109 Fun Facts About Statistics

Which fallacy is the belief that a coin is 'due' to land tails after a long run of heads?

The gambler's fallacy cost roulette players millions at Monte Carlo in 1913, when black came up 26 times running and they kept betting on red; the wheel has no memory, so each spin's odds stay the same.

Which measure of central tendency is the middle value when data are sorted?

The median is the middle value of sorted data; it resists distortion by a few huge values, which is why median income describes a typical household better than mean income.

What is the 'mode' of a data set?

The value that appears most often is the mode; it is the only one of the three averages that works for categories, so the most common eye colour in a class is a mode.

Which Greek letter is conventionally used for standard deviation?

Sigma (σ) stands for standard deviation, which is why physicists waited for a 'five-sigma' signal, the level their field demands for a discovery, before announcing the Higgs boson in 2012.

In a normal distribution, roughly what percentage of values fall within one standard deviation of the mean?

68% of values lie within one standard deviation of the mean, which is why IQ tests, scaled to a mean of 100 and a deviation of 15, put about two-thirds of people between 85 and 115.

The normal distribution is also named after which mathematician?

Carl Friedrich Gauss gave the normal distribution its other name, the Gaussian, after using it in 1809 to model measurement errors in astronomy.

Which theorem explains why averages of large samples follow a bell curve, whatever the shape of the data?

The central limit theorem says averages of large samples settle into a bell curve even when the raw data are lopsided, which is why methods built on the normal curve work on data that are far from normal.

Which law says the average of many independent trials settles at the expected value?

The law of large numbers is why casinos win in the long run: a single spin can cost the house money, but over thousands of spins its takings settle at a predictable percentage.

The t-distribution used for small samples was invented by a statistician working for which company?

Guinness employed William Sealy Gosset at its Dublin brewery, where he devised the t-distribution to judge small samples of barley; the firm made him publish under a pseudonym.

Under what pen name did William Sealy Gosset publish his t-distribution work?

Student was the alias because his employer let its researchers publish only if they hid their own names and the firm's; the 1908 paper gave science Student's t-test, still named for the pen name.

Who founded the world's first university statistics department, at University College London in 1911?

Karl Pearson formed the Department of Applied Statistics by merging the two laboratories he already ran at the college, one for biometrics and one for eugenics, and led it until he retired in 1933.

Which statistician developed analysis of variance (ANOVA) while analysing crop data at Rothamsted?

Ronald Fisher went to Rothamsted in 1919 on a temporary post to see what could be learned from crop records kept since the 1840s, stayed 14 years, and devised analysis of variance there.

Which threshold did the man behind ANOVA popularise as 'convenient' for judging statistical significance?

P = 0.05, or 1 in 20, was the level Statistical Methods for Research Workers called convenient in 1925; it stuck so firmly that 72 researchers jointly urged tightening it to 0.005 in 2017.

A p-value is the probability of results at least as extreme as those observed, assuming what is true?

The null hypothesis is what a p-value assumes: it gives the probability of data at least this extreme if the null were true, which the ASA's 2016 statement stressed is not the probability the hypothesis is true.

Which cousin of Charles Darwin developed the concept of correlation and promoted 'regression toward the mean'?

Francis Galton hit on regression through his 1875 sweet-pea breeding experiments and introduced the correlation coefficient in 1888, an idea later formalised by his protégé.

Regression toward the mean took its name from an 1880s study comparing the heights of whom?

Parents and their children: tall parents tended to have children nearer the average height, and short parents taller ones, an effect the study named 'regression towards mediocrity'.

Bayes' theorem became the basis of which email tool in the early 2000s?

The spam filter: a Bayesian filter learns how often each word turns up in junk and in real mail, then updates the odds that a new message is spam, the approach Paul Graham set out in 2002.

London's Bills of Mortality, the raw data for a founding 1662 study of demography, singled out deaths from which disease?

Plague deaths had a count of their own in the weekly bills; haberdasher John Graunt mined them in 1662 to build an early life table, the start of demography.

Which nursing pioneer was also a statistical innovator, famous for her polar area (rose) diagrams?

Florence Nightingale drew her polar area diagrams in 1858 to show that disease, not wounds, killed most British soldiers in the Crimean War.

William Playfair, inventor of the bar graph, introduced which familiar graphic in 1801?

The pie chart made its debut in Playfair's Statistical Breviary; many statisticians call it a poor way to show data, because the eye judges lengths better than areas, yet it became a staple of business and the media.

Which coiner of the term 'bit' introduced the box-and-whisker plot?

John Tukey's box-and-whisker plot draws a box around the middle half of a data set, so that stray extreme values stand out as lone points beyond the whiskers.

What is the interquartile range?

The difference between the 75th and 25th percentiles is the interquartile range; because it ignores the tails, it is a robust measure of spread unaffected by outliers.

A z-score tells you how many of what a value lies above or below the mean?

Standard deviations: a z-score of 2 means two standard deviations above the mean, which lets results from differently scaled tests, such as the SAT and ACT, be compared directly.

Whose paradox describes a trend that appears in several groups but reverses when the groups are combined?

Simpson's paradox: the classic example is UC Berkeley's 1973 graduate admissions, which looked biased against women overall but not within individual departments.

How many people are needed in a room for a better-than-even chance that two share a birthday?

With 23 people there are 253 possible pairs, which is why the odds climb so fast.

In the Monty Hall problem, what should the contestant do after the host reveals a goat?

Switch doors: switching wins the car two-thirds of the time, because the first pick is right only one time in three and the host's reveal moves the remaining odds to the other door.

Which columnist's 1990 answer to the Monty Hall problem drew thousands of angry letters?

Marilyn vos Savant gave her answer in Parade magazine; about 10,000 readers wrote in, most of them to say she was wrong and many signing as PhDs, and she was right.

According to Benford's law, about how often is 1 the leading digit in real-world numerical data?

30%: Benford's law puts the leading digit 1 at about 30% of entries (log10 2 ≈ 0.301), a pattern Simon Newcomb spotted in worn logarithm tables in 1881.

Which magazine mailed ten million ballots in 1936 and wrongly predicted Alf Landon would beat FDR?

The Literary Digest drew its mailing list from its own readers plus car and telephone owners, a sample skewed to the better-off; fewer than a quarter replied, and the magazine folded within two years.

Abraham Wald's WWII study of returning bombers is the classic example of correcting for which statistical error?

Survivorship bias: as the story is told, Wald reasoned that returning planes showed where hits could be survived, so armour belonged on the spots they came back without damage to.

Anscombe's quartet is four data sets that share nearly identical summary statistics but differ how?

They look completely different when graphed: Francis Anscombe built the four sets in 1973, each with nearly the same means, variances, correlation and regression line, to show why you must plot your data.

Twain popularised 'lies, damned lies, and statistics' by attributing it to which prime minister?

Benjamin Disraeli was credited by Mark Twain in his autobiography, but the phrase appears nowhere in Disraeli's works and its true origin is unknown.

The Poisson distribution was famously used in 1898 to model what in the Prussian army?

Deaths from horse kicks were rare and random, just what the Poisson distribution describes; Ladislaus Bortkiewicz's tally of 14 Prussian corps over 20 years became its textbook example.

Which Belgian statistician devised the 'average man' concept and the index later renamed BMI?

Adolphe Quetelet devised his index to describe the 'average man' of a whole population, not to judge any one person's health; Ancel Keys renamed it the body mass index in 1972.

BMI is calculated by dividing weight in kilograms by what?

Height in metres squared: a 70 kg adult 1.75 m tall scores about 22.9, but the sum cannot tell muscle from fat, so a very muscular athlete can come out as overweight.

Which statistician founded FiveThirtyEight and called 49 of 50 states correctly in the 2008 US election?

Nate Silver named FiveThirtyEight for the 538 electors in the US electoral college; he called all 50 states in 2012, and his final 2016 forecast gave Trump about a 29% chance.

Swedish physician Hans Rosling co-founded which foundation to animate global development data?

Gapminder, founded in 2005 by Hans Rosling with his son Ola and daughter-in-law Anna, built the Trendalyzer bubble charts he used in his TED talks, one of which ended with him swallowing a sword.

What is the first step in building a histogram?

Sorting values into bins comes first, and the choice matters: different bin widths can reveal different features of the same data, so it pays to try more than one.

What does a poll's 'margin of error' express?

The random sampling uncertainty is all a margin of error covers: about ±3 points for a sample of 1,000, with nothing added for errors from question wording or from people who decline to answer.

In World War II, Allied statisticians estimated Germany's tank production with startling accuracy from what?

Serial numbers on captured tanks pointed to about 246 built a month; spies had guessed 1,400, and German records found after the war showed 245.

Which campaign does Charles Minard's celebrated 1869 flow map depict?

Napoleon's Russian campaign is drawn as a band that thins from 422,000 men to 10,000; by Moscow it is already down to 100,000, so most of the losses came before the winter retreat.

Random sampling matters chiefly because it lets you do what?

Generalise to the whole population: random selection gives everyone the same chance of being picked, so about 1,000 people can stand in for a country of millions.

The product-moment correlation coefficient is abbreviated by which letter?

r runs from -1 to 1 but measures only straight-line relationships, so a perfect U-shaped pattern can still score an r of zero.

What is a variable called when it drives both the supposed cause and the effect?

A confounding variable, like hot weather driving both ice-cream sales and drownings, influences both sides of an apparent relationship and can make a correlation masquerade as a cause.

Which company, founded in 1935 by the man who called the 1936 election, became famous for opinion polls?

Gallup made its name by calling the 1936 election against a far bigger straw poll, then twelve years later famously forecast that Dewey would beat Truman.

The 1790s survey that helped bring the word 'statistics' into English questioned the minister of every parish in which country?

Scotland was surveyed parish by parish: Sir John Sinclair sent each minister a questionnaire and titled the results with a word he had picked up on his travels in Germany.

Which centuries-old test of the Royal Mint's coinage is an early example of statistical sampling?

The Trial of the Pyx puts a sample of each year's coins before a jury that checks their weight and metal, since no one could test every coin; the pyx is the box the samples travel in.

Which French mathematician first published the method of least squares, in a book on comet orbits?

Adrien-Marie Legendre set out the method of least squares in an appendix to a book on comet orbits, a few years before a rival claimed he had been using it since 1795.

Who first derived the normal curve, in 1733, as an approximation for many coin tosses?

Abraham de Moivre found the bell curve as a shortcut for the odds in many coin tosses; a Huguenot exile who never won a university chair, he lived by tutoring in London's coffee houses.

Which posthumous 1713 book by Jakob Bernoulli treated probability as a branch of mathematics?

Ars Conjectandi was seen into print by his nephew Nicolaus eight years after Jakob Bernoulli died; he had worked for some 20 years on the proof of its 'golden theorem' about long-run frequencies.

Which newly found body, lost in 1801, was relocated thanks to a young mathematician's orbit calculation?

Ceres, the first asteroid, was discovered by Giuseppe Piazzi on 1 January 1801 and lost in the Sun's glare; a 24-year-old German mathematician's orbit computation let astronomers find it again that December.

Thomas Bayes, whose theorem was published only after his death, worked as what?

A clergyman at the Presbyterian chapel in Tunbridge Wells, Bayes never published his theorem; his friend Richard Price found the essay among his papers and sent it to the Royal Society.

Which French mathematician estimated France's population from birth registers and a sample of districts?

Pierre-Simon Laplace found about 28 people per yearly birth in his sample districts, multiplied France's total births by that ratio, and even worked out how far wrong the answer might be.

Which newspaper trusted the polls and printed the famously wrong 1948 headline 'Dewey Defeats Truman'?

Chicago Daily Tribune editors had to go to press early because of a printers' strike and trusted the forecasts; the leading pollster had stopped surveying weeks before election day.

Which statistics journal did Raphael Weldon co-found in 1901?

Biometrika was the answer when Royal Society biologists resisted conclusions drawn from mathematical analysis of data, so Weldon's circle started a journal of its own.

Which pair introduced Type I and Type II errors and the power of a test?

Neyman and Pearson, Jerzy and Egon, set out the two error types and test power in papers from 1928 to 1933; power calculations are how researchers decide how many patients a clinical trial needs.

John Snow's 1854 map of deaths in London's Soho traced a cholera outbreak to what?

A water pump on Broad Street sat at the centre of the deaths Snow plotted; his evidence persuaded the parish authorities to disable it by removing the handle.

The 'lady tasting tea' experiment tested her ability to detect what about her cup?

Milk or tea poured first: Muriel Bristol claimed she could tell, so eight cups were prepared, four each way, and she was asked to sort them, a design described in The Design of Experiments (1935).

Which 1950s finding did the father of ANOVA publicly dispute, arguing correlation is not causation?

Smoking causes lung cancer was the finding he attacked in letters and a 1959 pamphlet, suggesting a common genetic cause; a pipe smoker, he also advised the tobacco industry.

Which numbers are purely nominal labels, so that averaging a list of them gives a meaningless result?

ZIP codes look like quantities but only identify delivery areas, so subtracting one from another tells you nothing. Numerals used purely as labels like this are known as nominal numbers.

Which temperature unit is ratio-level, so that 200 of them really is twice as hot as 100?

Kelvin readings track the kinetic energy of molecules, so doubling the reading doubles the energy. The unit honours a Belfast-born physicist who taught at Glasgow for 53 years.

The word 'nominal', as in the nominal level of measurement, comes from the Latin 'nomen', which means what?

Name is the meaning of 'nomen', and a nominal value is exactly that: a label with no size or order. The same Latin root gives English the word 'noun'.

Which hardness scale is only ordinal, since diamond at 10 is about four times as hard as corundum at 9?

Mohs numbers only record which mineral scratches which, so the steps are wildly uneven. A German mineralogist introduced the ten-step list in 1812, with talc at the soft end.

What does a ratio scale have that an interval scale lacks, making 'twice as much' a meaningful claim?

A true zero means none of the quantity is present, which is what lets 50 cents count as double 25 cents. Money passes the test: empty pockets hold no money at all.

Which survey format, running from 'strongly disagree' to 'strongly agree', yields classic ordinal data?

Likert scale answers can be ranked, but nobody can prove the step from 'agree' to 'strongly agree' matches any other step. Rensis Likert devised the format for his 1932 doctoral thesis.

Which of these is ordinal data, where the order is known but the size of each step is not?

Military ranks place a general above a colonel without saying by how much. Socioeconomic status is ordinal for the same reason.

Which of these is recorded at the nominal level, the lowest of the four levels of measurement?

Nationality only puts people into groups, with no passport outranking another. Counting how many fall into each group is the only arithmetic such data allow.

Which graph, drawn with gaps between its columns, is the standard display for counts of nominal data?

A bar chart gives every group its own column, and some authors insist on the gaps so that nobody mistakes it for a histogram of continuous measurements.

Stanley Smith Stevens, who devised the four levels of measurement, was a leading figure in which academic field?

Psychology needed the scheme because physicists of the 1930s had argued that mental qualities could not be measured at all. Stevens answered by widening what counts as measurement.

Which of these hospital readings is only ordinal data, so averaging it across a ward is questionable?

Pain score numbers only rank how bad things feel: easing from 7 to 6 need not be the same relief as easing from 3 to 2. Opinion and satisfaction ratings share the same limit.

A count, such as the number of books on a shelf, is always a whole number, making it which kind of data?

Discrete data come from counting, so they jump from one value to the next with nothing in between. Measurements such as call lengths can land anywhere, fractions included.

Which of these is ratio-level data, the most informative of the four levels of measurement?

Pulse rate can be compared by multiplying: 120 beats a minute is exactly double 60. Doctors' other ratio measurements include weight, dose and survival time.

Nominal and ordinal data, which sort cases into groups instead of measuring amounts, are both described as what?

Categorical data resist ordinary arithmetic even when the groups are coded as digits, because the digits are only labels. Interval and ratio data are the numerical kinds.

What is a nominal variable called when it has exactly two possible values, such as heads or tails?

Dichotomous comes from Greek words for cutting in two, and such yes-or-no variables are also called binary. Forcing a richer measurement into two camps is known as dichotomization.

Which arithmetic operation becomes legitimate only at the ratio level, the highest of the four levels?

Division needs numbers that are true amounts, which is why 10 kg is twice 5 kg while a size 10 shoe is not twice a size 5. Interval data stop at adding and subtracting.

Which of these do statisticians class as interval-level, one step below ratio-level?

Calendar date counts from an agreed epoch, so the year 2000 is not 'twice as late' as 1000. Gaps still behave: a decade is the same length in any century.

Which of these do many researchers class as ordinal at best, even though the figures are routinely averaged?

IQ scores only place people relative to one another: a 10-point gap need not mean the same thing at different points, and no score stands for an absence of intelligence.

Stanley Smith Stevens, author of the four levels of measurement, also proposed the sone as a unit of what?

Loudness as listeners judge it is what the sone captures: doubling the sone value means a sound seems twice as loud. Stevens proposed the unit in 1936.

At which Ivy League university did Stanley Smith Stevens work when he published the four levels of measurement?

Harvard took Stevens on as an instructor in 1936. A colleague there, the Nobel-winning physicist Percy Bridgman, shaped how he defined measurement.

In regression, what is a 0-or-1 stand-in for a nominal attribute such as 'smoker' called?

A dummy variable lets a model turn an influence on or off, so groups with no numerical meaning can sit beside real measurements. A factor with several groups needs a whole set of them.

Which statistics package, bought by IBM in 2009, asks users to set each variable's level of measurement?

SPSS first appeared in 1968 as the Statistical Package for the Social Sciences. Its Measure setting offers only three choices: nominal, ordinal and scale.

Ordinal data are best tested with which family of methods, home to the sign test and Wilcoxon's signed-rank test?

Nonparametric methods typically replace raw values with their ranks, which is all that ordinal data can honestly supply. They do not assume a bell-shaped distribution.

Which rank correlation coefficient, devised by a former British Army engineer, is the usual choice for ordinal data?

Spearman's rho correlates the ranks of two variables instead of their raw values. Charles Spearman served more than a decade as an army officer before turning to research on intelligence.

Which average, favoured for growth rates and investment returns, did Stevens permit only for ratio-level data?

Geometric mean multiplies the values together and takes a root, which demands numbers that can be multiplied meaningfully. Investors use it because gains compound instead of adding up.

Stanley Smith Stevens defined measurement as 'the assignment of numerals to objects or events according to' what?

Rules of any consistent kind were enough for Stevens, which is how merely labelling things came to count as measuring them. Critics have called the definition far too broad.

Which journal published Stevens's 'On the Theory of Scales of Measurement', the paper behind the four levels?

Science carried the paper in 1946, and it ran to just four pages. Stevens wrote it to answer physicists who doubted that perceptions could be measured at all.

Which rank-based test compares two independent groups of ordinal data, standing in for the two-sample t-test?

Mann–Whitney pools both groups, ranks every observation and asks whether one group's ranks run higher. Henry Mann and Donald Whitney set it out in a 1947 paper.

Which measure of relative spread did Stevens allow only for ratio data, since it breaks down on interval scales?

Coefficient of variation is the spread expressed as a share of the mean, which makes it handy for comparing variability in quantities measured in different units.

Which kind of count, like the one the US Constitution orders every ten years, covers a whole population, not a sample?

A census tries to reach everyone, which is why it is slow and costly and why most research settles for samples. The word comes from the Latin 'censere', to assess.

In the language of set theory, what is a sample in relation to the population it was drawn from?

A subset is all a sample can ever be, since every member comes from inside the population. Researchers settle for one because gathering everything is usually too slow or too dear.

Which kind of quality testing, such as crash-testing cars, forces makers to check a sample instead of every item?

Destructive tests ruin whatever they examine, so a full inspection would leave nothing to sell. Sampling plans for such checks became common during the Second World War.

What do statisticians call a number describing a whole population, such as the true average income of every household?

A parameter is usually unknowable in practice, so researchers estimate it with the matching figure from a sample, called a statistic. Handily, the initials pair up: P with P, S with S.

Which company's ratings have estimated America's television audience from a sample of households since the 1950s?

Nielsen was tracking about 1,700 metered homes, plus several hundred diary keepers, in the early 1980s to speak for every TV household in the country.

Ecologists gauge a wild animal population by tagging a sample, freeing it and sampling again, a method called what?

Mark and recapture works by proportion: if a tenth of the second catch carries tags, the tagged animals are taken to be about a tenth of the whole population.

Which branch of statistics draws conclusions about a population from a sample, in contrast to descriptive statistics?

Inferential statistics takes a leap that description never does: it says something about people nobody measured, and attaches a probability to being wrong.

In statistical formulas, what does a lowercase n conventionally stand for, in contrast to a capital N?

Sample size takes the small letter while the population gets the capital, so the share of a population that was surveyed is written n/N.

Pollsters whose sample has too few young people can correct the imbalance through which adjustment?

Weighting makes each under-represented respondent count for more, so the sample's mix of ages and sexes matches the population's. It cannot rescue a sample that missed a group entirely.

Which Greek letter, also the prefix for 'micro' in units, is the standard symbol for the mean of an entire population?

Mu, written μ and pronounced 'mew', marks a value that belongs to the whole population. The mean of a mere sample gets a Latin letter instead.

What is the gap between a sample's result and the true population figure, arising purely by chance, called?

Sampling error is no one's mistake: draw another sample and the answer shifts a little. Faulty questions or data entry belong to a separate family that survey experts call nonsampling errors.

A bar drawn over which letter is the usual symbol for the mean of a sample, as distinct from a population?

x with a bar on top, read aloud as 'x-bar', is the textbook shorthand for a sample's mean. Latin letters describe samples, while Greek ones are kept for whole populations.

Which survey problem arises when people chosen for a sample cannot be reached or decline to take part?

Nonresponse matters because those who decline often differ from those who answer, so simply surveying more people does not cure it. Advance letters and repeat calls are the usual remedies.

A sample average offered as a one-number best guess for a population average is known as what?

A point estimate commits to one number, so it is almost never exactly right. That is why careful reports attach a range of plausible values around it.

What is the working list from which a sample is actually drawn, such as an electoral register, called?

A sampling frame rarely matches the population exactly: a telephone directory, for instance, leaves out everyone without a listed number, and careful selection cannot put them back.

The n − 1 used when calculating a sample's variance is described as its number of what?

Degrees of freedom count the values still free to vary: once the sample's mean has been worked out, the last of the n observations is forced, leaving n − 1.

Dividing by n − 1 instead of n when estimating a population's variance from a sample is known as whose correction?

Bessel's correction offsets the way a sample clusters around its own mean more tightly than around the population's. Its namesake was the first astronomer to measure the distance to a star.

In 1999 the US Supreme Court decided that statistical sampling could not replace a full head count for which purpose?

Apportioning House seats needs an actual count, the justices held, after officials proposed using samples in 2000 to fix a chronic undercount of minorities, children and renters.

Which of these traits is nominal data, with groups that have no built-in order?

Handedness simply places people in the left, right or mixed camp, and no camp outranks another. Favourite colour and religion are nominal for the same reason.

Which of these school records is ordinal data, ranked but with no fixed distance from one step to the next?

Letter grades say an A beats a B without promising that the gap equals the one from B to C, which is why turning them into grade-point averages makes purists wince.

The monthly US unemployment rate is estimated from a sample of households, not a full count, through which programme?

Current Population Survey interviewers question tens of thousands of households every month, and the Bureau of Labor Statistics turns their answers into the jobless figure for the whole country.