σ√n, which grows with more tosses, so the gap between heads and half the tosses tends to widen. Only the proportion of heads settles, because its standard error σ/√n shrinks. Chapter 08 · The Law of Averages and the Central Limit Theorem
- Why is the Wilson interval not centred on the sample proportion?
- Because the standard error depends on p itself, the edges come from solving a quadratic rather than adding and subtracting a fixed amount. The result is pulled towards 0.5, so its midpoint lies between the sample proportion and one half. Chapter 09 · Unknown Proportions: Intervals and Z-tests
- When should I use a T-test instead of a Z-test for a mean?
- Use the Z-test only when the box SD is genuinely known before the data are collected. If you estimate the SD from the same sample with sd(), the extra randomness in the denominator means you need Student's t with n − 1 degrees of freedom. Chapter 10 · Unknown Means: T-statistic, T-test and Bootstrap
- When should I use var.equal = T in t.test()?
- Only when you are willing to assume the two boxes have the same SD, for example when the sample SDs are similar and the boxplots show comparable spread. Otherwise leave it out and R runs the Welch test, which is the safer default. Chapter 11 · Two-Sample and Paired T-tests
- Why is the chi-squared test always one-sided?
- Pearson's statistic grows with any departure from the null proportions, in either direction for any category. A value near zero means the counts sit close to the claim, which is agreement, so only the upper tail is evidence against the null. Chapter 12 · Chi-squared Tests of Goodness of Fit and Independence
How to study for the exam
Build one page of formulas that grows each week and is organised by the box model rather than by chapter: the mean and SD of the box, then the expected value and standard error of a sum, an average, a proportion and a difference. Most inference questions in this unit are that page plus a decision.
For every test, write the five HATPC lines in order, hypotheses, assumptions, test statistic, p-value, conclusion, even when the question only asks for a number. Practise the R functions as pairs that undo each other: pnorm turns a cut-off into an area, qnorm turns an area into a cut-off, and the same holds for pt and qt.
Rework the weekly exercise sheets and the extra multiple choice sets without notes, because both supervised tasks prohibit AI and the Week 8 Quiz is closed-book. Keep a list of the direction words that cost marks: above or below the line, left or right tail, reject or retain. Before the exam, rehearse an extended answer aloud, then ask Sia to set a fresh scenario with new numbers and check each step.
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