2.3. Box-and-Whisker Plots http://www.ck12.org
Lesson Summary
Thefive-numbersummary is useful collection of statistical measures consisting of the following in ascending order:
Minimum, lower quartile, median, upper quartile, maximum
ABox-and-Whisker Plotis a graphical representation of the five-number summary showing a box bounded by the
lower and upper quartiles and the median as a line in the box. The whiskers are line segments extended from the
quartiles to the minimum and maximum values. Each whisker and section of the box contains approximately 25%
of the data. The width of the box is theinterquartile range (IQR), and shows the spread of the middle 50% of the
data. Box-and-whisker plots are effective at giving an overall impression of the shape, center, and spread. While an
outlier is simply a point that is not typical of the rest of the data, there is an accepted definition of an outlier in the
context of a box-and-whisker plot. Any point that is more than 1.5 times the length of the box (IQR) from either
end of the box, is considered to be an outlier. Whenchanging unitsof a distribution, the center and spread will be
affected, but the shape will stay the same.
Points to Consider
- What characteristics of a data set make it easier or harder to represent it using dot plots, stem and leaf plots,
histograms, and box and whisker plots? - Which plots are most useful to interpret the ideas of shape, center, and spread?
- What effects do other transformations of the data have on the shape, center, and spread?
Review Questions
- Here is the 1998 data on the percentage of capacity of reservoirs in Idaho.
70 , 84 , 62 , 80 , 75 , 95 , 69 , 48 , 76 , 70 , 45 , 83 , 58 , 75 , 85 , 70 ,
62 , 64 , 39 , 68 , 67 , 35 , 55 , 93 , 51 , 67 , 86 , 58 , 49 , 47 , 42 , 75
a. Find the five-number summary for this data set.
b. Show all work to determine if there are true outliers according to the 1. 5 ∗IQR rule.
c. Create a box-and-whisker plot showing any outliers.
d. Describe the shape, center, and spread of the distribution of reservoir capacities in Idaho in 1998.
e. Based on your answer in part d., how would you expect the mean to compare to the median? Calculate
the mean to verify your expectation.- Here is the 1998 data on the percentage of capacity of reservoirs in Utah.
80 , 46 , 83 , 75 , 83 , 90 , 90 , 72 , 77 , 4 , 83 , 105 , 63 , 87 , 73 , 84 , 0 , 70 , 65 , 96 , 89 , 78 , 99 , 104 , 83 , 81
a. Find the five-number summary for this data set.
b. Show all work to determine if there are true outliers according to the 1. 5 ∗IQR rule.
c. Create a box-and-whisker plot showing any outliers.
d. Describe the shape, center, and spread of the distribution of reservoir capacities in Utah in 1998.
e. Based on your answer in part d., how would you expect the mean to compare to the median? Calculate
the mean to verify your expectation.- Graph the box plots for Idaho and Utah on the same axes. Write a few statements comparing the water levels
in Idaho and Utah by discussing the shape, center, and spread of the distributions.