Sec 4 · G1 (N(T))

Statistics

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1. Cumulative frequency

Marks	Frequency	Cumulative frequency
$0 < x \leq 10$	$2$	$2$
$10 < x \leq 20$	$6$	$8$
$20 < x \leq 30$	$14$	$22$
$30 < x \leq 40$	$22$	$44$
$40 < x \leq 50$	$11$	$55$
$50 < x \leq 60$	$5$	$60$

Cumulative frequency curve for 60 students, plotted at the upper end of each class. Reading across from 15, 30 and 45 gives the lower quartile about 25.7, the median about 33.7 and the upper quartile about 40.6.

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students in a test. Use the curve to estimate the number of students who scored more than $15$ marks.

Step 1 of 2

Go up from the mark on the horizontal axis to the curve, then across

The curve gives about

4

students with at most

15

marks.

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2. Median, quartiles and percentiles

From a curve ( $60$ values)	Read across from	Here
lower quartile $Q_1$	$\dfrac{1}{4} \times 60$ , so $15$	$25.7$
median	$\dfrac{1}{2} \times 60$ , so $30$	$33.7$
upper quartile $Q_3$	$\dfrac{3}{4} \times 60$ , so $45$	$40.6$
interquartile range	$Q_3 - Q_1$	$14.9$
$p$ th percentile	$\dfrac{p}{100} \times 60$	80th: across from $48$

From a list: 39, 42, 44, 46, 48, 50, 52, 55, 57, 61

\displaystyle n = 10

median: between the 5th and 6th, $\dfrac{48 + 50}{2}$ , so $49$
lower half $39, 42, \mathbf{44}, 46, 48$ : $Q_1 = 44$
upper half $50, 52, \mathbf{55}, 57, 61$ : $Q_3 = 55$

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students. Use the curve to estimate the median mark.

Step 1 of 2

Which cumulative frequency do you go across from?

\mathstrut \dfrac{60}{2} = 30

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1. Cumulative frequency

Marks	Frequency	Cumulative frequency
$0 < x \leq 10$	$2$	$2$
$10 < x \leq 20$	$6$	$8$
$20 < x \leq 30$	$14$	$22$
$30 < x \leq 40$	$22$	$44$
$40 < x \leq 50$	$11$	$55$
$50 < x \leq 60$	$5$	$60$

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students in a test. Use the curve to estimate the number of students who scored more than $15$ marks.

Step 1 of 2

Go up from the mark on the horizontal axis to the curve, then across

The curve gives about

4

students with at most

15

marks.

Practise this →

2. Median, quartiles and percentiles

From a curve ( $60$ values)	Read across from	Here
lower quartile $Q_1$	$\dfrac{1}{4} \times 60$ , so $15$	$25.7$
median	$\dfrac{1}{2} \times 60$ , so $30$	$33.7$
upper quartile $Q_3$	$\dfrac{3}{4} \times 60$ , so $45$	$40.6$
interquartile range	$Q_3 - Q_1$	$14.9$
$p$ th percentile	$\dfrac{p}{100} \times 60$	80th: across from $48$

From a list: 39, 42, 44, 46, 48, 50, 52, 55, 57, 61

\displaystyle n = 10

median: between the 5th and 6th, $\dfrac{48 + 50}{2}$ , so $49$
lower half $39, 42, \mathbf{44}, 46, 48$ : $Q_1 = 44$
upper half $50, 52, \mathbf{55}, 57, 61$ : $Q_3 = 55$

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students. Use the curve to estimate the median mark.

Step 1 of 2

Which cumulative frequency do you go across from?

\mathstrut \dfrac{60}{2} = 30

Practise this →