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A student scores far above the class average in one mock test and lower in the next. A coach praises a team after a terrible game and it does better the following week. Often nothing caused the change: extreme results tend to be followed by less extreme ones. Francis Galton was the first to measure this effect. This post covers his work, a worked example, the mistakes it causes, and an honest look at his wider legacy.
Who was Francis Galton?
According to his MacTutor biography at the University of St Andrews, Galton was born near Birmingham on 16 February 1822 and died in Surrey on 17 January 1911. He was a cousin of Charles Darwin. He explored south-west Africa, coined the meteorological term anticyclone, and built the fingerprint classification later developed by Edward Henry and used at Scotland Yard from 1901. The Royal Society awarded him its Copley Medal in 1910.
MacTutor also describes him as famous, or infamous, as the father of eugenics, the idea of improving humans through selective breeding. Eugenics was later used to justify discrimination and grave harm, and it is now discredited. His statistical tools outlived it, and students should know both parts of the story.
What did Galton find with sweet peas and heights?
In his 1886 paper, Regression Towards Mediocrity in Hereditary Stature, Galton first recalls experiments on sweet pea seeds that he described in a Royal Institution lecture on 9 February 1877. The offspring did not resemble their parent seeds in size but were more mediocre: smaller if the parents were large, larger if the parents were very small.
He then turned to people. The full text of the paper says his data were the heights of 930 adult children and their parents, 205 sets in number. To compare men and women he multiplied every female height by 1.08. He averaged the two parents into a mid-parent height and measured everything as a deviation from the average.
His conclusion, in his words: "the height-deviate of the offspring is, on the average, two-thirds of the height-deviate of its mid-parentage." He also warned that the converse is "very far from being its numerical opposite": it does not follow that parents of tall children are even taller.
MacTutor notes he first called the effect reversion and later regression, and that his work led directly to the correlation coefficient, developed by Karl Pearson. For that next step, read Karl Pearson and correlation.
How do you predict with regression? A worked example
Galton's rule: predicted child deviation = 2/3 × mid-parent deviation.
Using sample numbers for illustration, suppose the average adult height is 170 cm, and two parents' mid-parent height, after his adjustment, is 179 cm. The deviation is 179 − 170 = 9 cm. The predicted child deviation is 2/3 × 9 = 6 cm, so the predicted height is 170 + 6 = 176 cm. Tall, but less tall than the parents.
The same logic works for tests. These are sample numbers for a hypothetical class: two mock tests both have a mean of 60 and the same spread, and the correlation between them is r = 0.6. A student scores 90 in the first. Their deviation is 30, so the predicted second score is 60 + 0.6 × 30 = 60 + 18 = 78. A student who scored 35 has a deviation of −25 and a predicted second score of 60 + 0.6 × (−25) = 60 − 15 = 45. Both move towards 60 without anything changing in how they studied.
What mistakes does regression to the mean cause?
Crediting a fix for a natural rebound. If you pick students for extra help because they scored lowest, they will tend to improve even if the help does nothing. You need a comparison group to know.
Blaming praise for a slump. A top score followed by a lower one is expected, not proof that praise made someone careless.
Reading the rule backwards. As Galton warned, regression runs both ways; tall children do not imply even taller parents.
Forgetting the spread. The simple rule mean + r × deviation needs both tests on the same scale. Revise mean and variance if this is unclear.
Regression is not in the JEE Main syllabus, whose 2024 statistics unit stops at mean, median, mode, variance and standard deviation, so treat this as enrichment and check the current bulletin. It does matter for reading any claim that something improved results.
What can a student learn from how Galton worked?
Galton tested an idea on seeds first, where he could grow hundreds, before trying it on people, and he gathered family height records by offering prizes for them, taking care not to reveal what he was looking for. He also looked for the converse and found it was not symmetric. When you track your own mock scores, judge your progress by the trend across several tests rather than one jump, and look for causes only when a change lasts.
If you want to see the steps in a regression or correlation problem laid out, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning.
Plot your last five mock scores, mark the average, and check whether your highest score was followed by a lower one.
