Week 8 · Quartiles from a frequency table: ascending cumulative frequency, the quartile class and the formula, and the graphical method · 3 lessons
3
Finding quartiles from the cumulative frequency curve and checking the graph against the formula
Statistics Sec 3 Term 1, Unit 2, Lesson 2
GoalDraw the ascending cumulative frequency curve, read the quartiles from it and check the readings with the formula.
New words
cumulative frequency curve · the points (upper boundary, cumulative frequency) joined by straight segmentsreading · from the height of the order go across to the curve, then down to the horizontal axis
Explanation
The ascending cumulative frequency curve is drawn from the cumulative table. Each point has the upper boundary of a class as its x-value and the cumulative frequency up to that boundary as its y-value, and the first point is the lowest boundary with height 0. Join the points by straight segments. To find a quartile: mark its order on the vertical axis (n ÷ 4, n ÷ 2, 3n ÷ 4), go across to the curve, then straight down to the horizontal axis, and read the value. The picture shows 40 parcels weighed in a post office (invented data): classes of width 2 kg with frequencies 4, 8, 12, 9, 5 and 2. The orders are 10, 20 and 30. A reading from a drawing is only about right, to the nearest tenth or so, so we check with the formula; the curve and the formula agree because both treat the values as evenly spread inside each class.
Worked examples
Drawing the curve. The points are (0, 0), (2, 4), (4, 12), (6, 24), (8, 33), (10, 38) and (12, 40). The curve only rises (or stays level), and it ends at the height n = 40.
Reading. At the height 10 the curve is above x = 3.5; at 20 above x = 5.3; at 30 above x = 7.3. So Q₁ ≈ 3.5 kg, Q₂ ≈ 5.3 kg and Q₃ ≈ 7.3 kg. The sign ≈ means “about”.
Checking by the formula. Q₁: order 10 in the class 2– (4 before, frequency 8): 2 + (10 − 4) ÷ 8 × 2 = 3.5. Q₂: order 20 in the class 4– (12 before, frequency 12): 4 + (20 − 12) ÷ 12 × 2 = 5.33. Q₃: order 30 in the class 6– (24 before, frequency 9): 6 + (30 − 24) ÷ 9 × 2 = 7.33. They agree with the readings.
Class activity
Take the 40 school-journey times of the earlier activity. Draw the cumulative curve on squared paper, read the three quartiles, then check them with the formula and say whether your readings were within 0.5 minute.
Practice
1
For the classes 0–, 10–, 20–, 30–, 40–, 50–60 with frequencies 3, 9, 18, 15, 10, 5, which points are plotted?
