Cumulative Frequency

Statistics

Median, quartiles and percentiles

Everything read off a cumulative frequency diagram is an estimate, because the original values are unknown
Find the total, n, from the highest point the curve reaches, at the top right

Reading the median and quartiles off a cumulative frequency curve of 100 phone calls: horizontal red lines drawn across at 25, 50 and 75 on the cumulative frequency axis, each dropping down to the time axis to give a lower quartile of 4.2 minutes, a median of 6.2 and an upper quartile of 8.2 — Source: Interpreting cumulative frequency diagrams by Save My Exams

Then work from the cumulative frequency axis across to the curve and down to the horizontal axis

Statistic	Position on the cumulative frequency axis
Lower quartile	n ÷ 4
Median	n ÷ 2
Upper quartile	3n ÷ 4
p-th percentile	p ÷ 100 × n

Note that the median position here is n ÷ 2, not (n + 1) ÷ 2 as it is for a list of values, because the curve treats the data as spread smoothly
The interquartile range is the upper quartile minus the lower quartile, both read off in the same way
Draw the horizontal and vertical lines on the diagram, since they show where the reading came from

Exam tip

Plot each point at the upper bound of its class, never the midpoint — that single mistake ruins every reading afterwards. Join with a smooth curve, not straight lines. State the position you read from on the cumulative frequency axis (30 for the median of 60 values) and leave the lines you drew on the diagram, because they are the working. On a curve the median sits at n ÷ 2, not the (n + 1) ÷ 2 used for a list.

Worked example

Quartiles from a cumulative frequency diagram

The table in section 1 records the times of 60 people. A cumulative frequency curve is drawn from it. Explain how to estimate the median, the quartiles and the interquartile range.

Solution:

The total is n = 60, which is where the curve finishes at the top right
The lower quartile is at 60 ÷ 4 = 15 on the cumulative frequency axis
The median is at 60 ÷ 2 = 30
The upper quartile is at 3 × 60 ÷ 4 = 45
For each one, draw a horizontal line from that value across to the curve, then a vertical line down to the time axis and read the value
The cumulative frequency reaches 15 exactly at the end of the second class, so the lower quartile is 20 minutes
The reading for 30 falls between 20 and 30 minutes, giving a median of roughly 27 minutes
The reading for 45 falls between 30 and 40 minutes, giving an upper quartile of roughly 36 minutes
Interquartile range = upper quartile − lower quartile, so about 36 − 20 = 16 minutes

How many are above or below a value

To find how many are below a given value, go up from that value on the horizontal axis to the curve, then across to read the cumulative frequency
To find how many are above it instead, subtract that reading from the total

The same curve used the other way round: a red line goes up from 12 minutes on the time axis to the curve and across to 90 on the cumulative frequency axis, so 90 of the 100 calls were shorter than 12 minutes and the remaining 10 were longer — Source: Interpreting cumulative frequency diagrams by Save My Exams

Read the question carefully: "more than", "at least" and "fewer than" all need the same reading but different final steps

Worked example

How many took longer than a given time

Using the same curve for 60 people, estimate how many took longer than 35 minutes.

Solution:

Start at 35 on the horizontal time axis and go straight up to the curve
Now read horizontally to the vertical scale and note the running total there
35 minutes lies halfway between the points (30, 36) and (40, 52), so the reading is roughly 44
That means about 44 people took 35 minutes or less
The question asks for longer than 35 minutes, so subtract from the total
60 − 44 = about 16 people
Always check whether the question wants the reading itself or the total minus the reading