110 Fun Facts About Statistics

Which 18th-century German scholar gave the field its name, Statistik?

Gottfried Achenwall, a Göttingen professor, used the word Statistik in 1749 for the systematic description of a state's affairs, and the name stuck as the field turned mathematical.

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; a data set can have two modes (bimodal) or none at all if every value occurs just once.

Which Greek letter is conventionally used for standard deviation?

Sigma (σ) denotes the population standard deviation; a low sigma means data cluster tightly around the mean, while a high sigma means they are spread out.

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

68%: about 68% of values in a normal distribution lie within one standard deviation of the mean, and about 99.7% lie within three.

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.

What does the central limit theorem say about the spread of sample means?

They tend toward a normal curve as samples grow: the central limit theorem says the distribution of sample means approaches normality whatever the population's shape, which is why normal-based methods work on non-normal data.

Which law says the average of many independent samples converges to the true expected value?

Large numbers law, better known as the law of large numbers, is why casinos win in the long run: over thousands of spins the average result settles at the expected value.

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 and stout; the firm made him publish under a pseudonym.

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

Student was the pen name Gosset used for his 1908 paper 'The Probable Error of a Mean', which introduced the t-distribution for small samples.

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

Karl Pearson founded the Department of Applied Statistics at University College London in 1911, the world's first university statistics department, and also coined the term 'histogram'.

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

Ronald Fisher developed analysis of variance during his years at Rothamsted Experimental Station (1919-1933), where he analysed decades of crop-yield records.

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

P = 0.05, or 1 in 20, was the level called 'convenient' in the 1925 book Statistical Methods for Research Workers, and it became the default cut-off for significance.

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.

Who developed the statistical 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.

What does 'regression toward the mean' predict about an extreme measurement?

The next one is likely to be closer to average: an extreme score mixes skill with luck, and the luck rarely repeats, which is why the 'Sports Illustrated cover jinx' needs no curse.

Bayes' theorem lets you calculate the probability of a cause given what?

Its effect: Bayes' theorem inverts conditional probability, turning the likelihood of an effect given a cause into the probability of the cause given its effect, updating a prior belief with evidence.

Who is called the founder of demography for his 1660s study of London's mortality bills?

John Graunt, a London haberdasher, published Natural and Political Observations on the Bills of Mortality in 1662 and built the first crude life table from them.

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.

Who is credited with inventing the pie chart, in his 1801 Statistical Breviary?

William Playfair, a Scottish engineer, had already introduced line and bar charts in his 1786 Commercial and Political Atlas before drawing the first pie charts in 1801.

Who introduced the box-and-whisker plot in 1970?

John Tukey introduced the box-and-whisker plot in 1970 and popularised it in his 1977 book Exploratory Data Analysis; the box spans the middle half of the data.

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 is the value minus the mean, divided by the standard deviation, so z = 2 means two standard deviations above the mean.

Which name is given to 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 answered the problem in her Parade 'Ask Marilyn' column in 1990; the puzzle had first been posed by Steve Selvin in The American Statistician in 1975.

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's 1936 poll of ten million people wrongly predicted Alf Landon would beat FDR?

The Literary Digest mailed ten million ballots drawn from its subscribers plus car and telephone lists, a sample skewed to the better-off, and folded within two years of the fiasco.

How large a sample did the American Institute of Public Opinion use for the 1936 election?

50,000: the American Institute of Public Opinion polled roughly 50,000 people chosen by quota sampling and correctly forecast Roosevelt's win, while a far larger mail-in poll got it wrong.

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

Survivorship bias: Wald reasoned the returning planes showed where hits could be survived, so armour belonged where they were undamaged, the places hits had downed the missing bombers.

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 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, tallied across Prussian cavalry corps over two decades, were shown by Ladislaus Bortkiewicz in 1898 to follow the Poisson distribution almost perfectly.

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

Adolphe Quetelet introduced his 'average man' in his 1835 book on social physics; Ancel Keys renamed the Quetelet Index the body mass index in 1972.

BMI is calculated by dividing weight in kilograms by what?

Height in metres squared: BMI = kg/m², so a 70 kg adult 1.75 m tall scores about 22.9; 25 to 29.9 counts as overweight and 30 or more as obese.

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

Nate Silver founded FiveThirtyEight in 2008 and called 49 of 50 states that year, then all 50 in 2012; his final 2016 forecast gave Trump roughly 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: a histogram divides the range into intervals (bins), counts the values in each, and draws bars whose heights show those counts.

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

The random sampling uncertainty is what a margin of error expresses: for a simple random sample of 1,000 people it is roughly ±3 percentage points at the usual confidence level.

Which confidence level is used by default when none is stated?

95% is the conventional confidence level, corresponding to about 1.96 standard errors either side of the estimate, so roughly one interval in twenty misses the true value.

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

Napoleon's Russian campaign of 1812 is the subject of Minard's 1869 flow map, whose shrinking band shows the army falling from 422,000 men to 10,000.

Random sampling matters in statistics chiefly because it lets you do what?

Generalise from the sample to the population: random selection gives every member a known chance of inclusion, so the sample's results can be projected to the whole with a calculable margin of error.

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

r, the product-moment correlation coefficient, ranges from -1 (perfect negative) through 0 (no linear relation) to 1 (perfect positive).

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?

George Gallup's institute made its name calling the 1936 election that humiliated a rival straw poll.

Who introduced the word 'statistics' into English in 1791 with his Statistical Account of Scotland?

Sir John Sinclair borrowed the German term for his 21-volume Statistical Account of Scotland (1791-1799), a parish-by-parish survey compiled from questionnaires sent to every Church of Scotland minister.

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

The Trial of the Pyx, first recorded in 1282, tests a sample of newly struck coins for weight and purity; the pyx was the box that held the coins in Westminster Abbey's Pyx Chamber.

Which French mathematician first published the method of least squares, in 1805?

Adrien-Marie Legendre published the method of least squares in 1805 in a work on determining comet orbits, four years before a rival claimed to have used it earlier.

Who first plotted the normal curve, in 1733, while studying coin tosses?

Abraham de Moivre derived the bell-shaped curve in 1733 as the limit of the binomial distribution for many coin tosses, in a note later folded into his Doctrine of Chances.

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

Ars Conjectandi, published in 1713 eight years after Jakob Bernoulli's death, contains his 'golden theorem' on long-run frequencies and introduced the Bernoulli numbers.

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.

Who transmitted Thomas Bayes's proof to the Royal Society in 1763, after Bayes's death?

Richard Price, a Welsh minister and friend of Bayes, edited the essay and sent it to the Royal Society in 1763, two years after Bayes died.

In what year did Laplace estimate France's population at 28,328,612 from births and sample towns?

1802: Laplace took the population-to-births ratio in a sample of départements and applied it to France's total annual births, an early ratio estimator, to reach 28,328,612.

Which statistician proposed the 'representative method' of sampling to the ISI in 1895?

Anders Nicolai Kiær, head of Norway's statistical bureau, urged the 'representative method' in 1895, surveying a cross-section instead of a full census; a 1934 paper later put random stratified sampling on a firm footing.

Which journal was founded in 1901 by Walter Weldon and two other pioneers of biometry?

Biometrika was founded in 1901 by Walter Weldon and two colleagues to publish statistical studies of biology; its first editor ran it until his death in 1936.

Which pair introduced the ideas of Type II error, test power and confidence intervals?

Egon Pearson and Jerzy Neyman laid out the framework of Type I and Type II errors and test power in papers from 1928 to 1933, with Neyman adding confidence intervals in the 1930s.

In which year was the term 'variance' introduced, in a paper on Mendelian inheritance?

1918: the paper 'The Correlation Between Relatives on the Supposition of Mendelian Inheritance' coined 'variance' and founded quantitative genetics by reconciling Mendel with biometric data.

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.