# Boxplots

CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.
LO 4.4: Using appropriate graphical displays and/or numerical measures, describe the distribution of a quantitative variable in context: a) describe the overall pattern, b) describe striking deviations from the pattern
LO 4.7: Define and describe the features of the distribution of one quantitative variable (shape, center, spread, outliers).
Video: Boxplots (2 videos, 7:02 total)

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## Introduction

Now we introduce another graphical display of the distribution of a quantitative variable, the boxplot.

## The Five Number Summary

So far, in our discussion about measures of spread, some key players were:

• the extremes (min and Max), which provide the range covered by all the data; and
• the quartiles (Q1, M and Q3), which together provide the IQR, the range covered by the middle 50% of the data.

Recall that the combination of all five numbers (min, Q1, M, Q3, Max) is called the five number summary, and provides a quick numerical description of both the center and spread of a distribution.

## EXAMPLE: Best Actress Oscar Winners

We will continue with the Best Actress Oscar winners example (Link to the Best Actress Oscar Winners data).

34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33

The five number summary of the age of Best Actress Oscar winners (1970-2001) is:

min = 21,    Q1 = 32,     M = 35,     Q3 = 41.5,      Max = 80

To sketch the boxplot we will need to know the 5-number summary as well as identify any outliers. We will also need to locate the largest and smallest values which are not outliers.  The stemplot below might be helpful as it displays the data in order. Learn By Doing: 5-Number Summary

Now that you understand what each of the five numbers means, you can appreciate how much information about the distribution is packed into the five-number summary. All this information can also be represented visually by using the boxplot.

## The Boxplot

LO 4.17: Explain the process of creating a boxplot (including appropriate indication of outliers).

The boxplot graphically represents the distribution of a quantitative variable by visually displaying the five-number summary and any observation that was classified as a suspected outlier using the 1.5(IQR) criterion.

## EXAMPLE: Constructing a boxplot

1. The central box spans from Q1 to Q3. In our example, the box spans from 32 to 41.5. Note that the width of the box has no meaning. 2. A line in the box marks the median M, which in our case is 35. 3. Lines extend from the edges of the box to the smallest and largest observations that were not classified as suspected outliers (using the 1.5xIQR criterion). In our example, we have no low outliers, so the bottom line goes down to the smallest observation, which is 21. Since we have three high outliers (61,74, and 80), the top line extends only up to 49, which is the largest observation that has not been flagged as an outlier. 4. outliers are marked with asterisks (*). To summarize: the following information is visually depicted in the boxplot:

• the five number summary (blue)
• the range and IQR (red)
• outliers (green) Learn By Doing: Boxplots
Did I Get This?: Boxplots

## Side-By-Side (Comparative) Boxplots

LO 4.18: Compare and contrast distributions (of quantitative data) from two or more groups, and produce a brief summary, interpreting your findings in context.

As we learned earlier, the distribution of a quantitative variable is best represented graphically by a histogram. Boxplots are most useful when presented side-by-side for comparing and contrasting distributions from two or more groups.

## EXAMPLE: Best Actress/Actor Oscar Winners

So far we have examined the age distributions of Oscar winners for males and females separately. It will be interesting to compare the age distributions of actors and actresses who won best acting Oscars. To do that we will look at side-by-side boxplots of the age distributions by gender. Recall also that we found the five-number summary and means for both distributions. For the Best Actress dataset, we did the calculations by hand. For the Best Actor dataset, we used statistical software, and here are the results:

• Actors: min = 31, Q1 = 37.25, M = 42.5, Q3 = 50.25, Max = 76
• Actresses: min = 21, Q1 = 32, M = 35, Q3 = 41.5, Max = 80

Based on the graph and numerical measures, we can make the following comparison between the two distributions:

Center: The graph reveals that the age distribution of the males is higher than the females’ age distribution. This is supported by the numerical measures. The median age for females (35) is lower than for males (42.5). Actually, it should be noted that even the third quartile of the females’ distribution (41.5) is lower than the median age for males. We therefore conclude that in general, actresses win the Best Actress Oscar at a younger age than actors do.

Spread: Judging by the range of the data, there is much more variability in the females’ distribution (range = 59) than there is in the males’ distribution (range = 45). On the other hand, if we look at the IQR, which measures the variability only among the middle 50% of the distribution, we see more spread in the ages of males (IQR = 13) than females (IQR = 9.5). We conclude that among all the winners, the actors’ ages are more alike than the actresses’ ages. However, the middle 50% of the age distribution of actresses is more homogeneous than the actors’ age distribution.

Outliers: We see that we have outliers in both distributions. There is only one high outlier in the actors’ distribution (76, Henry Fonda, On Golden Pond), compared with three high outliers in the actresses’ distribution.

## EXAMPLE: Temperature of Pittsburgh vs. San Francisco

In order to compare the average high temperatures of Pittsburgh to those in San Francisco we will look at the following side-by-side boxplots, and supplement the graph with the descriptive statistics of each of the two distributions. Statistic Pittsburgh San Francisco
min 33.7 56.3
Q1 41.2 60.2
Median 61.4 62.7
Q3 77.75 65.35
Max 82.6 68.7

When looking at the graph, the similarities and differences between the two distributions are striking. Both distributions have roughly the same center (medians are 61.4 for Pitt, and 62.7 for San Francisco). However, the temperatures in Pittsburgh have a much larger variability than the temperatures in San Francisco (Range: 49 vs. 12. IQR: 36.5 vs. 5).

The practical interpretation of the results we obtained is that the weather in San Francisco is much more consistent than the weather in Pittsburgh, which varies a lot during the year. Also, because the temperatures in San Francisco vary so little during the year, knowing that the median temperature is around 63 is actually very informative. On the other hand, knowing that the median temperature in Pittsburgh is around 61 is practically useless, since temperatures vary so much during the year, and can get much warmer or much colder.

Note that this example provides more intuition about variability by interpreting small variability as consistency, and large variability as lack of consistency. Also, through this example we learned that the center of the distribution is more meaningful as a typical value for the distribution when there is little variability (or, as statisticians say, little “noise”) around it. When there is large variability, the center loses its practical meaning as a typical value.

## Let’s Summarize

• The five-number summary of a distribution consists of the median (M), the two quartiles (Q1, Q3) and the extremes (min, Max).
• The five-number summary provides a complete numerical description of a distribution. The median describes the center, and the extremes (which give the range) and the quartiles (which give the IQR) describe the spread.
• The boxplot graphically represents the distribution of a quantitative variable by visually displaying the five number summary and any observation that was classified as a suspected outlier using the 1.5(IQR) criterion. (Some software packages indicate extreme outliers with a different symbol)
• Boxplots are most useful when presented side-by-side to compare and contrast distributions from two or more groups.