Cumulative Frequency and Boxplots

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Cumulative Frequency and Boxplots Revision

Cumulative Frequency and Boxplots

A cumulative frequency graph can be used to estimate the interquartile range.

A boxplot is a graph that shows the median, quartiles, highest/lowest values and outliers.

Make sure you are happy with the following topics before continuing.

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Cumulative Frequency Graphs

Suppose we have a table showing classes and frequencies, such as for histograms. We can find the cumulative frequency by adding on a column in which we add up the frequencies as we go. To create a cumulative frequency graph, we plot these cumulative frequencies as y values against the top of each class as x values, then join the points up with straight lines.

To estimate the interquartile range from these graphs, we draw lines from the y axis at \dfrac{n}{4} and \dfrac{3n}{4} and find the corresponding x values, which are our quartiles.

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Boxplots

Boxplots are diagrams that look like this:

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Example 1: Cumulative Frequency Graph

Draw a cumulative frequency graph from this data and estimate the interquartile range.

[4 marks]

Step 1: Add a high point column and cumulative frequency column to the table.

Step 2: Plot cumulative frequency against high point.

Step 3: Draw lines at cumulative frequencies of 25 (first quartile) and 75 (third quartile) and read off the x values.

Step 4: The estimate for the interquartile range is 26.5-10=16.5

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Example 2: Boxplots

Create a boxplot for the following data: 1,9,11,12,15,16,18 where a point is considered to be an outlier if it is more than the interquartile range lower than the first quartile or higher than the third quartile.

[6 marks]

There are 7 data points.

\dfrac{7}{4}=1.75 so the first quartile is the second data point, which is 9.

\dfrac{3\times 7}{4}=5.25 so the third quartile is the sixth data point, which is 16.

The interquartile range is therefore 16-9=7

So points less than 9-7=2 or greater than 16+7=23 are outliers. So there is one outlier in our data set, at the point 1

The lowest non-outlier value is 9, and the highest non-outlier value is 18

Finally, the median is 12.

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Cumulative Frequency and Boxplots Example Questions

Add high point and cumulative frequency columns.

 

 

Plot high point against cumulative frequency.

 

There are 20 data points.

 

\dfrac{20}{4}=5 so the first quartile is the midpoint between the fifth and sixth point, which is \dfrac{22+24}{2}=23

 

\dfrac{3\times 20}{4}=15 so the third quartile is the midpoint between the 15th and 16th point, which is \dfrac{46+48}{2}=47

 

\dfrac{20}{2}=10 so the second quartile (median) is the midpoint between the 10th and 11th point, which is \dfrac{36+40}{2}=38

 

The lowest and highest values are 5 and 68.

 

This gives the following box plot:

Add high point and cumulative frequency onto the table.

 

 

Plot high point against cumulative frequency on the graph.

 

Use the graph to determine:

First quartile =1.3

Median =2.6

Third quartile =3.3

 

Create the box plot.

 

Additional Resources

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Exam Tips Cheat Sheet

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Formula Booklet

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