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A table of 12 test marks tells you little at a glance. A box plot of the same marks shows the middle, the spread and the odd one out in a few seconds. The box plot comes from John Tukey, a chemist turned mathematician who argued that statisticians should look at data before testing it. This post covers his life from reliable sources, builds a box plot step by step with sample numbers, and lists the mistakes students make when reading one.
Who was John Tukey?
Princeton University's 2000 news release on his death records that John Wilder Tukey was born in New Bedford, Massachusetts, on 16 June 1915. He earned bachelor's and master's degrees in chemistry at Brown University in 1936 and 1937, moved to Princeton for graduate work in mathematics, and finished his PhD in two years. He spent the war years in the government's Fire Control Research Office in Princeton, was a full professor by 1950, and helped found Princeton's statistics department in 1966. He also worked for decades at Bell Labs and received the National Medal of Science in 1973.
David Brillinger's memoir, John W. Tukey: his life and professional contributions in the Annals of Statistics (2002), adds that he was taught at home until college and died on 26 July 2000. It lists words Tukey coined or gave new meanings, including boxplot, five-number summary, hinge, fences, stem-and-leaf and exploratory data analysis.
Did Tukey really name the bit?
Yes. Claude Shannon's 1948 paper A Mathematical Theory of Communication says that with base-2 logarithms the units of information "may be called binary digits, or more briefly bits, a word suggested by J. W. Tukey." Princeton also credits him with the word "software". In 1965 he and James Cooley published the fast Fourier transform method in An algorithm for the machine calculation of complex Fourier series.
What is exploratory data analysis?
The US National Institute of Standards and Technology's Engineering Statistics Handbook calls Tukey's 1977 book *Exploratory Data Analysis* "the seminal work in EDA". NIST describes EDA as an approach that postpones the usual assumptions about what kind of model the data follow and lets the data reveal their structure, mostly through simple graphs.
The box plot is the best known of those graphs. In Variations of Box Plots (1978), Robert McGill, Tukey and Wayne Larsen explain that box plots conventionally use five values: the two extremes, the upper and lower hinges (quartiles) and the median.
Worked example: building a box plot
Here are sample marks out of 100 for 12 students, already in order:
38, 45, 48, 52, 55, 58, 60, 63, 66, 70, 74, 99
Median. With 12 values, take the average of the 6th and 7th: (58 + 60) ÷ 2 = 59.
Lower quartile, Q1. The median of the lower six values (38 to 58): (48 + 52) ÷ 2 = 50.
Upper quartile, Q3. The median of the upper six (60 to 99): (66 + 70) ÷ 2 = 68.
Interquartile range. IQR = Q3 − Q1 = 68 − 50 = 18.
Fences. NIST's page on outliers sets inner fences at Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. Here 1.5 × 18 = 27, so the fences are 50 − 27 = 23 and 68 + 27 = 95.
Outliers. 99 is above 95, so it is a mild outlier. The outer fence is Q3 + 3 × IQR = 68 + 54 = 122, so 99 is not an extreme outlier.
Draw it. A box from 50 to 68 with a line at 59, whiskers out to the most extreme marks inside the fences (38 and 74), and a separate dot at 99.
Compare the mean: the 12 marks add to 728, so the mean is 728 ÷ 12 ≈ 60.7. One high score pulls the mean above the median, while the box barely moves.
What mistakes do students make with box plots?
Reading the box as the full range. The box holds the middle half of the data; the whiskers and dots show the rest.
Expecting one quartile method. Textbooks and software use slightly different rules for quartiles, so small differences in Q1 and Q3 are normal. Say which method you used.
Deleting outliers automatically. An outlier is a reason to check, not to delete. The 99 might be a typing error, or the strongest student in the class.
You will meet the median of grouped data in our post on statistics in Class 10.
What can a student learn from how Tukey worked?
Brillinger quotes a 1982 overview Tukey wrote with Colin Mallows: the stronger the analyst's qualitative understanding of the field the data come from, the better, "just so long as he does not take it too seriously". Tukey moved from chemistry to topology to statistics and kept asking what the numbers in front of him actually looked like before reaching for formulas. In your own lab work, plot the readings before you average them.
If you like seeing an idea before calculating with it, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
Try it now: add a 13th mark of 20 to the list, find the new median, quartiles and fences, and decide whether 20 is an outlier.
