Theory question 114
StatisticsThe table shows the frequency distribution of marks scored by some 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: 2, 5, 8, 18, 20, 15, 5, 4, 2, 1
(a) Draw the cumulative frequency curve of the distribution.
(b) Use your graph to estimate the: (i) Semi − interquartile range of the distribution; (ii) percentage of candidates who passed with distinction if the least mark for distinction was 72.
Show the model answer
(a) Upper class boundaries 9.5, 19.5, ..., 99.5 against cumulative frequencies 2, 7, 15, 33, 53, 68, 73, 77, 79, 80; join with a smooth curve
(b)(i) N = 80: Q1 = 20th mark ≈ 29.5 + (5/18) × 10 ≈ 32.3; Q3 = 60th mark ≈ 49.5 + (7/15) × 10 ≈ 54.2
Semi-interquartile range = (54.2 − 32.3)/2 ≈ 10.9 (accept about 10.5 to 11 from the graph)
(ii) Read the cumulative frequency at 71.5: about 73.8, so about 80 − 73.8 ≈ 6 candidates scored 72 or more
Percentage ≈ 6.2/80 × 100 ≈ 7.8% (about 8%)
Quartiles are at N/4 and 3N/4 on the cumulative frequency axis.