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# Drawing Box Plots

One way to make statistical data more accessible is to use charts. One type of chart that shows the median and the range of a data set is a box plot.

## Five-Number Summary

A five-number summary contains five pieces of information, and is one way of describing the distribution of data in a data set.
• minimum value
• maximum value
• median
• first quartile
• third quartile

Box plots are visual illustrations of this information.

## Box Plot

A box plot can be used to illustrate the distribution of a data set. In the figure, the minimum, maximum, median, and first and third quartiles ( and ) are shown.

A box plot provides a visual approximation of the distribution of a data set since each segment of the chart contains one quarter, or of the data. Note that the center of the data lies inside the box. The further apart the segments are, the greater the spread is for that quarter of the data.

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Exercise

Thanks a Latte Café measured the temperature of cappuccinos. The box plot below shows the results in F.

A good cappuccino, according to the experts, has a temperature between F and F. Approximately how many of the cappuccinos could be considered "good"?

Show Solution
Solution

Each part of a box plot contains of the data. This means the box contains of the data.

The first quartile is located at degrees and the third is at degrees. This means that about of the cappuccionos were in the ideal temperature range. This gives Approximately cappuccinos can be considered "good".

## Drawing a Box Plot

Because a box plot shows the minimum, maximum, median, and first and third quartiles these need to be identified from the data set. The following data set gives the test scores for a grade.

### 1

Order the data set, then find the minimum and maximum

Sometimes, the data will be given in ascending order. When it is not, it's necessary to order the data so the quartiles can be found. With an ordered data set, the minimum and maximum are easily identifiable. Here, the minimum is and the maximum is These are marked above a number line with a line segment, indicating the range.

### 2

Determine the median

Since there are values, the median is the mean of the numbers at the th and th position. Now, the median can be determined by calculating the mean of and The median is This is marked as a vertical line segment in the range. Remember that the line for the median falls inside the box.

### 3

Determine the quartiles
Lastly, find the first and third quartiles. The median divides the set into two smaller sets each with values. The first quartile is the middle value in the lower set: The third quartile is the median of the upper set:

{\displaystyle \begin{aligned} \ \ \ 11.5 \ \ \ 12\ \ \ 12 \ \ \ 12.5 \ \ \ 13 \ \ \ 13 \ \ \ {\color{#0000FF}{13.5}} \ \ \ 13.5 \ \ \ &15 \ \ \ 15 \ \ \ 15.5 \ \ \ 15.5 \ \ \ 16 \end{aligned}}

The quartiles are marked as the sides of the box plot. Now the box plot is complete.