Mathematical statistics

Reasoning from a sample back to the thing that produced it. Where probability-theory starts from a known mechanism and derives what data it will produce, statistics runs the arrow the other way: given the data, what can be said about the mechanism, and with what confidence.

The corpus holds two accounts, and they sit at different altitudes. larsen-marx-mathematical-statistics derives the machinery and closed the spoke’s statistics-and-inference growth edge. herzog-understanding-statistics (Herzog, Francis & Clarke, open access) assumes much less mathematics, skips the derivations, and spends the space on what the machinery does to a research literature when many people use it at once.

The two halves

Estimation — recovering a parameter. Maximum likelihood and the method of moments as recipes; then the harder question of what makes one estimator better than another, answered through unbiasedness, sufficiency, consistency, and the Cramér–Rao lower bound, which sets a floor on the variance of any unbiased estimator and so tells you when to stop looking. Interval estimation replaces the point answer with a range. Bayesian estimation appears as its own section rather than an appendix.

Hypothesis testing — deciding between accounts. The decision rule, the two error types, and the generalized likelihood ratio as the principle that generates good tests rather than a catalogue of tests to memorize. Everything after it in Larsen & Marx is that machinery applied: one-sample, two-sample, goodness-of-fit, regression, ANOVA, blocked designs, and the nonparametric versions for when the distributional assumptions fail.

Where it sits under this spoke

It is probability-theory pointed backwards, and it needs the machinery the rest of the corpus supplies. Sampling distributions are transformations of random variables; the t and F statistics are derived, not posited (larsen-marx-mathematical-statistics ch. 7). Regression and ANOVA are linear-algebra — least squares is a projection, and the analysis of variance is a decomposition of one. The Central Limit Theorem that licenses most of the normal-theory inference is a real-analysis convergence result.

So statistics is the corpus’s first subject that is downstream of three others, which is a different shape from the founding texts and worth watching as the spoke grows.

The honesty thread

The distinguishing feature of the corpus’s source is that it spends a section per chapter on how its own methods mislead — significance versus practical significance, how not to read a correlation coefficient, outliers, why a sample is never “valid.”

That connects outward. ../quant-trading-wiki holds the corpus’s sharpest instance of the failure these sections warn about: backtest overfitting, where enough hypotheses tested against one dataset guarantee a significant result. The statistics is not in dispute between the two spokes; what differs is that one is teaching the caveat and the other is documenting an industry that ignored it.

What happens before the data

Everything above begins once numbers exist. measurement-theory (2026-08-06) is the layer under that assumption: mari-measurement-across-sciences argues that a measurement result is built through a model rather than read off the world, that measurement is not the same as quantification, and that the physical sciences answer imperfect measurement with uncertainty while the human sciences answer it with validity — how far off the number is, versus whether the procedure measures the intended thing at all. A t-test does not ask whether its input sits on an interval scale; that decision was made off the page, and this is the subject that makes it.

larsen-marx-mathematical-statistics · probability-theory · linear-algebra · real-analysis · measurement-theory · mathematics

When the machinery is used by a whole field at once

Everything above describes what a test does on one dataset. herzog-understanding-statistics adds the layer above it, and it changes how a reader should treat a published result.

The Test for Excess Success is the argument in one line: if an effect is real but power is moderate, sampling alone guarantees that some experiments fail, so a set of studies should succeed at roughly the rate its power predicts. Sum the power across the set and compare with the number of significant results reported. A precognition set of ten experiments expected 6.27 successes and reported 9, giving a probability of 0.058 — the paper’s own effect size makes the paper’s own success rate implausible. Nineteen bystander-effect studies expected 10.77 and reported 10, at probability 0.76, so the test separates a believable literature from an implausible one rather than condemning everything.

The consequence for inference as taught above: “replication cannot be the final arbiter for science when hypothesis testing is used, unless experimental power is very high.” At moderate power a failed replication is sometimes just sampling, and an unbroken run of successes is evidence of a problem. Power stops being a planning parameter and becomes a reading parameter — a number you need in order to interpret someone else’s results, not just to design your own.

The same subject taught to people who will never derive it (added 2026-08-12)

illowsky-introductory-statistics covers a route that overlaps this page’s second half almost exactly — sampling distributions, the central limit theorem, confidence intervals, one- and two-sample hypothesis tests, chi-square, regression, one-way ANOVA — and reaches all of it without calculus and without a single proof. Its preface says so directly: “focuses on statistics application over theory”, for “students majoring in fields other than math or engineering”, assuming intermediate algebra.

Set beside larsen-marx-mathematical-statistics, which arrives at the same tests through estimation theory, Cramér–Rao and the generalized likelihood ratio, the pair is the cleanest natural experiment this spoke has on what a derivation is for. Both books teach a reader to run a t-test. One can say where the t distribution comes from and why the ratio has that law under the null; the other supplies a table, a calculator keystroke sequence, and a decision rule.

The honest reading is that they are not competing accounts of one subject but two different subjects sharing a vocabulary. Larsen & Marx teach why the procedures are the right ones; Illowsky & Dean teach which procedure to reach for and how to carry it out. Whether the second can be done well without the first is exactly the question the corpus cannot settle from inside, and it is the practical form of the spoke’s standing research-methods fork.

One thing the applied book does hold that the theoretical one does not: an experimental apparatus. Its statistics labs have students collect their own data and interpret their own analysis, which is where measurement-theory‘s off-the-page decisions get made in practice — and which no derivation-first text here asks a student to do.