Theory question 108
Measures of Central TendencyMarks: 1, 2, 3, 4, 5
Number of students: m + 2, m − 1, 2m − 3, m + 5, 3m − 4
The table shows the distribution of marks scored by some students in a test. (a) If the mean mark is 36/23, find the value of m. (b) Find the: (i) interquartile range; (ii) probability of selecting a student who scored at least 4 marks in the test.
Show the model answer
Model answer
(a) Σf = 8m − 1; Σfx = (m + 2) + 2(m − 1) + 3(2m − 3) + 4(m + 5) + 5(3m − 4) = 28m − 9
(28m − 9)/(8m − 1) = 75/23 → 644m − 207 = 600m − 75 → m = 3
(b) Frequencies 5, 2, 3, 8, 5 (23 students); cumulative 5, 7, 10, 18, 23
(i) Q₁ = 6th mark = 2; Q₃ = 18th mark = 4; interquartile range = 4 − 2 = 2
(ii) At least 4 marks: 8 + 5 = 13 students; probability = 13/23
Explanation
Write the mean as an improper fraction 75/23 before cross-multiplying.