Theory question 112
Measures of DispersionThe table below shows the frequency distribution of the marks of 800 candidates 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: 10, 40, 80, 140, 170, 130, 100, 70, 40, 20
(a)(i) Construct a cumulative frequency table, (ii) Draw the ogive: (iii) Use your ogive to determine the 50th percentile: (b) The candidates that scored less than 25% are to be withdrawn from the institution, while those that scored more than 75% are to be awarded scholarship. Estimate the number of candidates that will be retained, but who will not enjoy the award.
Show the model answer
(a)(i) Upper class boundaries 9.5, 19.5, …, 99.5 with cumulative frequencies 10, 50, 130, 270, 440, 570, 670, 740, 780, 800.
(ii) Plot cumulative frequency against upper class boundary and join with a smooth S-shaped curve.
(iii) 50th percentile = median = the 400th candidate: about 39.5 + (130/170) × 10 ≈ 47 marks.
(b) From the ogive: about 94 candidates scored below 25%; about 709 scored up to 75%, so about 91 scored above 75%.
Retained without award ≈ 709 − 94 ≈ 615 candidates (accept roughly 600–630 from the graph).
Plot against upper class boundaries, not class midpoints.