Question

The Environmental Protection Agency provides fuel economy and pollution information on over 2000 car models. Here is a boxplot of Combined Fuel Economy (using an average of driving conditions) in miles per gallon by vehicle Type (car, van, or SUV). Summarize what you see about the fuel economies of the three vehicle types.

Short answer

Cars generally have the highest median fuel economy, followed by vans and then SUVs, with SUVs showing the most variability.

Step-by-step solution

Identify Vehicle Types

Review the boxplot and identify the three vehicle types being compared: cars, vans, and SUVs. This distinction is crucial as each type offers different utility and performance characteristics.

Examine the Median Values

Locate the line inside each box on the plot; this represents the median fuel economy for each vehicle type. Compare these medians to see which vehicle type, on average, offers better fuel economy.

Assess the Range and Spread

Look at the length of each box as well as the whiskers extending from the boxes. The range of fuel economy for each vehicle type can be determined by the position of the whiskers, which illustrates the spread from the minimum to maximum values.

Identify the Interquartile Range (IQR)

The IQR is the distance between the first and third quartiles (the ends of the box). It represents the middle 50% of the data. Understand which vehicle types have a smaller or larger IQR, indicating variability in fuel economy.

Analyze Outliers

Check for any points plotted outside the whiskers as these represent outliers. Identify which vehicle types have outliers and consider what this indicates about their fuel efficiency distributions.

Compare Fuel Economies

Synthesize the observations: Compare medians, ranges, and outliers to summarize the overall fuel economy differences across vehicle types, noting which tends to have higher or more variable fuel economies.

Key concepts

Boxplot Interpretation

Boxplots are a powerful visualization tool used to summarize data distributions at a glance. In a boxplot, data is divided into quartiles, making it easy to identify key statistical figures such as median, range, and interquartile range (IQR).

To interpret a boxplot, focus on its main components:

• Box: The central rectangle shows where the middle 50% of data resides, known as the interquartile range (IQR).
• Median: The line within the box marks the median of the data. It's the middle value when data points are arranged in order.
• Whiskers: They extend from the box to the smallest and largest observations within a credible range. These show the spread of the data.
• Outliers: Sometimes, individual data points fall outside the whiskers, indicating they lie an unusual distance away from the other data points.

By analyzing these elements, one can quickly assess central tendencies, variability, and potential anomalies within any dataset.

Fuel Economy Comparison

Comparing fuel economies using boxplots involves examining the distribution of miles per gallon for various vehicle types such as cars, vans, and SUVs. Each vehicle type will have its own boxplot, allowing for an easy visual comparison.

• Medians: The position of each median line within a box is crucial. A higher median indicates better fuel economy for that vehicle type. For instance, if the median of cars is higher than that of vans and SUVs, cars are generally more fuel-efficient.
• Spread: The length of the whiskers helps determine how varied the fuel economies are within a vehicle category. Longer whiskers indicate a wider range of fuel consumption, suggesting more variability in vehicle performance.
• IQR: By looking at the width of the boxes, you can see which vehicle type has the most consistent fuel economy. A smaller IQR indicates that the data points are tightly clustered around the median, showing less variability.
• Outliers: These influence the perception of fuel economy consistency. An SUV with extreme values might suggest particular models that significantly deviate from typical performance.

Using these comparisons, one can synthesize a broader understanding of which vehicle types tend to be more uniform in fuel efficiency and which offer more variable fuel economy options.

Interquartile Range (IQR)

The interquartile range (IQR) is a key measurement of variability in statistics, often used to assess the spread of a dataset's middle 50%. This concept is particularly important when interpreting boxplots.

To find the IQR, locate the first quartile (Q1), which marks the 25th percentile, and the third quartile (Q3), at the 75th percentile. The IQR is calculated as: \[ IQR = Q3 - Q1 \]

• A small IQR suggests that the data points are closely packed, indicating less variability in the dataset.
• A large IQR implies a wide range of values and greater variability.

When comparing the IQRs of different datasets, like the fuel economy of cars, vans, and SUVs, a smaller IQR for one vehicle type would mean that its fuel economy is more consistent compared to others. This is helpful in identifying which vehicles generally maintain a steady miles per gallon rate under varying conditions.

Understanding the IQR helps in gauging data distribution and is a solid base for making informed comparisons across different datasets.