Statistics Course Guide: Basic to Advanced

A statistics course at this level has to do something unusual: teach the computation properly and then spend equal time on what each result licenses you to say. The second part is where almost all the value sits, and it is the part most courses skip.

What a p-value is not

A p-value is the probability of observing data at least as extreme as yours, assuming the null hypothesis is true.

It is not the probability that the null hypothesis is true. It is not the probability your result arose by chance. It is not a measure of effect size — a tiny, meaningless difference produces a minuscule p-value given a large enough sample.

These three misreadings are everywhere, including in published work, and a student who holds the correct interpretation reads research very differently. We treat this as core content rather than as a cautionary footnote.

Confidence intervals say less than you think

A 95 per cent confidence interval is not the range in which the true value probably lies, though that is how nearly everyone reads it.

It is produced by a procedure that captures the true parameter in 95 per cent of repeated samples. The particular interval you computed either contains the parameter or it does not; there is no probability attached to that specific interval in the frequentist framework.

The interpretation people actually want — a probability statement about the parameter given this data — is the Bayesian credible interval, which is one good reason to do the Bayesian unit rather than treating it as optional.

Sampling distributions: the idea everything rests on

If a student does not understand the sampling distribution of a statistic, nothing after it makes sense; they are following procedures and hoping.

The move is this: a statistic computed from a sample is itself a random variable, with its own distribution across the samples that might have been drawn. Standard error, confidence intervals and every hypothesis test are consequences of that single idea.

Students habitually rush this to get to tests, and then find the tests arbitrary. A good statistics course does not allow that, because the cost appears later and is hard to diagnose.

Regression, with the assumptions checked

Simple and multiple linear regression, least squares, residual analysis, R-squared and its limits.

The emphasis belongs on diagnostics, because fitting a model is easy and checking whether it was appropriate is the part that gets skipped. A regression with excellent R-squared and patterned residuals is telling you the model is wrong, and a student who never plots residuals never hears it.

The causation question deserves honest treatment too: what correlation does not establish, and what kind of design or argument would. That distinction is the difference between a student who can run a regression and one who can interpret it.

Estimation and what makes an estimator good

Point estimation, bias, consistency and efficiency, maximum likelihood, and the method of moments.

This is where statistics stops being a collection of recipes. Asking why one estimator is preferable to another — and discovering that unbiased is not automatically better than biased — changes how a student thinks about every method that follows.

Maximum likelihood in particular is worth the effort, because it is the single idea that unifies most of modern statistics and because it reappears immediately in any machine learning course.

Bayesian inference as a genuine alternative

Prior, likelihood and posterior; conjugate priors including beta-binomial and normal-normal; credible intervals.

This belongs in a statistics course as a coherent framework rather than an advanced extra. The Bayesian posterior answers the question people instinctively ask — what should I now believe about this parameter — at the cost of requiring a prior.

That trade is worth understanding rather than inheriting a tribal position about. Students who meet both frameworks properly are better at each.

Why so much published research does not replicate

The final unit is the one students remember, and it follows directly from techniques taught in every introductory course.

Test twenty hypotheses at the 5 per cent level and you expect one significant result from pure noise. Decide which analysis to run after seeing the data — the garden of forking paths — and you can reach significance almost at will without any conscious dishonesty. Add publication bias, where null results are never submitted, and the published literature becomes a biased sample of what was actually found.

We cover Bonferroni and false discovery rate control, pre-registration, and what a reader should look for. A student who has done this unit evaluates evidence better than most graduates.

Where statistics is actually used

Worth saying plainly, because students sometimes treat this as the least interesting of the topic courses.

Statistics is the mathematics of every field that argues from data: medicine, economics, psychology, public policy, sport, and all of machine learning. The techniques in the advanced stage — regularisation, cross-validation, the bias-variance tradeoff — are the direct foundation of modern predictive modelling, and they appear again in our quant trading course with specific failure modes attached.

What to expect from the first term

Statistics tends to feel easy early and then abruptly does not, which catches students out.

Descriptive measures and basic probability are familiar enough that confidence builds quickly. Sampling distributions then arrive and require a genuine conceptual shift: the object of study stops being the data and becomes the behaviour of a statistic across samples that were never collected.

Students who find that disorienting are responding correctly to a real change in the subject. Those who do not notice it have usually not understood it, and will struggle later when every test turns out to depend on it.

Who this suits

Students from Class 9 upward through undergraduate study with school algebra and basic probability,

Who this suits

Students from Class 9 upward through undergraduate study with school algebra and basic probability, which is reviewed within the course. It pairs naturally with our probability course, which is the same mathematics approached from the other direction.