Cumulative frequency graphs

Median, quartiles, percentiles and counts

In 3 of 20 recent exams
Paper 1, no calculator
1 to 4 marks

How it is set in the exam

Paper 1 (no calculator) mostly, 1–4 marks a part, usually as a whole Section B question built on one graph. Typical (3 of 20 exam versions, 2 of them on Paper 1 (no calculator); 12 logged parts).

Spot it in the exam

Exam phrases to spot (5)

“use the graph to estimate …”
“the top $p$ % … find the least …”
“complete the frequency table”
“estimate how many … in the whole population”
“how many reached at least a given percentage of the maximum”

In disguise (2)

a percentile named by number ("the 90th percentile"): read across from $90$ % of $n$
a threshold given as a fraction of a capacity or of the largest value: turn it into a value first, then read up

Variants

asked: the median, a quartile, a percentile from the upper tail, a count above a threshold, a count in an interval (the difference of two readings), the missing frequencies of a grouped table, a sample proportion scaled up to a population

Looks like it, but isn't (2)

"the top $p$ %" of a normal model: Find the value from a probability (inverse normal, not a graph
quartiles read off a box plot: Box plots

Worked example

The median, the top 15 %, and a count above a threshold

Paper 1 (no calculator)

Question

The distance, $d$ km, walked by each of the $120$ people who took part in a sponsored charity walk is recorded. The cumulative frequency graph of the distances is a curve through the points in this table.

$d$ (km)	Cumulative frequency
$0$	$0$
$5$	$18$
$10$	$45$
$15$	$60$
$20$	$84$
$25$	$102$
$30$	$111$
$40$	$118$
$50$	$120$

$d$ (km)	$0$	$5$	$10$	$15$	$20$	$25$	$30$	$40$	$50$
Cumulative frequency	$0$	$18$	$45$	$60$	$84$	$102$	$111$	$118$	$120$

(a) Use the graph to find the median distance. [2]
(b) The $15$ % of walkers who walked furthest receive a certificate. Find the least distance walked by a walker who receives a certificate. [3]
(c) The longest distance walked was $50$ km. Walkers who walked more than $40$ % of the longest distance receive a free T-shirt. Find the number of walkers who receive a T-shirt. [3]

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Steps

Decide the direction.
A count → a value: read across from the count, then down. A value → a count: read up from the value, then across. The graph always counts the values below.
Turn the request into a cumulative count:
the median at $\frac{n}{2}$ , $Q_1$ at $\frac{n}{4}$ , $Q_3$ at $\frac{3n}{4}$ . For "the top $p$ %", the count below the cut-off is $n - \frac{p}{100}n$ . Show the count (or a line on the graph): that is the method mark. For the median, markschemes accept $\frac{n}{2}$ or $\frac{n+1}{2}$ (Averages uses $\frac{n+1}{2}$ for a short list; on a graph either works), and a reading anywhere in a small range; a value between plotted points is read off the curve.
Counts above a value
are $n$ minus the reading.
Table frequencies
are differences of the readings at the class boundaries; the last one can come from the total.
Scale up to a population:
the proportion in the sample $\times$ the population size.

Common mistakes

"The top $p$ %" read at $p$ % of $n$
that finds the bottom $p$ %. The count below the cut-off is $n - \frac{p}{100}n$ .
Giving the reading as the count above
the graph counts below. The number above is $n$ minus the reading.
Plotting at the midpoints
of the classes: cumulative frequencies go at the upper boundaries.

More mistakes (2)

Copying cumulative frequencies into a frequency table
the frequencies are the differences.
Scaling with the wrong total
the proportion is out of the sample ( $200$ ); the answer is out of the population ( $5000$ ).

Easy to confuse with

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