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Box-and-Whisker Plots

Plotting the five-number summary for ascending data.

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Box-and-Whisker Plots

Let's Think About It

License: CC BY-NC 3.0

Mr. Wilson created a box-and-whisker plot with the test scores from the most recent math test. Identify the median, quartiles, and extremes. What can you infer from the box-and-whisker plot?

In this concept, you will learn how to draw a box-and-whisker plot from a set of data.

Guidance

A box-and-whisker plot or box plot is a graph that represents the distribution of a data set. The key parts needed to draw a box-and-whisker plot are:

  1. Median - the middle number of a data set that is ordered from least to greatest.
  2. Lower and upper quartiles - values that divide the data set into four sections
  3. Lower and upper extremes - the smallest and largest values in the data set. 

Here is the data set from a survey of the number of hours worked by teenagers with part-time jobs.

6, 8, 8, 8, 10, 10, 11, 11, 12, 15, 16, 16, 20

The median is 11. The lower quartile is 8 and the upper quartile is 15.5. The lower extreme is 6 and the upper extreme is 20.

Here are the steps to drawing a box-and-whisker plot:

  1. Draw a number line labeled to show the range of data from least to greatest.
  2. Identify the median, the upper quartile, the lower quartile, the lower extreme and the upper extreme on the number line.
  3. Draw in a box around the quartiles. The median is the middle line of the two boxes.
  4. Then draw in the whiskers. These are lines that extend from each quartile to the upper and lower extremes.

Here is a picture of a number line with a completed box-and-whisker plot on it.

The first box goes from the lower quartile 8 to the median 11. The second box goes from the median 11 to the upper quartile 15.5. The whiskers extend out from the lower quartile to the lower extreme of 6, and from the upper quartile to the upper extreme of 20.

Now that you know how to draw a box-and-whisker plot using measure of data, you can look at a box-and-whisker plot and identify the median, quartiles and extremes. Look at the box-and-whisker plot below. 

Use this chart to examine the data. The median divides the two boxes. The median here is 200. The lower quartile is 100 and the upper quartile is 300. The lower extreme is 50 and the upper extreme is 400. The wider the quartile range, the more varied the data set. 

You can compare two sets of data using box-and-whisker plots. 

License: CC BY-NC 3.0

These box-and-whisker plots show the number of miles James and Angie ran per day in a month.

James and Angie ran a similar range of miles. James ran between 2 and 17 miles a day. Angie ran between 2 and 15 miles.

The width of the quartile boxes tells you that James more varied distances than Angie. James ran between 4 to 10 miles. Angie ran between 9 to 12 miles on most days. Also, look at size of each quartile box. Notice that the box to the left of Jame's median is smaller than the box to the right of the median. This tells you that James ran fewer than 5 miles more often. Angie ran more than 11 miles more often.

Guided Practice

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.

What are the highest and lowest weights represented on the box-and-whisker plot?

The lowest value or weight is 127 pounds. The highest value or weight is 201 pounds.

What is the median weight for a black bear?

The median weight is 163 pounds.

What is an observation you can make about the weight of the black bears in this survey?

It appears that there are a lot of bears on the high and low end of the spectrum. 

Examples

Answer the following questions using this box-and-whisker plot.

Example 1

What is the smallest value on the plot?

The lower extreme is 34.

Example 2

What is the greatest value on the plot?

The upper extreme is 58.

Example 3

What is the median of the whole data set?

The median is 49.

Follow Up

License: CC BY-NC 3.0

Remember Mr. Wilson's box-and-whisker plot?

Mr. Wilson created a box-and-whisker plot with the test scores from the most recent math test. Identify the median, quartiles, and extremes. What can you infer from the box-and-whisker plot?

The median is 85. The lower quartile is around 79 and the upper quartile is around 83. The lower extreme is around 68 and the upper extreme is around 93. 

You can infer that the lowest grade was a 68 and the highest grade was a 98. The quartile boxes are not too far from the median, so you can also infer that the average for the class is probably a B.  

Video Review

Explore More

Use the box-and-whisker plot to answer the following questions.

1. What is the median score in this box-and-whisker plot?

2. What is the lower quartile?

3. What is the upper quartile?

4. What is the range of the data?

5. What is the lower extreme?

6. What is the upper extreme?

7. How many values are in the data?

Use the data to answer the following questions.

25, 26, 30, 18, 24, 26, 19, 21, 22

8. Draw a box-and-whisker plot.

9. Write the data in order from least to greatest.

10. What is the median score?

11. What is the lower quartile?

12. What is the upper quartile?

13. What is the lower extreme?

14. What is the upper extreme?

15. What is the range of the data?

Vocabulary

five point summary

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

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

Observed data are the values that result from computations performed on the input variable.
Outlier

Outlier

In statistics, an outlier is a data value that is far from other data values.
skewed

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

Statistical correlation is a representation of possible related changes in values between the two sets of data.
trends

trends

Trends in data sets or samples are indicators found by reviewing the data from a general or overall standpoint
uniform

uniform

A uniform shaped histogram indicates data that is very consistent; the frequency of each class is very similar to that of the others.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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