IGCSE · Practice questions · Cumulative frequency diagrams

Read across, read down.

Extended. Six original Cambridge-style questions on cumulative frequency tables and curves.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Running totals at upper boundaries.

Table: add the frequencies as you go; the last total is \(n\).
Plot each total at the upper class boundary, as a cross, and join with a smooth curve.
Read across at \(\dfrac{n}{4}\), \(\dfrac{n}{2}\), \(\dfrac{3n}{4}\) (or \(p\%\) of \(n\)), then down.
IQR = UQ \(-\) LQ, both read from the horizontal axis.
More than: total \(-\) reading.

Recall

[2 marks]

Complete the cumulative frequency table for the masses of 50 parcels.

Mass \(m\) (kg)	\(0 < m \le 20\)	\(20 < m \le 40\)	\(40 < m \le 60\)	\(60 < m \le 80\)	\(80 < m \le 100\)
Frequency	5	13	22	7	3

Cumulative frequencies: 5, 18, 40, 47, 50. ✓
Check: the last value equals the total, 50. ✓

Calculation

[2 marks]

Write down the coordinates of the points you would plot for a cumulative frequency curve of the parcels in the previous question.

(0, 0), (20, 5), (40, 18), (60, 40), (80, 47), (100, 50). ✓
Each total goes at the upper boundary of its class. ✓

Calculation

[3 marks]

A cumulative frequency curve for the heights of 120 plants passes through these points.

Height (cm)	10	15	20	25	30	35	40
Cumulative frequency	0	12	30	60	90	108	120

Find the median and the interquartile range.

Median: read at \(\dfrac{120}{2} = 60\), giving 25 cm. ✓
LQ: read at 30, giving 20 cm. UQ: read at 90, giving 30 cm. ✓
IQR \(= 30 - 20 = 10\) cm. ✓

Analysis

[2 marks]

For the plants in the previous question, find (a) the 90th percentile (b) the number of plants taller than 35 cm.

(a) 90% of 120 = 108, which the curve reaches at 35 cm. ✓
(b) 108 plants are 35 cm or shorter, so \(120 - 108 = 12\) are taller. ✓

Analysis

[2 marks]

A second batch of plants, grown in shade, has median 28 cm and IQR 16 cm. Compare the two batches, using the results from the earlier question (median 25 cm, IQR 10 cm).

The shade plants are taller on average (median 28 cm against 25 cm). ✓
The first batch is more consistent in height (IQR 10 cm against 16 cm). ✓

Analysis

[4 marks]

The cumulative frequency curve for the masses of 200 apples passes through (100, 15), (120, 62), (140, 130), (160, 178) and (180, 200), where the first number is the mass in grams.

Apples heavier than 140 g are graded 'large'.

(a) How many apples are large? (b) What percentage of the apples are large? (c) How many apples have a mass between 120 g and 160 g?

(a) 130 apples are 140 g or less, so \(200 - 130 = 70\) are large. ✓
(b) \(\dfrac{70}{200} \times 100 = 35\%\). ✓
(c) \(178 - 62 = 116\) apples. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.