Comparative Analysis of Examination Performance

The Mathematics Preliminary Examination results for students across schools \(X\) and \(Y\) were recorded. Schools \(X\) and \(Y\) have \(200\) and \(150\) math students respectively.

The following cumulative frequency curves show the distribution of the marks.

Cumulative frequency curve for student's marks in Math Prelim Examination in School X and Y

1. Find the median mark for both schools \(X\) and \(Y\).
2. Calculate the interquartile range for both schools \(X\) and \(Y\).
3. A teacher analysed the data above and commented: "School \(X\) performs better than school \(Y\) because the cumulative frequency curve of \(X\) is always above \(Y\)." Use your results in (a) and (b) to argue if you agree with the teacher's analysis.
4. A distinction grade is given for students who scored \(80\) marks or more.
   
   Which school has a greater percentage distinction for the Math preliminary examination?