Question

In the seasons that followed his 2001 record-breaking season, Barry Bonds hit \(46,45,45,5,\) and 26 homers, respectively (www.espn.com). \(^{14}\) Two boxplots, one of Bond's homers through 2001 , and a second including the years 2002-2006, follow. The statistics used to construct these boxplots are given in the table. $$ \begin{array}{lccccccc} \text { Years } & \text { Min } & a_{1} & \text { Median } & a_{3} & \text { IQR } & \text { Max } & n \\ \hline 2001 & 16 & 25.00 & 34.00 & 41.50 & 16.5 & 73 & 16 \\ 2006 & 5 & 25.00 & 34.00 & 45.00 & 20.0 & 73 & 21 \end{array} $$ a. Calculate the upper fences for both of these boxplots. b. Can you explain why the record number of homers is an outlier in the 2001 boxplot, but not in the 2006 boxplot?

Short answer

Question: Calculate the upper fences for the 2001 and 2006 boxplots, and explain why the record number of homers (73) is considered an outlier in the 2001 boxplot but not in the 2006 boxplot. Answer: The upper fences for the 2001 and 2006 boxplots are 66.25 and 75, respectively. The record number of homers (73) is an outlier in the 2001 boxplot because it is greater than the Upper Fence (66.25), but it is not considered an outlier in the 2006 boxplot because it is not greater than the Upper Fence (75). This indicates that the distribution of homers changed with the inclusion of the additional years (2002-2006), resulting in a higher range of values and making the record number of homers no longer an outlier in the 2006 dataset.

Step-by-step solution

Calculate the Upper Fences

For the 2001 dataset, we have Q3 = 41.5 and IQR = 16.5. Using the formula for Upper Fence, we get: $$Upper Fence_{2001} = 41.5 + 1.5 * 16.5 = 41.5 + 24.75 = 66.25$$ For the 2006 dataset, we have Q3 = 45 and IQR = 20. Using the formula for Upper Fence, we get: $$Upper Fence_{2006} = 45.00 + 1.5 * 20 = 45.00 + 30 = 75$$ Hence, the Upper Fence for 2001 boxplot is 66.25 and for 2006 boxplot is 75.

Determine the outlier

Recall that the record number of homers is 73. Now let's compare the Upper Fences with this value: For the 2001 dataset, 73 is greater than the Upper Fence (66.25), so the record number of homers is considered an outlier in the 2001 boxplot. For the 2006 dataset, 73 is not greater than the Upper Fence (75), so the record number of homers is not considered an outlier in the 2006 boxplot. In conclusion: a. The upper fences for 2001 and 2006 boxplots are 66.25 and 75, respectively. b. The record number of homers (73) is an outlier in the 2001 boxplot because it is greater than the Upper Fence (66.25) but not in the 2006 boxplot because it is not greater than the Upper Fence (75). This means that with the inclusion of the additional years (2002-2006), the distribution of homers has changed, resulting in a higher range of values, and thus, the record number of homers is no longer considered as an outlier in the 2006 dataset.

Key concepts

Upper Fence Calculation

The upper fence in a boxplot serves as a critical cutoff to determine whether a data point is an outlier or not. Understanding how to calculate it is essential. The formula for the upper fence is given as \(Q3 + 1.5 \times IQR\), where \(Q3\) is the third quartile, and \(IQR\) is the interquartile range.

For the 2001 dataset in our exercise:
\(Upper Fence_{2001} = 41.5 + 1.5 \times 16.5 = 66.25\)
And for the 2006 dataset:
\(Upper Fence_{2006} = 45.00 + 1.5 \times 20 = 75\)

This calculation demonstrates how we determine the boundary for identifying outliers within our dataset; any data point lying above this value can significantly affect the interpretation of our data.

Interpreting Boxplots

A boxplot, or box-and-whisker plot, visually summarizes the distribution of a dataset. Important features of boxplots include the median, quartiles, and potentially any outliers. The box represents the middle 50% of the data, showing the interquartile range, while the 'whiskers' extend to the smallest and largest values within the fences.

When interpreting boxplots, look for the positioning of the median, the spread of the quartiles (which tells us about the variability in the data), and the length of the whiskers. If any data points are plotted outside the 'fences', they are potential outliers which can indicate extremes that may warrant further investigation.

Identification of Outliers

Outliers are data points that fall outside the expected range of a dataset, and they can be easily identified using a boxplot. When the data points exceed the upper fence or fall below the lower fence (which is \(Q1 - 1.5 \times IQR\)), they are considered outliers. In our example with Barry Bonds’ homers:

In 2001, the number of homers (73) exceeds the upper fence (66.25), qualifying it as an outlier.
In 2006, however, the number of homers (73) does not exceed the upper fence (75), hence it is not an outlier.

Outliers can influence the mean of a dataset significantly and may indicate a need for further analysis to understand why they are different from the rest of the data.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion and is calculated as the difference between the third quartile (\(Q3\)) and the first quartile (\(Q1\)) of a dataset. It tells us about the spread of the middle 50% of the data.

In the context of our exercise:
The IQR for 2001 is \(Q3 - Q1 = 41.5 - 25 = 16.5\)
The IQR for 2006 is \(Q3 - Q1 = 45.0 - 25 = 20.0\)

The IQR is crucial for determining the upper and lower fences and thereby identifying outliers. Any significant changes in the IQR can alter the calculation of the fences and potentially change which data points are considered outliers.