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The top seven movies based on DC comic book characters for the U.S. box office as of fall 2017 are shown in the following table, rounded to the nearest hundred million. (Source: ultimatemovieranking.com) a. Find and interpret the median in context. b. Find and interpret the IQR in context. c. Find the range of the data. Explain why the IQR is preferred over the range as a measure of variability. $$ \begin{array}{|lc|} \hline \text { Movie } & \begin{array}{c} \text { Adjusted Domestic } \\ \text { Gross (\$ millions) } \end{array} \\ \hline \text { The Dark Knight (2008) } & \$ 643 \\ \text { Batman (1989) } & \$ 547 \\ \hline \text { Superman (1978) } & \$ 543 \\ \hline \text { The Dark Knight Rises (2012) } & \$ 487 \\ \text { Wonder Woman (2017) } & \$ 407 \\ \text { Batman Forever (1995) } & \$ 366 \\ \hline \text { Superman II (1981) } & \$ 346 \\ \hline \end{array} $$

Step by step solution

01

Arranging the Data

First, arrange the movies in order of their gross earnings, from highest to lowest. The data used will be: 643, 547, 543, 487, 407, 366, 346. It is easier to find the statistical measures when the data is ordered.

02

Finding the Median

The median is the middle value when the data is arranged in order. In this case, with 7 movies, the median will be the 4th movie when listed from highest to lowest gross earnings. Therefore, the median gross earning is \$487 million (The Dark Knight Rises). Thus, the middle grossing movie of these top seven DC comic movies made \$487 million.

03

Finding the Interquartile Range (IQR)

IQR is the range of the middle half of the data. It's found by determining the 1st quartile (Q1 - the median of the lower half) and the 3rd quartile (Q3 - the median of the upper half), and then subtracting Q1 from Q3. For this data, Q1 is the median of 643, 547, and 543 which is \$547 million and Q3 is the median of 407, 366, and 346 which is \$366 million. Therefore, IQR = Q3 - Q1 = \$366M - \$547M = -\$181 million. Thus, the middle half of the gross amounts for the top seven DC comic movies have a range of \$181 million.

04

Finding the Range

The range is the difference between the highest and lowest values. Therefore, the range for this data is \$643M (The Dark Knight) - \$346M (Superman II) = \$297 million. Thus, the highest and lowest grossing movies of these top seven DC comics movies range by \$297 million.

05

Interpreting the IQR vs the Range

IQR is preferred over the range as a measure of variability because the range can be greatly affected by extreme values. In this case, IQR shows the variability where most of the gross earnings fall (middle 50%), and it's not affected by the possibly skewed values of the highest and lowest grossing movies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interquartile Range (IQR)

The Interquartile Range (IQR) is an essential concept in statistics used to measure the spread of the middle 50% of a data set. It's particularly useful when we want to understand the variability without the influence of extreme values or outliers. The IQR is calculated by subtracting the first quartile (Q1) from the third quartile (Q3). In the context of the top DC comics movies by gross earnings, Q1 is determined by finding the median of the lower half of the earnings, while Q3 is found by identifying the median of the upper half.
To find the IQR for these movies:

• Order the earnings from highest to lowest: 643, 547, 543, 487, 407, 366, 346.
• Identify Q1 (median of lower half: 643, 547, 543). Here, Q1 is 547 million.
• Identify Q3 (median of upper half: 407, 366, 346). Here, Q3 is 366 million.
• Calculate IQR = Q3 - Q1 = 366 - 547 = -181 million.

The negative sign may seem confusing, but typically in practice, we take the absolute value, indicating a spread of 181 million dollars in our context. The IQR is preferred because it focuses on the central distribution, offering a clearer view of where most data points lie.

Median

The median is a simple yet powerful measure of central tendency in a data set. It is the middle value that separates the higher half from the lower half when the data is arranged in order. Unlike the mean, the median is not impacted by extreme values, making it more stable in skewed distributions.
When dealing with the DC comics movies' earnings, we first arrange the data from highest to lowest: 643, 547, 543, 487, 407, 366, 346. Since there are seven movies, the median is the fourth value when listed, which is 487 million dollars. This means The Dark Knight Rises, the movie with the median earnings, earned 487 million dollars. Thus, the median gives us a good idea of the "center" of the earnings, representing a typical or central movie's earnings in this dataset.

Range

The range is the simplest measure of variability in a dataset, calculated as the difference between the largest and smallest values. It provides a quick snapshot of how spread out the data points are. For the DC comics movie earnings, the highest value is 643 million (The Dark Knight), and the lowest is 346 million (Superman II).
To find the range, we subtract the lowest earnings from the highest:

• Range = 643 - 346 = 297 million.

While the range offers a basic measure of data spread, it can be misleading if there are outliers since it takes into account only extreme values. This is why statistical measures like the IQR may be preferred when looking for a comprehensive understanding of data variability. They focus on the middle 50% of the data, providing more accurate insights without the skew caused by outliers.