Theory question 111
Measures of DispersionThe table below shows the frequency distribution of the marks scored by fifty students in an examination.
Marks (%): 0-9, 10-19, 20-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89, 90-99
Frequency: 2, 3, 4, 6, 13, 10, 5, 3, 2, 2
(a) Draw a cumulative frequency curve for the distribution.
(b) Use your curve to estimate the: (i) upper quartile; (ii) pass mark if 60% of the students passed.
Show the model answer
(a) Upper class boundaries 9.5, 19.5, …, 99.5 with cumulative frequencies 2, 5, 9, 15, 28, 38, 43, 46, 48, 50. Plot them (start at (−0.5, 0)) and draw a smooth ogive.
(b)(i) Upper quartile at the ¾ × 50 = 37.5th student: about 59 marks (49.5 + 9.5/10 × 10)
(ii) 60% passed, so 40% (20 students) failed. The pass mark is the mark at cumulative frequency 20: about 43 (39.5 + 5/13 × 10 = 43.3)
The pass mark is the score below which the failing 40% lie.