Cumulative Frequency Diagrams

Cumulative Frequency

What is cumulative frequency?

Cumulative means “running total" or "adding up as you go along”
So in a table of grouped data cumulative frequency means all of the frequencies for the different groups totalled up to the end of that group
When working out cumulative frequencies you may see tables presented in two ways
- A regular grouped data table with an extra column for cumulative frequencies
  - e.g. rows labelled 0 ≤ x < 20, 20 ≤ x < 40, 40 ≤ x < 60, etc
- or a separate table where every group is relabelled as starting at the beginning (often zero)
  - e.g. rows labelled x < 20, x < 40, x < 60, etc

Drawing Cumulative Frequency Diagrams

How do I draw a cumulative frequency graph?

This is best explained with an example
- The times taken to complete a short general knowledge quiz taken by 50 students are shown in the table below:

Time taken ( $s$ seconds)	Frequency
$25 \leq s < 30$	3
$30 \leq s < 35$	8
$35 \leq s < 40$	17
$40 \leq s < 45$	12
$45 \leq s < 50$	7
$50 \leq s < 55$	3
Total	50

- Then the cumulative frequency is the running total of the frequencies

Time taken ( $s$ seconds)	Frequency
$s < 30$	3
$s < 35$	3 + 8 = 11
$s < 40$	11 + 17 = 28
$s < 45$	28 + 12 = 40
$s < 50$	40 + 7 = 47
$s < 55$	47 + 3 = 50
Total	50

We can now draw the cumulative frequency graph
- The most crucial part is that cumulative frequency is plotted against the end ( upper bound) of the class interval
  - For the above example the first two points to plot would be (30, 3) and (35, 11)
  - To explain this, consider the second row ( $30 \leq s < 35$ )
    the 11 students in this group could have taken any time between 30 and 35 seconds
    they cannot all be guaranteed to have been accounted for until we reach 35 seconds
- Once all points from the table are plotted, a point for the start needs adding
  - this will be at the lowest time from the table - so at 25 seconds
    with a cumulative frequency of 0
    so plot the point (25, 0)
- Join points up with a smooth curve (this takes some practice)
  - in general a cumulative frequency graph has a stretched-S-shape appearance
  - a cumulative frequency graph will never come back towards the x-axis
Here is the final cumulative frequency graph for the quiz times

Interpreting Cumulative Frequency Diagrams

How do I use and interpret a cumulative frequency graph?

A cumulative frequency graph provides a way to estimate key facts about the data
- median, lower and upper quartiles and interquartile range
- percentiles
These values will be estimates as the original raw data is unknown
- cumulative frequency graphs are used with grouped (summarised) data
- points are joined by a smooth curve meaning the data is assumed to be smoothly spread out over each interval
The median and quartiles are also key features of a box plot (aka box-and-whisker diagram)
- It is possible to draw a box plot from a cumulative frequency graph
- This may allow for an easier comparison of two data sets
  - e.g. Comparing male data and female data

How do I find the median, lower quartile and upper quartile from a cumulative frequency graph?

This is all about understanding how many data values are in represented by the cumulative frequency graph
- This may be stated in words within the question
- If not, it will be the highest value on the frequency ( y-) axis that the graph reaches
  - This should be "top right" on a cumulative frequency graph as they have that general stretched-S shape
MEDIAN
STEP 1
Find the position of the median
For $n$ data values, this will be $\frac{n}{2}$
(You do not need to worry about $n$ being odd or even, nor using $n + 1$ as you may have seen elsewhere. Cumulative frequency graphs are used for large data sets and our answer will be an estimate anyway)
STEP 2
Draw a horizontal line from $\frac{n}{2}$ on the cumulative frequency axis until it hits the curve
STEP 3
Draw a vertical line from the curve down to the horizontal ( x-) axis and take a reading
This reading will be the median
LOWER QUARTILE
STEP 1
Find the position of the lower quartile using $\frac{n}{4}$
STEP 2
Draw a horizontal line from $\frac{n}{4}$ on the cumulative frequency axis until it hits the curve
STEP 3
Draw a vertical line from the curve to the horizontal ( x-) axis and take a reading
This reading will be the lower quartile
UPPER QUARTILE
STEP 1
Find the position of the upper quartile using $\frac{3 n}{4} (i . e . 3 \times \frac{n}{4})$
STEP 2
Draw a horizontal line from $\frac{3 n}{4}$ on the cumulative frequency axis until it hits the curve
STEP 3
Draw a vertical line from the curve to the horizontal ( x-) axis and take a reading
This reading will be the upper quartile

How do I find a percentile from a cumulative frequency graph?

Percentiles split the data into 100 parts
- So the 50 ^th percentile is another way of describing the median
- The 25 ^th and 75 ^th percentiles would be the lower and upper quartiles (respectively)
^{$p^{th}$} PERCENTILE
STEP 1
Find the position of the percentile
For $n$ data values, this will be $\frac{n p}{100} (i . e . \frac{n}{100} \times p)$
STEP 2
Draw a horizontal line from $\frac{n p}{100}$ on the cumulative frequency axis until it hits the curve
STEP 3
Draw a vertical line from the curve down to the horizontal ( x-) axis and take a reading
This reading will be the $p^{th}$ percentile

Worked example

A company is investigating the length of telephone alls customers make to its help centre.
The company randomly selects 100 phone calls from a particular day and the results are displayed in the cumulative frequency graph below.

(a)

Estimate the median, the lower quartile and the upper quartile.

There are 100 pieces of data, so $n = 100$ .

$\begin{array}{rcl} \frac{n}{2} & = & \frac{100}{2} = 50 \\ \frac{n}{4} & = & \frac{100}{4} = 25 \\ \frac{3 n}{4} & = & 3 \times 25 = 75 \end{array}$

So the median is the 50 ^th value, the lower quartile the 25 ^th value and the upper quartile the 75 ^th value.
Draw horizontal lines from these on the cumulative frequency axis until they hit the curve, then draw vertical lines down to the time of calls axis and take readings.

Median = 6.2 minutes (6 m 12 s)
Lower quartile = 4.2 minutes (4 m 12 s)
Upper quartile = 8.2 minutes (8 m 12 s)

There is no need to convert to minutes and seconds unless the question asks you to.
However, writing 6 m 2 s or 6 m 20 s would be incorrect.

(b)

The company is thinking of putting an upper limit of 12 minutes on calls to its help centre.
Estimate the number of these 100 calls would have been beyond this limit?

Draw a vertical line up from 12 minutes on the time of calls axis until it hits the curve.
Then draw a horizontal line across to the cumulative frequency axis and take a reading, in this case, 90.
This tells us that up to 12 minutes, 90 of the calls had been accounted for.
The question wants the number of calls that were greater than 12 minutes so subtract this from the total of 100.

100 - 90 = 10

10 (out of 100) calls were beyond the 12 minute limit

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