We began by saying that probability goes from a model to data, and statistics goes from data back to the world. We asked three questions: is it real or is it noise, how should we estimate, and how does randomness enter the data? Here is what the chapter’s seven questions showed.
1. Is it real or is it noise?
In Section 13.1, four headlines turned numbers into claims. Comparing totals without denominators, reading a trend into one year or one day, and hiding a trend inside an average were all errors, and the Poisson model gave a yardstick for the natural variation of counts. A rise of \(8\) standard deviations was almost certainly real; seven deaths on a Tuesday almost certainly was not. Regression to the mean and the need for fair comparisons completed the picture.
2. How should we estimate?
Three questions developed the theory of estimation.
Maximum likelihood (Section 13.2): choose the value of the unknown that makes the observed data most probable. For capture–recapture it gave the natural estimate \(\frac{200 \times 200}{11} \approx 3636\), with a broad likelihood showing how uncertain it was.
Bias and variance (Section 13.3): two unbiased estimators of the number of submarines, one based on the maximum and one on the mean. The maximum, a sufficient statistic, was far more precise, with variance shrinking like \(\frac{1}{n^2}\) rather than \(\frac1n\). The same reasoning outwitted wartime intelligence about German tanks.
Efficiency and robustness (Section 13.4): for uniform data the mean is three times as efficient as the median, but for heavy-tailed data the median wins, and it is robust to wild errors.
3. How does randomness enter the data?
Randomness can be added deliberately: randomised response (Section 13.5) protects each respondent’s privacy while still estimating the population, provided each person randomises independently; a single shared coin would produce a two-humped mixture that tells us almost nothing. Randomness also enters through processing: rounding errors cancel and grow like \(\sqrt n\), while truncation errors accumulate linearly, as the Vancouver Stock Exchange discovered (Section 13.6). And randomness can be engineered: independent ingredients, combined by the Cholesky construction, produce variables with any valid set of correlations (Section 13.7), but not every set of correlations is valid.
4. A toolkit for statistics
| Method | Where you used it |
|---|---|
| Poisson variation | Headlines (13.1) |
| Likelihood and its maximum | Voles (13.2) |
| Unbiasedness and variance | Serial numbers (13.3); median (13.4) |
| Sufficient statistics | Serial numbers (13.3) |
| Normal approximation | 13.4, 13.5, 13.6 |
| Law of total variance | Randomised response (13.5) |
| Bias versus random error | Rounding (13.6) |
| Linearity of covariance | Correlation (13.7) |
Where next
Bayes’ theorem (Section 12.2) is the basis of Bayesian statistics, the main alternative to the methods of this chapter.
The central limit theorem (Section 12.24) justifies the normal approximations used throughout; the Cauchy distribution (Section 12.22) shows when they fail.
Correlation matrices are positive semi-definite matrices, studied with the eigenvectors of Section 7.2.
David Spiegelhalter, The Art of Statistics: the best introduction to statistical thinking for a general reader.
Tim Harford, How to Make the World Add Up: how to read numbers in the news.
John Rice, Mathematical Statistics and Data Analysis: a first university course in mathematical statistics.
Stephen Stigler, The Seven Pillars of Statistical Wisdom: the big ideas of statistics and where they came from.
George Box said that all models are wrong, but some are useful. This chapter has tried to show how to make models useful: by knowing how much variation to expect, by judging methods by how they behave over many repetitions, and by thinking about how the data came to be. These are the habits that turn numbers into knowledge.
A last word
This is the end of the book. Over thirteen chapters, we have used STEP questions as doorways into the mathematics of a first university course: number theory and analysis, calculus and complex numbers, algebra and linear algebra, differential equations and dynamics, combinatorics, mechanics, probability and statistics. Each question was an invitation to ask what lies behind it, and almost every time the answer was a whole subject. If you have worked through even a few of these sections with a pencil in hand, asking the questions in the margins and trying the problems at the end, you have already begun to think like a mathematician. The rest of the journey is yours.