Question

In the seasons that followed his 2001 record-breaking season, Barry Bonds hit \(46,45,45,5,26,\) and 28 homers, respectively, until he retired from major league baseball in 2007 (www.ESPN,com). \(^{16}\) Two box plots, one of Bond's homers through \(2001,\) and a second including the years \(2002-2007\) follow. The statistics used to construct these box plots are given in the table. \begin{tabular}{lccccccc} Years & Min & \(a_{1}\) & Median & \(a_{3}\) & IQR & Max & \(n\) \\ \hline 2001 & 16 & 25.00 & 34.00 & 41.50 & 16.5 & 73 & 16 \\ 2007 & 5 & 25.00 & a. Calculate the upper fences for both of these box plots. b. Can you explain why the record number of homers is an outlier in the 2001 box plot, but not in the 2007 box plot?34.00 & 45.00 & 20.0 & 73 & 22 \end{tabular}

Short answer

Question: Calculate the upper fences for the 2001 and 2007 box plots and explain why the record number of homers is an outlier in the 2001 box plot, but not in the 2007 box plot. Answer: The upper fence for the 2001 box plot is 66.25, while the upper fence for the 2007 box plot is 75.00. The record number of homers (73) is an outlier in the 2001 box plot because it is greater than the upper fence (66.25). However, it is not an outlier in the 2007 box plot because it lies within the range of the upper fence (75.00). This indicates that the distribution of home runs widened over the years and the record number of homers became less exceptional compared to the rest of the data.

Step-by-step solution

Determine the IQR for the 2001 box plot

The IQR (interquartile range) is the difference between the third quartile \(a_3\) and the first quartile \(a_1\). From table, we see that \(a_3 = 41.50\) and \(a_1 = 25.00\). So, the IQR is \(41.50 - 25.00 = 16.50\).

Calculate the upper fence for the 2001 box plot

The upper fence is found by adding 1.5 times the IQR to the third quartile, \(a_3\). Therefore, the upper fence is given by \(41.50 + 1.5 \times 16.5\). Let's calculate: \(41.50 + 1.5 \times 16.5 = 41.50 + 24.75 = 66.25\). So, the upper fence for the 2001 box plot is 66.25. 2. Calculating the upper fence for the 2007 box plot

Determine the IQR for the 2007 box plot

The IQR (interquartile range) is the difference between the third quartile \(a_3\) and the first quartile \(a_1\). From table, we see that \(a_3 = 45.00\) and \(a_1 = 25.00\). So, the IQR is \(45.00 - 25.00 = 20.0\).

Calculate the upper fence for the 2007 box plot

The upper fence is found by adding 1.5 times the IQR to the third quartile, \(a_3\). Therefore, the upper fence is given by \(45.00 + 1.5 \times 20.00\). Let's calculate: \(45.00 + 1.5 \times 20 = 45.00 + 30 = 75.00\). So, the upper fence for the 2007 box plot is 75.00. 3. Explaining the outlier in the 2001 and 2007 box plot

Check if the record number of homers is an outlier in the 2001 box plot

The record number of homers in the 2001 data is 73. As we calculated earlier, the upper fence for the 2001 box plot is 66.25. Since 73 is greater than 66.25, the record number of homers is an outlier in the 2001 box plot.

Check if the record number of homers is an outlier in the 2007 box plot

The record number of homers is the same in the 2007 data, 73. As we calculated earlier, the upper fence for the 2007 box plot is 75.00. Since 73 is less than 75.00, the record number of homers is not considered an outlier in the 2007 box plot.

Interpret the results

The record number of homers is an outlier in the 2001 box plot because it is greater than the upper fence, while it is not an outlier in the 2007 box plot because it lies within the range of the upper fence. This shows that the home runs distribution widened over the years, and the record number of homers became less exceptional compared to the rest of the data.

Key concepts

Box Plot

A box plot, also known as a whisker plot, provides a visual summary of data that helps in understanding its distribution and variability. A box plot is made up of a box, which spans from the first quartile (Q1) to the third quartile (Q3), with a line inside the box indicating the median. This box captures the middle 50% of the data. The 'whiskers' extend from the box to the smallest and largest values within 1.5 times the interquartile range (IQR) from the quartiles. Data points outside the whiskers are potential outliers and are typically indicated by dots or stars. The beauty of a box plot is its ability to clearly show if a distribution is skewed and identify potential outliers at a glance.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of statistical dispersion, which is the spread between the first quartile (Q1) and the third quartile (Q3). It essentially demonstrates the range within which the central 50% of the data falls. To calculate the IQR, you simply subtract the value at Q1 from the value at Q3, i.e., \( IQR = Q3 - Q1 \). The IQR is particularly useful in identifying the variability in the data while ignoring the extreme values. It is also utilized to determine outliers. Data points lying more than 1.5 times the IQR beyond the quartiles are considered potential outliers. By focusing on the central portion of the dataset, IQR allows a more robust analysis of data distributions.

Outlier Detection

Outliers are data points that differ significantly from the rest of the dataset, indicating variability in the measurements, errors, or novel variations. Detecting outliers is crucial as they can skew and potentially mislead statistical analyses. In the context of box plots, any data point that lies beyond 1.5 times the IQR from the first or third quartile is marked as an outlier. A formula is used where:

• Upper boundary = \( Q3 + 1.5 \times IQR \)
• Lower boundary = \( Q1 - 1.5 \times IQR \)

Values beyond these boundaries are potential outliers. It is important to scrutinize these outliers further, as they can provide insights into errors in data collection or reveal unexpected phenomena. However, not all outliers are spurious, and in some cases, they represent legitimate variations.