Statistics

• 1. Introduction

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  "Statistics is both the science of uncertainty and the technology of extracting information from data." That line appears in the International Encyclopedia of Statistical Science, and it captures a discipline that touches almost everything. Statistics concerns the collection, organization, analysis, interpretation, and presentation of data. It draws its very name from the idea of a state. The word descends from the Latin Status, meaning situation or condition in society, which in late Latin came to mean state itself. So how did a word about governments and tax rolls become the mathematics of doubt? Why does a field built on numbers insist that it can never prove anything true? And what does a lighting experiment in a factory have to do with the way scientists now design their studies? The answers run from cryptographers of the medieval Islamic world to a tea-tasting experiment that gave us the phrase null hypothesis.

• 2. From States To Science

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  Gottfried Achenwall, a German political scientist, coined the word statistik to mean a summary of how things stand. In 1749 he began using the term for a collection of quantitative information, in the modern sense for the science. The Italian scholar Girolamo Ghilini had introduced the term statistic earlier, in 1589, to mean a collection of facts about a state. The word crossed into English in 1770 by way of German, where it referred to the study of political arrangements. Its modern meaning arrived in the 1790s through the works of John Sinclair. Early statistical thinking served the needs of states that wanted to base policy on demographic and economic data, which is exactly why the stat- root sits at the heart of the name. The earliest European writing containing statistics dates to 1663, when John Graunt published Natural and Political Observations upon the Bills of Mortality. In the early 19th century the scope widened beyond matters of state to include the collection and analysis of data in general. In those days the name referred mainly to matters of state, and British statisticians were often called statists, a forerunner to today's reach into government, business, and the natural and social sciences.

• 3. The Science Of Uncertainty

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  Probability and statistics were once paired as a single subject, yet they are conceptually distinct, and the difference defines what statistics actually does. Probability starts with the given parameters of a total population and deduces probabilities about samples. Statistical inference moves the other way, inferring from samples to the parameters of a larger or total population. This inductive direction is why statistics is called the science of uncertainty, since it must cope with measurement and sampling error and with uncertainty in modelling. The Mathematics Subject Classification indexes statistics at 62, a subclass of probability theory and stochastic processes. Mathematical statistics sits in the range 276-280 of subclass QA in the Library of Congress Classification. Some treat statistics as a distinct mathematical science rather than a branch of mathematics, because while many investigations use data, statistics is concerned with data in the context of uncertainty and decision-making in the face of uncertainty. Two broad approaches summarize data. Descriptive statistics summarize a sample using indexes such as the mean or standard deviation. Inferential statistics interpret data from a population sample to make statements and predictions about the population, the engine that turns a handful of observations into a claim about the world.

• 4. Watching And Manipulating

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  The Hawthorne study set out to learn whether brighter lights made workers faster, and instead it taught statisticians a lasting lesson about observation. Researchers at the Hawthorne plant of the Western Electric Company first measured productivity, then changed the illumination in part of the plant, then checked whether the change affected output. Productivity did improve under the experimental conditions. The study is heavily criticized today for the lack of a control group and blindness. The Hawthorne effect now names the finding that an outcome can change due to observation itself, because workers became more productive not from the lighting but from being watched. This is one of two major kinds of causal study. An experimental study takes measurements, manipulates the system, then takes further measurements with different levels using the same procedure. An observational study involves no manipulation at all. The classic example explores the association between smoking and lung cancer. Researchers might use a cohort study to collect observations of smokers and non-smokers and count lung cancer cases in each group. A case-control study instead invites people with and without the outcome and collects their exposure histories. Statisticians recommend that experiments compare at least one new treatment against a standard treatment or control, which allows an unbiased estimate of the difference in treatment effects.

• 5. The Logic Of The Null

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  A criminal trial offers the clearest picture of how a null hypothesis works. The null hypothesis, H0, asserts that the defendant is innocent, while the alternative, H1, asserts guilt. H0 stands as the status quo and is maintained unless H1 is supported by evidence beyond a reasonable doubt. Failure to reject H0 does not imply innocence. It means only that the evidence was insufficient to convict, so the jury fails to reject H0 rather than accepting it. The null hypothesis cannot be proven true because it is already assumed to be true when the test is conducted. From this setup come two broad categories of error. A Type I error falsely rejects the null hypothesis, a false positive. A Type II error fails to reject the null when an actual difference exists, a false negative. Interpretation hangs on statistical significance, often the p-value, the probability of observing a result at least as extreme as the test statistic assuming the null is true. Yet a difference that is highly statistically significant can have no practical meaning. In a large study a drug might show a statistically significant but very small benefit, too small to help a patient noticeably. To express how closely a sample estimate matches the true population value, statisticians often report 95% confidence intervals, a range that would include the true value in 95% of all possible repeated datasets.

• 6. Inventing The Discipline

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  Formal discussion of inference reaches back to the cryptographers of the Islamic Golden Age, between the 8th and 13th centuries. Al-Khalil, who lived from 717 to 786, wrote the Book of Cryptographic Messages, containing one of the first uses of permutations and combinations. Al-Kindi described how to use frequency analysis to decipher encrypted messages, an early example of statistical inference. Ibn Adlan, who lived from 1187 to 1268, contributed on the use of sample size in frequency analysis. The mathematical foundations later grew from debates about games of chance among Gerolamo Cardano, Blaise Pascal, Pierre de Fermat, and Christiaan Huygens. Probability theory took shape at the very end of the 17th century, especially in Jacob Bernoulli's posthumous work, the first book to combine games of chance with the realm of the probable. Adrien-Marie Legendre first described the method of least squares in 1805, though Carl Friedrich Gauss presumably used it a decade earlier in 1795. The Belgian scientist Adolphe Quetelet, who lived from 1796 to 1874, introduced the notion of the average man, l'homme moyen, to understand crime, marriage, and suicide rates. In 1853 Quetelet organised the First International Statistical Congress in Brussels to unify measurement in statistical research.

• 7. Three Waves Of Modern Statistics

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  Francis Galton and Karl Pearson led the first wave at the turn of the 20th century, transforming statistics into a rigorous mathematical discipline for science, industry, and politics. Galton introduced standard deviation, correlation, and regression analysis, applying them to human traits such as height, weight, and eyelash length. Pearson developed the Pearson product-moment correlation coefficient, the method of moments, and the Pearson distribution. Together the two founded Biometrika as the first journal of mathematical statistics and biostatistics, and Pearson founded the world's first university statistics department at University College London. William Sealy Gosset began the second wave in the 1910s and 1920s, which reached its height in Ronald Fisher. Fisher's 1918 paper, The Correlation between Relatives on the Supposition of Mendelian Inheritance, was the first to use the term variance. His 1925 Statistical Methods for Research Workers and his 1935 The Design of Experiments defined the academic discipline. Fisher coined the term null hypothesis during the Lady tasting tea experiment, which he said is never proved or established, but is possibly disproved, in the course of experimentation. The final wave came from the collaboration of Egon Pearson and Jerzy Neyman in the 1930s, who introduced Type II error, the power of a test, and confidence intervals. In 1934 Neyman showed that stratified random sampling was generally better than purposive or quota sampling.

• 8. When Numbers Mislead

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  "There are three kinds of lies: lies, damned lies, and statistics." That quotation captures a mistrust that misuse of statistics can earn, errors subtle enough that even experienced professionals make them. The stakes are real, since social policy, medical practice, and the reliability of structures like bridges all depend on the proper use of statistics. Darrell Huff's book How to Lie with Statistics outlines a range of considerations and proposes questions to ask in each case. Who says so, and do they have an axe to grind. How do they know. What is missing. Did someone change the subject. Does it make sense. Huff warned that the dependability of a sample can be destroyed by bias, and he urged allowing oneself some degree of skepticism. Correlation is especially prone to confusion. A study of annual income and age of death might find that poor people tend to have shorter lives than affluent people, yet the two variables may or may not cause one another. A third, previously unconsidered phenomenon, called a lurking or confounding variable, could be the real driver. The basic skills and skepticism people need to handle such information in everyday life is referred to as statistical literacy, the quiet defense against being fooled by a well-drawn bar graph.