On this page
Every time a bank flags an odd transaction or a model spots an outlier, there is a good chance it is using an idea named after a physicist from Kolkata. Prasanta Chandra Mahalanobis trained in the physical sciences, turned to statistics, and then built the institutions that made statistics part of how India plans. This post covers what he did, explains his distance with a worked example, and draws out what a student can learn from how he worked.
Who was P. C. Mahalanobis?
The Indian Statistical Institute's page on its founder gives his dates as 29 June 1893 to 28 June 1972. He was born in Kolkata, studied at Presidency College and then at the University of Cambridge, and his interest moved from physics to statistics.
He set up a Statistical Laboratory at Presidency College in the 1920s. According to the ISI timeline, the Indian Statistical Institute was founded there on 17 December 1931 and registered as a learned society on 28 April 1932. The same page says that into the mid-1930s nearly all statistical work in India was done by Mahalanobis himself, on problems from rainfall and floods to soil and agriculture. Later, scholars including R. C. Bose, S. N. Roy and C. R. Rao joined him. In 1959 Parliament passed the Indian Statistical Institute Act, declaring the institute "an institution of national importance". He remained its director, and editor of its journal Sankhyā, until his death.
What is the Mahalanobis distance?
In 1936 he published "On the generalised distance in statistics", reprinted in Sankhyā A in 2018. The ISI describes the resulting Mahalanobis distance as a measure of the distance between a point and a distribution that, unlike simple distance measures, "accounts for correlations between variables".
Worked example. Suppose height and weight in a group of students are correlated, with correlation 0.8. Measure each in standard deviations from the mean (z-scores). Two students:
Student A is 1 SD taller and 1 SD heavier than average: z = (1, 1).
Student B is 1 SD taller and 1 SD lighter than average: z = (1, −1).
Ordinary distance treats them alike: both are √2 ≈ 1.41 from the mean. For two variables with correlation ρ, the squared Mahalanobis distance is
D² = (z₁² + z₂² − 2ρz₁z₂) / (1 − ρ²).
A: D² = (1 + 1 − 1.6) / 0.36 = 0.4 / 0.36 ≈ 1.11, so D ≈ 1.05.
B: D² = (1 + 1 + 1.6) / 0.36 = 3.6 / 0.36 = 10, so D ≈ 3.16.
In this group taller students are usually heavier, so A fits the pattern and B does not. The Mahalanobis distance captures that; ordinary distance does not. Standard deviation and correlation are both Class 11 and 12 ideas, covered in our guide to mean, variance and standard deviation.
Why did his sample surveys matter?
India in the 1930s and 1940s needed reliable figures on crops and land, and a full census of every field was too slow and costly. Mahalanobis designed large-scale sample surveys instead. His long paper On large-scale sample surveys, published in the Philosophical Transactions of the Royal Society in 1944, states the planning problem directly: choose the size and number of sample units so that "the precision is a maximum for any assigned cost, or ... the cost is a minimum for any assigned precision".
The Royal Society records his election as a Fellow on 22 March 1945, and the ISI timeline says it was his research on sample surveys that won him the fellowship. The ISI also credits his methods with laying the foundation for India's National Sample Survey.
What did he do for planning?
After independence, the Government of India gave the ISI the task of preparing the draft Second Five-Year Plan, with Mahalanobis as its chief architect, according to the ISI timeline; the model behind it became known as the Mahalanobis growth model. In his honour, the Government of India designated his birthday, 29 June, as Statistics Day, to inspire younger people about the role of statistics in planning and policy.
What can a student take from how he worked?
Move between fields. He came to statistics with training in physics, and he used it on rainfall, floods and crops. Your physics and maths are tools for problems outside the exam too.
State the trade-off. His survey paper frames the task as precision against cost. When you plan revision, ask the same question: which use of the next hour cuts the most uncertainty?
Build things that last. He started with a single laboratory in a college. Habits work the same way: a small weekly routine kept for years beats a burst of effort.
Ask what normal looks like. His distance asks whether a point fits the pattern of the data. When an answer seems strange, check it against what you would expect.
To see where his ideas went next, read our profile of C. R. Rao, who joined him at the ISI: C. R. Rao and modern statistics. If you want to understand correlation and standard deviation properly before reading him, the Fermi Concept Library covers every concept in mathematics with its own lesson.
Try the worked example again with a correlation of 0.5 instead of 0.8, and see how the two distances change.
