Marks vs. Percentile data of the JEE (Main): A study to assess the preparation level of the examinees

In the present study, I have carried out a mathematical formulation to find a simple way to determine the mean and standard deviation of marks and also the most probable marks obtained by examinees in an examination like the Joint Entrance Examination - Main [or, JEE (Main)], using Marks vs. Percentile data. Observing the nature of variation of the percentile score as a function of marks, based on information obtained from examinees, I have chosen an empirical expression for the percentile score (in terms of marks) for this study. This expression represents the behavior of real data, with sufficient accuracy, for a certain combination of values of the constant parameters associated with the expression. A parameter among them has been identified as one representing the difficulty level of the question paper. Using my empirical expression for percentile score, I have derived an expression for the probability of scoring marks of a certain value. I have defined a cumulative probability here, which represents the probability of scoring marks equal to or more than a certain value. Findings of this study have been depicted graphically. The mathematical expressions, chosen or derived in this study, can be used for making predictions whose accuracy depends upon the largeness of the sample of Marks vs. Percentile data used for determination of constant parameters.