# Cumulative Frequency and Boxplots

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## 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

Question 1: Create a cumulative frequency graph from the following table:

[3 marks]

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Add high point and cumulative frequency columns.

Plot high point against cumulative frequency.

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Question 2: Make a boxplot from the following data:

$5,11,17,21,22,24,28,31,32,36\\40,41,44,45,46,48,51,54,58,68$

[5 marks]

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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 $15$th and $16$th point, which is $\dfrac{46+48}{2}=47$

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

The lowest and highest values are $5$ and $68$.

This gives the following box plot:

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Question 3: Create a box plot by estimating the quartiles from a cumulative frequency graph for the following data:

[8 marks]

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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.

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