Statistics: Basic to Advanced

The subject everyone uses and few interpret correctly

Statistics is unusual in that the computational side is widely taught and the interpretive side is widely misunderstood — including by people who have passed statistics examinations.

This course covers the mathematics properly, from descriptive measures to generalised linear models, and spends unusual time on what each result licenses you to say. That second part is where the value is.

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 will produce 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. 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.

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.

The move is this: a statistic computed from a sample is itself a random variable, with its own distribution across possible samples. Standard error, confidence intervals and every hypothesis test are consequences of that single idea. Students habitually rush this unit to get to tests, and then find the tests arbitrary. We do not let them.

Regression, with the assumptions checked

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

The emphasis is 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. We also take the causation question seriously: what correlation does not establish, and what kind of design or argument would.

Bayesian inference, as a genuine alternative

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

We teach this as a coherent framework rather than as an advanced extra, and we are explicit about where the two approaches genuinely differ. 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.

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.

Who this suits

Students from Class 9 upward through undergraduate study with school algebra and basic probability, which is reviewed within the course. It suits students heading for economics, psychology, medicine, data science or any field that argues from data — which is most of them now.

Classes are live and online in groups of at most eight, with written work corrected individually.