Statistics - Percentiles and Quartiles from Cumulative Frequency Curves

The cumulative frequency curve below illustrates the marks obtained by \(120\) students in a Science test for Class \(A\).

Cumulative frequency curve for Science test marks

1. Use the graph to find,
   1. the \(80^{th}\) percentile.
   2. the interquartile range.
      In this test, only \(15\)% of the students scored a distinction.
   3. Find the least marks for a student to be considered for distinction.
      Two student are chosen at random.
   4. Find the probability that both the students scored above \(80\) marks.
2. The cumulative frequency curve can be represented using the following box-and-whisker diagram.
   
   1. State the value of \(x_{3}\).
   2. State the value of \(x_{5}-x_{1}\). What does \(x_{5}-x_{1}\) represent?
3. The box-and-whisker diagram below illustrates the marks obtained by \(120\) students in a Science test for Class \(B\).
   
   Compare the test results for the two classes in two different ways.
4. The results of the science test of Class \(C\) of \(120\) students have the same median but a smaller standard deviation. Describe how the cumulative frequency curve will differ from the given curve.