Cumulative Frequency and Boxplots

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 $26.5-10=16.5$

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 $9$.

$dfrac{3times 7}{4}=5.25$ 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 $12$.