{"id":17226,"date":"2026-06-23T05:23:04","date_gmt":"2026-06-23T05:23:04","guid":{"rendered":"https:\/\/www.smilefoundationindia.org\/blog\/?p=17226"},"modified":"2026-06-26T06:03:00","modified_gmt":"2026-06-26T06:03:00","slug":"pc-mahalanobis-distance","status":"publish","type":"post","link":"https:\/\/www.smilefoundationindia.org\/blog\/pc-mahalanobis-distance\/","title":{"rendered":"Understanding PC Mahalanobis’ Contribution to Data Science: Mahalanobis Distance"},"content":{"rendered":"\n

PC Mahalanobis<\/strong> gave the world a statistical tool that is, by almost any measure, more useful today than when he first proposed it in the 1930s. The Mahalanobis distance \u2014 a method for measuring how far a data point lies from a distribution, while accounting for the shape and spread of that data \u2014 has become a foundational technique in modern data science, machine learning and anomaly detection.<\/p>\n\n\n\n

This blog explains what the Mahalanobis distance is, how it works, where it is used and why the man behind it deserves a more prominent place in the history of data science than he typically receives.<\/p>\n\n\n\n

Who is PC Mahalanobis \u2014 Life and Legacy<\/strong><\/h2>\n\n\n\n

Prasanta Chandra Mahalanobis was born on 29 June 1893 in Calcutta, into a family with deep roots in Bengal’s intellectual and reform traditions. He studied physics at Presidency College, Calcutta, where his teachers included Jagadish Chandra Bose, before travelling to King’s College, Cambridge, to continue his education in mathematics and physics.<\/p>\n\n\n\n

It was during this period in England that a passing introduction to statistics changed the course of his life. He returned to India with a new conviction: that rigorous data collection and statistical reasoning could be applied to real, pressing problems in agriculture, anthropology, economics and public policy.<\/p>\n\n\n\n

What followed was a career of extraordinary breadth. PC Mahalanobis<\/strong> founded the Indian Statistical Institute in Calcutta in 1931, established India’s first statistical journal and built the survey methods that laid the groundwork for India’s National Sample Survey \u2014 systems that continue to inform government policy to this day. He also played a central role in shaping India’s Second Five Year Plan, where his macro-economic model prioritised heavy industry investment as the foundation for long-term growth.<\/p>\n\n\n\n

He was elected a Fellow of the Royal Society in 1945, received the Weldon Medal from Oxford University in 1944 and was awarded the Padma Vibhushan in 1968. He died in June 1972, one day before what would have been his seventy-ninth birthday.<\/p>\n\n\n\n

Founding of the Indian Statistical Institute<\/strong><\/h3>\n\n\n\n

The Indian Statistical Institute began in December 1931, growing out of an informal statistical laboratory that Mahalanobis had set up in his own room at Presidency College. Initially registered as a non-profit learned society in April 1932, it operated on a budget of just 238 rupees in its first year.<\/p>\n\n\n\n

Over the following decades, ISI expanded into a globally recognised centre for statistical and quantitative research. In 1933, it launched Sankhya \u2014 India’s first academic statistics journal, still published today \u2014 which carried many of Mahalanobis’s landmark papers, including the original formulation of what would become the Mahalanobis distance.<\/p>\n\n\n\n

ISI’s Kolkata campus became a hub for interdisciplinary research spanning economics, genetics, computer science and sociology. It trained generations of statisticians who went on to shape statistical practice in India and internationally. The Institute remains one of the most rigorous quantitative research institutions in the world.<\/p>\n\n\n\n

What is Mahalanobis Distance \u2014 Simple Explanation<\/strong><\/h2>\n\n\n\n

The Mahalanobis distance<\/a> answers a deceptively simple question: <\/p>\n\n\n\n

how unusual is this data point, given everything we know about the distribution it comes from?<\/p>\n\n\n\n

Imagine you are looking at the heights and weights of a large group of people. You want to know whether a specific individual’s measurements are typical or unusual. The challenge is that height and weight are correlated \u2014 taller people tend to weigh more \u2014 and they are measured in completely different units. A simple measurement of distance from the average would ignore both of these facts.<\/p>\n\n\n\n

The Mahalanobis distance solves this by accounting for the correlation between variables and the scale of each variable simultaneously. It transforms the space in which the measurement exists, stretching and rotating it to reflect the actual shape of the data, and then measures distance within that corrected space. The result is a single number that tells you, in a statistically meaningful way, how far a point lies from the centre of its distribution.<\/p>\n\n\n\n

A Mahalanobis distance of zero means the point is exactly at the mean of the distribution. The larger the value, the more unusual the observation \u2014 regardless of how many variables are involved.<\/p>\n\n\n\n

The Formula and How It Works<\/strong><\/h3>\n\n\n\n
\"PC<\/figure>\n\n\n\n

The Mahalanobis distance between a point x and a distribution with mean vector mu and covariance matrix S is defined as:<\/p>\n\n\n\n

D(x) = sqrt [ (x – mu)^T * S^-1 * (x – mu) ]<\/p>\n\n\n\n

Breaking this down:<\/p>\n\n\n\n