The answer
(a)(i) median \(\approx 115\) cm
(ii) IQR \(\approx 20\) cm
(b) \(h \approx 121\) cm
(c)(i) \(115.75\) cm
(ii) \(12.1\) cm
O-Level E-Math 2023 Paper 1 Question 8 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 8 of the O-Level E-Math 2023 Paper 1. It tests cumulative frequency curve (median, in the Dispersion / Central tendency area. It is worth 6 marks: (a) 1 + 2, (b) 1, (c) 1 + 1. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
The cumulative frequencies at \(100, 110, 120, 130, 140\) are \(3, 16, 24, 34, 40\).
(a)(i) Median \(=\) value at \(\mathrm{CF} = 20\). Reading the curve at \(\mathrm{CF}=20\) gives \(\approx 115\) cm.
(ii) Lower quartile at \(\mathrm{CF} = 10\): \(\approx 106\) cm; upper quartile at \(\mathrm{CF} = 30\): \(\approx 126\) cm. IQR \(\approx 126 - 106 = 20\) cm. *(The annotated key reads \(\approx 19\) cm; both are acceptable graph estimates.)*
(b) "Only 15 allowed" means 15 children have height \(\geqslant h\), so \(40 - 15 = 25\) have height \(< h\). Reading the curve at \(\mathrm{CF} = 25\) gives \(h \approx 121\) cm. *(The annotated key reads \(122\) cm, within graph tolerance.)*
(c)(i) Using midpoints \(95, 105, 115, 125, 135\): mean \(= \dfrac{3(95) + 13(105) + 8(115) + 10(125) + 6(135)}{40} = \dfrac{4630}{40} = 115.75\) cm.
(ii) \(\text{s.d.} = \sqrt{\dfrac{\sum f x^2}{40} - \bar{x}^2} = \sqrt{\dfrac{541\,800}{40} - 115.75^2} = \sqrt{13\,545 - 13\,398.06} = \sqrt{146.94} = 12.1\) cm (3 s.f.).
Answer: (a)(i) median \(\approx 115\) cm
(ii) IQR \(\approx 20\) cm
(b) \(h \approx 121\) cm
(c)(i) \(115.75\) cm
(ii) \(12.1\) cm
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a cumulative frequency curve (median question from Dispersion / Central tendency, worth 6 marks: (a) 1 + 2, (b) 1, (c) 1 + 1.
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