Theory question 111
Measures of DispersionThe frequency distribution shows the marks of 100 students in a Mathematics test.
Marks: 1-10, 11-20, 21-30, 31-40, 41-50, 51-60, 61-70, 71-80, 81-90, 91-100
No. of Students: 2, 4, 9, 13, 18, 32, 13, 5, 3, 1
(a) Draw a cumulative frequency curve for the distribution
(b) Use your curve to estimate; (i) the median; (ii) the lower quartile; (iii) the 60th percentile
Show the model answer
(a) Upper boundaries 10.5, 20.5, …, 100.5 with cumulative frequencies 2, 6, 15, 28, 46, 78, 91, 96, 99, 100; plot (0.5, 0) and these points and draw a smooth curve.
(b)(i) Median at 50: about 50.5 + 4/32 × 10 ≈ 51.8.
(ii) Lower quartile at 25: about 30.5 + 10/13 × 10 ≈ 38.2.
(iii) 60th percentile at 60: about 50.5 + 14/32 × 10 ≈ 54.9.
(Accept values read from the graph within about ±1.)
For 100 students, read across at 25, 50 and 60 on the cumulative frequency axis.