CAT 2016 Result: IIM-Bangalore may have done the Math Wrong!

The CAT 2022 Result was announced on Monday by the Indian Institute of Management (IIM), Bangalore. Close to 195,000 students took the Common Admission Test 2022, of which 20 candidates secured '100 percentile.' Given that the IIMs award percentiles up to two decimal places, it is a mathematical oddity that 20 out of 195,000 students secured 100 percentile.

'Percentile' is defined as the percentage of data points that have a value strictly less than the given data point. Hence, 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. As per 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.

Therefore, the way to interpret the '100 percentile' that IIMs have been releasing since 2003 is in terms of rounding off. Since a large number of aspirants write CAT each year, IIMs need to divide the applicants into over a 100 categories. For this, they use decimals to further distinguish between candidates. 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.

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 CAT exam (anything lower gets rounded off to a number lower than 100.00). In other words, not more than 0.005% of all the candidates can mathematically get 100.00 percentile. Since 195,000 candidates took CAT in 2022, this number cannot be more than 10 (0.005% of 195,000 is 9.75).

So according to this, it's very unusual that 20 candidates have secured '100 percentile.' The only way this can be possible (mathematically) is if there was an 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 like CAT), the likelihood of an 11-way tie for the 10th spot is almost impossible.

It seems like IIM-Bangalore (which conducted 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). This is rather unfortunate, considering how much emphasis IIMs place on quantitative ability among incoming students.