The results of the Common Admission Test (CAT), used for admissions to the Indian Institutes of Management (IIMs) and other business schools, were announced on Monday. Some 195,000 students took the test, according to a statement by the Indian Institute of Management-Bangalore, out of which 20 candidates were awarded “100 percentile". Given that IIMs award percentiles up to two decimal places, 20 out of 195,000 students getting “100 percentile" is a mathematical oddity.

“Percentile" is defined as the percentage of data points that have a value strictly less than the given data point. So a data point is in the “50th percentile" if 50% of all data points in the dataset lie below it. A data point in the 99th percentile has 99% of all data points below it. From this point of view, the concept of “100 percentile" is unusual, since it is impossible that 100% of all data points lie strictly below a particular data point.

Thus, the way to interpret the “100 percentile" that IIMs have been releasing since 2003 is in terms of rounding off. Thanks to the large number of candidates that write CAT each year, IIMs need to divide the applicants into more than a 100 categories, so they use decimals to further distinguish between candidates (normally, decimals are seldom used with percentiles. “50th percentile" is usual, but “50.1 percentile" is undefined outside of CAT scorecards). Traditionally, IIMs have used two places of decimals for this purpose. And the way to calculate a value to two places of decimals is to calculate it to three places, and then round off the least significant digit.

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Looking at it this way, we can define “100 percentile" as candidates who have done better than at least 99.995% of all candidates who took the test (anything lower gets rounded off to a number lower than 100.00). In other words, not more than 0.005% of all test takers can mathematically get 100.00 percentile. And given that 195,000 candidates took CAT in 2016, this number cannot be more than 10 (0.005% of 195,000 is 9.75).

From this point of view, it is bizarre that 20 candidates who took the test were awarded “100 percentile". The only way this were to be mathematically possible is if there was a 11-way tie for the 10th position in the exam. However, considering how thin the “upper tail" of the distribution of test scores generally are (especially for a tough examination such as CAT), the likelihood of a 11-way tie for the 10th spot is extremely low.

In other words, it seems like IIM-Bangalore (which has been responsible for conducting CAT this year) has messed up on the formula to calculate percentiles (the most plausible explanation is that everyone above 99.990 has been rounded up to 100, but that is mathematically wrong). Considering how much emphasis IIMs place on quantitative ability among incoming students, this is rather unfortunate.

Karthik Shashidhar is an alumnus of IIM-Bangalore (batch of 2006), and had secured “100 percentile" in CAT 2003-04.

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