Question

American League baseball teams play their games with the designated hitter rule, meaning that pitchers do not bat. The League believes that replacing the pitcher, typically a weak hitter, with another player in the batting order produces more runs and generates more interest among fans. Following are the average number of runs scored in American League and National League stadiums for the first half of the 2001 season: $$ \begin{array}{c|c|c|c} \text { Average Runs } & \text { League } & \text { Average Runs } & \text { League } \\ \hline 11.1 & \text { American } & 14.0 & \text { National } \\ 10.8 & \text { American } & 11.6 & \text { National } \\ 10.8 & \text { American } & 10.4 & \text { National } \\ 10.3 & \text { American } & 10.9 & \text { National } \\ 10.3 & \text { American } & 10.2 & \text { National } \\ 10.1 & \text { American } & 9.5 & \text { National } \\ 10.0 & \text { American } & 9.5 & \text { National } \\ 9.5 & \text { American } & 9.5 & \text { National } \\ 9.4 & \text { American } & 9.5 & \text { National } \\ 9.3 & \text { American } & 9.1 & \text { National } \\ 9.2 & \text { American } & 8.8 & \text { National } \\ 9.2 & \text { American } & 8.4 & \text { National } \\ 9.0 & \text { American } & 8.3 & \text { National } \\ 8.3 & \text { American } & 8.2 & \text { National } \\ & & 8.1 & \text { National } \\ & & 7.9 & \text { National } \end{array} $$ a) Create an appropriate graphical display of these data. b) Write a few sentences comparing the average number of runs scored per game in the two leagues. (Remember: shape, center, spread, unusual features!) c) Coors Field in Denver stands a mile above sea level, an altitude far greater than that of any other major league ball park. Some believe that the thinner air makes it harder for pitchers to throw curveballs and easier for batters to hit the ball a long way. Do you see any evidence that the 14 runs scored per game there is unusually high? Explain.

Short answer

Graphical displays show a higher variance in the National League runs. The 14.0 run average at Coors Field appears unusually high, suggesting unique factors at play.

Step-by-step solution

Organize the Data

First, organize the data into two groups: American League and National League. List the average runs for each league separately to help visualize the differences between the two.

Graphical Display

Create a graphical display (like a boxplot or dot plot) to compare the two groups. Plot the average runs for the American League and the National League to visually assess differences in the distribution.

Analyze the Graphical Distribution

Examine the graphical display for each league. Look for the shape (symmetrical, skewed), center (median), spread (range, interquartile range), and any unusual features like outliers in the data. Specifically, take note of the location of the '14.0' for the National League.

Compare Centers and Spread

Compare and contrast the centers of distribution (mean/median) and spread (range, IQR) between the American League and the National League, noticing any significant differences.

Investigate Unusual Features

Identify any unusual features from the graphical display. In this case, check if the '14.0' stands out significantly from other reported numbers, indicating it may be an outlier linked to specific conditions like Coors Field's altitude.

Key concepts

Understanding Boxplots

A boxplot, also known as a box-and-whisker plot, is a fantastic tool for visualizing data distribution. It gives a simple yet informative summary that helps us quickly understand the underlying trends in data. This plot shows five key statistics:

• Minimum
• First Quartile (Q1)
• Median (Q2)
• Third Quartile (Q3)
• Maximum

The 'box' represents the interquartile range (IQR), which is the range between Q1 and Q3. The 'whiskers' extend from the box to the minimum and maximum values that are not considered outliers. The beauty of a boxplot lies in its ability to show the data’s spread and central tendency, while quickly highlighting any outliers, represented as individual points outside the whiskers.

The Dynamics of Data Distribution

In the realm of statistical analysis, understanding the distribution of data is crucial. Knowing how data points are spread across different values helps us make informed judgements about the sample. We typically look at several key features in data distribution:

• Shape: Is the distribution symmetrical, skewed, or uniform? Shape affects how we interpret the central tendency and spread.
• Center: The median in a boxplot helps us grasp the middle value around which all other data points orient.
• Spread: This tells us how varied the data is, with the range and interquartile range acting as key indicators.

By evaluating these aspects, we compare different data sets, like the runs scored in baseball leagues. Such comparisons reveal underlying patterns or anomalies, shaping our understanding of different conditions affecting data.

Identifying Outliers in Data

Outliers are data points that significantly differ from other observations. Detecting outliers is an important step in statistical analysis since they might indicate variability in measurement, experimental errors, or novel insights about the data source.

• In a boxplot, any point beyond the whiskers is considered a potential outlier.
• They can be a result of unique conditions, as seen with Coors Field's high elevation affecting baseball games.
• Outliers can influence mean and standard deviation; hence, it's beneficial to consider them when analyzing data.

Understanding and managing outliers help in making accurate interpretations and decisions, especially in fields relying heavily on data-driven analysis. In our American vs National League analysis, addressing outliers provides crucial insights into unusual data behavior and condition-based performance. By determining the reasons behind outliers, such as environmental factors,"`` influence, we enhance our understanding of the data narrative.