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If you have ever read a headline saying two things are "linked", someone probably calculated a correlation coefficient. The number most often used is Pearson's r. This post explains where it came from, how to compute it by hand, the mistakes that trip students up, and why Pearson himself is remembered with mixed feelings.
Who was Karl Pearson?
Pearson was born in London on 27 March 1857 and died in Coldharbour, Surrey, on 27 April 1936, according to his MacTutor biography. He graduated from Cambridge in 1879 as Third Wrangler in the Mathematical Tripos, was called to the Bar, and in 1885 became Goldsmid Professor of Applied Mathematics at University College London, where he spent most of his career.
Two events turned him to statistics: Francis Galton's book *Natural Inheritance* (1889) and the appointment of Weldon as Professor of Zoology at UCL in 1890. From 1893 to 1912 Pearson wrote 18 papers under the title *Mathematical Contributions to the Theory of Evolution*. MacTutor notes that these include work on regression, the correlation coefficient and the chi-square test of significance (1900). He coined the term "standard deviation" in 1893, was elected a Fellow of the Royal Society in 1896 and co-founded the journal *Biometrika* with Weldon and Galton.
Did Pearson invent correlation?
The idea came first from Galton. The historian Stephen Stigler, in Francis Galton's account of the invention of correlation (1989), dates Galton's invention to late 1888. Pearson's contribution was to give it full mathematical treatment. His 1896 memoir in the *Philosophical Transactions*, Regression, heredity, and panmixia, says it will be devoted largely to "the expansion and fuller development of Mr. Galton's ideas". It also states his method plainly: "we must proceed by the method of statistics, rather than by the consideration of typical cases."
What does the correlation coefficient measure?
The NCERT Class 11 Statistics for Economics chapter on correlation calls it Karl Pearson's coefficient, also known as the product moment correlation coefficient. It gives a value between −1 and +1 for how closely two variables follow a straight-line relationship. The formula, with x and y as deviations from the means, is:
r = Σxy / √(Σx² × Σy²)
NCERT adds two warnings worth memorising. Correlation "measures covariation, not causation". And the coefficient should be used only when the relationship is linear, so you should look at the scatter diagram first.
Worked example 1: computing r by hand
These are sample numbers for practice. Five students record hours of revision (X) and test marks (Y): (1, 40), (2, 55), (3, 50), (4, 65), (5, 70).
Means. X̄ = 15/5 = 3. Ȳ = 280/5 = 56.
Deviations. x = −2, −1, 0, 1, 2. y = −16, −1, −6, 9, 14.
Products xy. 32, 1, 0, 9, 28. So Σxy = 70.
Squares. Σx² = 4 + 1 + 0 + 1 + 4 = 10. Σy² = 256 + 1 + 36 + 81 + 196 = 570.
Coefficient. r = 70 / √(10 × 570) = 70 / √5700 = 70 / 75.50 ≈ 0.93.
A value of 0.93 means the points lie close to a rising straight line. It does not mean revision caused the marks; with five students, many other explanations are possible.
Worked example 2: when r = 0 hides a pattern
Take the sample points (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), which lie exactly on y = x².
X̄ = 0, Ȳ = 10/5 = 2.
x = −2, −1, 0, 1, 2 and y = 2, −1, −2, −1, 2.
Σxy = −4 + 1 + 0 − 1 + 4 = 0, so r = 0.
The variables are perfectly related, yet r is zero, because the relationship is a curve. This is exactly why NCERT tells you to draw the scatter diagram first.
Common mistakes and how to avoid them
Forgetting to subtract the means. Use deviations, or use the equivalent formula with raw sums carefully.
Arithmetic slips in Σy². Write the squares in a column and add twice.
Claiming causation. A high r between ice-cream sales and drowning reflects summer, not ice cream.
Getting |r| > 1. That is impossible, so it always signals an error.
On the syllabus: correlation is taught in NCERT's Class 11 Statistics for Economics. The JEE (Main) 2026 syllabus lists mean, median, mode, standard deviation, variance and mean deviation, but not correlation, so check the syllabus for your own exam and year. Our guide to mean and variance covers the standard deviation that Pearson named.
If the idea of covariance is still hazy, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can review it before computing r.
Pearson's legacy, honestly told
MacTutor records that Pearson, with Galton, supported eugenics, and that he was the first Galton Professor of Eugenics at UCL from 1911 to 1933. Eugenics, the idea of "improving" human populations by controlling who has children, has been rejected and was used to justify serious harm. His statistical methods are still in daily use; his eugenic views are not. Both are part of the record. MacTutor also describes his serious disputes with Ronald Fisher, whose story is in our post on Fisher and the design of experiments.
What can a student learn from Pearson?
Pearson insisted on large sets of measured data and exact methods rather than impressions. The useful habit is to measure, compute and then question the number: check the plot, check the arithmetic, and ask what else could explain the pattern.
Practice routine:
Recompute worked example 1 with the last mark changed to 50 and see how r falls.
Make a data set of six points with r close to −1.
Sketch three scatter plots, one each for r near +1, 0 and −1.
Next step: collect five pairs of numbers from your own week, such as hours of sleep and problems solved, and compute r by hand.
