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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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

A box plot or box and whisker plot can be used to illustrate the distribution of a data set. A box plot has three parts.

- A rectangular box that extends from the first to the third quartiles $(Q_{1}$ and $Q_{3})$ with a line in the middle indicating the position of the median.
- A segment attached to the left of the box that extends from the first quartile to the minimum of the data set.
- A segment attached to the right of the box that extends from the third quartile to the maximum of the data set.

A boxplot is a scaled figure, usually presented above a number line.

A box plot provides a visual illustration of the five-number summary of a data set. Each segment of the chart contains one quarter, or $25%$ of the data, and the center $50%$ of the data lies inside the box. The further apart the segments are, the greater the spread is for that quarter of the data.Thanks a Latte Café measured the temperature of $800$ cappuccinos. The box plot below shows the results in $_{∘}$F.

A good cappuccino, according to the experts, has a temperature between $131_{∘}$F and $140_{∘}$F. Approximately how many of the cappuccinos could be considered "good"?

Show Solution

Each part of a box plot contains $25%$ of the data. This means the box contains $50%$ of the data.

The first quartile is located at $129$ degrees and the third is at $140$ degrees. This means that **about** $50%$ of the $800$ cappuccionos were in the ideal temperature range. This gives
$0.50⋅800=400.$ Approximately $400$ cappuccinos can be considered "good".

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.
$8.511161310.551515.513.5 12.51115.5127158988.56127.51310.511.513.5 $
### 1

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. $5677.510.511111313.513.5 888.58.5910.511.5121212.513151515.515.516 $ With an ordered data set, the minimum and maximum are easily identifiable. Here, the minimum is $5$ and the maximum is $16.$ These are marked above a number line with a line segment, indicating the range.

### 2

Since there are $26$ values, the median is the mean of the numbers at the $13$th and $14$th position. $5677.510.511111313.513.5 888.58.5910.511.5121212.513151515.515.516 $ Now, the median can be determined by calculating the mean of $11$ and $11.5.$ $211+11.5 =11.25$ The median is $11.25.$ This is marked as a vertical line segment in the range. Remember that the line for the median falls inside the box.

### 3

Lastly, find the first and third quartiles. The median divides the set into two smaller sets each with $13$ values. The first quartile is the middle value in the lower set: $8.5.$ $5677.5888.58.5910.510.51111 $
The third quartile is the median of the upper set: $13.5.$

Order the data set, then find the minimum and maximum

Determine the median

Determine the quartiles

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

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

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