# Box-and-Whisker Plots

## Plotting the five-number summary for ascending data.

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Box-and-Whisker Plots

A track and field coach, Mr. Watson was measuring shot put distances for his varsity and junior varsity teams. Here is his data, in feet, that he put in order from least to greatest.

Varsity:

36.8, 43.5, 45.8, 46.2, 49.1, 50.7, 52.7, 54.3, 54.4, 55.8, 56.0, 58.5

Junior Varsity:

33.2, 35.4, 36.2, 37.0, 37.6, 39.4, 40.6, 40.8, 41.3, 42.1, 44.5, 50.3

Mr. Watson wants to present this information to both of his teams. He wants to compare them. How can Mr. Watson create a display that will communicate what he wants to tell his team?

In this concept, you will learn to create box-and-whisker plots.

### Box and Whisker Plots

At times it is useful to get a general idea of how data clusters together. Box-and-whisker plots display the distribution of data items along a number line. The data are divided into four equal parts, separated by points called quartiles. A box-and-whisker plot also displays the smallest data point (the extreme minimum) and the largest data point (the extreme maximum).

A box-and-whisker plot is created by determining five points. This is called a five-point summary.

First, place the data in order from smallest to largest.

Next, create a number line that shows the range of the data using equal intervals. The median will be used as the middle point on the box-and-whisker plot and to split the data in half.

Next, the median of each half, the quartile, is then calculated. These separate the data into quarters.

Then, use the highest data point and the lowest data point as the endpoints or extremes. Boxes are drawn between the quartiles, and whiskers are drawn to the extremes.

Now let’s apply these steps to an example.

In this example, each step will be shown. Later all of the data for the box-and-whisker will be used to draw the graph.

Draw a box-and-whisker plot for the given data.

16, 51, 32, 16, 24, 37, 7, 22, 19, 40, 10, 31, 29, 38, 21, 11

First, put the data in order from smallest to largest.

7, 10, 11, 16, 16, 19, 21, 22, 24, 29, 31, 32, 37, 38, 40, 51

Next, draw a number line that includes the extremes, 7 and 51. In this case, use a number line from 5 to 55 using intervals of 5.

Then, determine the median of the data. The middle points in the data are 22 and 24 so the median is 23. Mark the median with a point beneath the number line.

Then, the median separates the data into two groups as shown below:

7, 10, 11, 16, 16, 19, 21, 22                                               24, 29, 31, 32, 37, 38, 40, 51

Find the median of each of these groups. These medians are the quartiles which are 16 and 34.5. These divide the data into four groups. Mark the quartiles as you did the median, with a point.

Then, draw boxes between the quartiles and the median.

Then, mark the extremes, the smallest and largest numbers, with points. In this case, the extremes are 7 and 51.

Then, draw whiskers, or horizontal lines, to connect the quartiles to the extremes.

It can be seen from the box-and-whisker plot that half of the data will be found between the first quartile and the third quartile. A quarter of the data is between the minimum and the first quartile and the last quarter is between the third quartile and the maximum. The median, of course, marks the half-way point between the data.

In this particular situation, the second half of the data is stretched out over a further area than the first half and about half way is between 15 and 35.

Double plots or graphs can be made when there are two factors being compared. A double box-and-whisker plot can be made by drawing the second factor beneath the first factor. This will allow both factors to be visible on the same plot.

### Examples

#### Example 1

Earlier, you were given a problem about the coach and the shot put distances. The coach’s data, in feet, for the shot put distances that he put in order from least to greatest are:

Varsity:

36.8, 43.5, 45.8, 46.2, 49.1, 50.7, 52.7, 54.3, 54.4, 55.8, 56.0, 58.5

Junior Varsity:

33.2, 35.4, 36.2, 37.0, 37.6, 39.4, 40.6, 40.8, 41.3, 42.1, 44.5, 50.3

Make a double box-and-whisker plot of this data. How does the data compare?

First, find the minimum, maximum, median, and the first and third quartiles for each set of data. These will give you the five-point summary for both the varsity and junior varsity teams.

 Varsity Junior Varsity Minimum extreme 36.8 33.2 Maximum extreme 58.5 50.3 Median 51.7 40.0 First Quartile 46.0 36.6 Third Quartile 55.1 41.7

Next, draw the double box-and-whisker plot.

Then, analyze the double box-and-whisker plot.

From this box-and-whisker plot, the coach can tell that the team’s results are what he expected. The varsity shot put distances are generally better than those of the junior varsity. There are a number of players whose results overlap. The highest junior varsity player is better than the entire first quartile of the varsity team. It is also apparent that the results are more dispersed, or spread out, in the varsity team than in the junior varsity team.

#### Example 2

The data values below depict the number of televisions sold at a department store each month for nine months. Create a box-and-whisker plot to display the data.

 April May June July August September October November December 110 98 91 102 89 95 108 118 152

First, to determine the median of the set of data, arrange the data in order from least to greatest. Identify the data value in the middle of the data set.

89, 91, 95, 98, 102, 108, 110, 118, 152

For this set of data, 102 is the median.

Next, identify the median for the lower quartile. Again, since two data values share the middle position, find their mean.

89, 91, 95, 98,             102,             108, 110, 118, 152

The median for the lower quartile is \begin{align*}\frac{91+95}{2}=93\end{align*}.

Then, identify the median of the upper quartile. Remember to find the mean of the two data values that share the middle position.

The median of the upper quartile is \begin{align*}\frac{110+118}{2}=114\end{align*}.

Then, draw a number line. The first value on the number line should be near the smallest number in the data set. In this case, the smallest number is 89. Therefore, the number line will start at 80. The last value on the number line should be near the largest number in the set of data. The largest number in the data set is 152. Therefore, the number line will end at 160. In this case, label the number line by tens.

#### Example 3

What is the minimum extreme of this box-and-whisker plot?

The minimum extreme is the point furthest to the left which is 34.

#### Example 4

What is the maximum extreme of this box-and-whisker plot?

The maximum extreme is the point furthest to the right which is 58.

#### Example 5

What is the median?

The median is the middle value found where the line lies in the box of the box-and-whisker which is 49.

### Review

Use each data for each set of instructions.

90, 104, 98, 156, 140, 85, 122, 129, 142, 138, 131, 81, 151, 147, 130, 156

1. Create a box-and-whisker plot for the data.

2. Identify the minimum extreme.

3. Identify the maximum extreme.

4. Identify the median.

The weight of bears varies between species. Weight also varies within species as a result of habitat and diet. The box-and-whisker plot was created after recording the weight (in pounds) of several black bears across the country. Use the box-and-whisker plot to answer the questions below.

5. What is the minimum extreme?

6. What is the maximum extreme?

7. What is the median?

8. What is the value of the first quartile?

9. What is the value of the third quartile?

A group of dog sled drivers collected the following data about the number of dogs who lead sled teams. Here is the data in a box-and-whisker plot.

10. What is the minimum extreme?

11. What is the maximum extreme?

12. What is the median?

13. What is the value of the first quartile?

14. What is the value of the third quartile?

15. How many dogs do most sled teams have?

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
five point summary The numbers needed to construct a box-and-whisker plot are called the five-point-summary. The five points are the minimum, the lower median (Q1), the median, the upper median (Q3), and the maximum.
line of fit A line of fit is a straight or continuously curved line representing the trend of changes in the comparison of two data sets (or one set of bivariate data).
observed data Observed data are the values that result from computations performed on the input variable.
Outlier In statistics, an outlier is a data value that is far from other data values.
skewed As with the horizontal skewing of a histogram, stem plots with a obvious skew toward one end or the other tend to indicate an increased number of outliers either lesser than or greater than the mode.
statistical correlation Statistical correlation is a representation of possible related changes in values between the two sets of data.
trends Trends in data sets or samples are indicators found by reviewing the data from a general or overall standpoint
uniform A uniform shaped histogram indicates data that is very consistent; the frequency of each class is very similar to that of the others.

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