Question

We use data from HollywoodMovies introduced in Data 2.7 on page \(95 .\) The dataset includes information on all movies to come out of Hollywood between 2007 and 2013 . The variable AudienceScore in the dataset HollywoodMovies gives audience scores (on a scale from 1 to 100 ) from the Rotten Tomatoes website. The five number summary of these scores is (19,49,61,74,96) . Are there any outliers in these scores, according to the \(I Q R\) method? How bad would an average audience score rating have to be on Rotten Tomatoes to qualify as a low outlier?

Short answer

Using the IQR method for outlier detection, there are no outliers in the Audience Score data. The lower outlier boundary is 11.5 which means a movie rating would have to be lower than 11.5 to be considered a low outlier.

Step-by-step solution

Calculation of the Interquartile Range (IQR)

The IQR can be calculated as Q3 - Q1. From the given data, Q1 is 49 and Q3 is 74. Therefore, IQR = 74 - 49 = 25.

Calculation of the Lower and Upper Outlier Boundaries

The lower and higher boundaries for outliers can be calculated as Q1 - 1.5*IQR and Q3 + 1.5*IQR respectively. For the lower boundary, replace Q1 and IQR with their calculated or given values, to get Lower Boundary = 49 - 1.5*25 = 11.5. So any rating below 11.5 would be considered a low outlier.

Check if there are any outliers

Now that the boundaries for outlier detection have been calculated, a comparison is made between minimum and maximum values in the dataset and the calculated boundaries. From the given data, minimum value is 19 and maximum value is 96. Both of these values lie within the calculated boundaries, so there are no outliers in this dataset according to the IQR method.

Key concepts

Interquartile Range (IQR)

The Interquartile Range, or IQR, is a measure of statistical dispersion and is considered a very robust tool for identifying the spread of the middle 50% of a dataset. In simpler terms, the IQR indicates the range within which the central half of the scores in a dataset lie.

To calculate IQR, one must first understand what quartiles are. Quartiles divide a rank-ordered dataset into four equal parts. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half. The IQR is the difference between Q3 and Q1, effectively covering the range from the 25th to the 75th percentile.

When considering Rotten Tomatoes audience scores or similar datasets, the IQR helps us see the span from moderately low to moderately high scores, excluding extremes which could skew our perception of the data's distribution.

Five Number Summary

The five number summary is a concise statistical description of a dataset. It consists of five numbers: the minimum value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum value. These numbers together provide a quick overview of the data's distribution, helping identify the center, spread, and shape in a clear and easy-to-understand manner.

For instance, the five number summary of Rotten Tomatoes audience scores includes:

• The minimum score (the lowest score obtained)
• Q1, representing the median of the lower half of the scores
• The median, which divides the dataset into two equal halves
• Q3, which is the median of the upper half of the scores
• The maximum score (the highest score obtained)

With these five measures, one can quickly grasp the range of scores and detect any potential asymmetry or outliers within the data.

Rotten Tomatoes Audience Scores

Rotten Tomatoes is a popular review-aggregation website for film and television. The audience scores on Rotten Tomatoes are particularly valuable because they reflect the opinions of regular viewers, not just critics. These scores are typically displayed on a 0 to 100 scale and can greatly influence a movie's public perception and success.

When analyzing data like the Rotten Tomatoes audience scores, using statistical tools like the IQR and the five number summary allows us to understand how well a movie was received by audiences. Since these scores are based on user-submitted ratings, they can vary widely, which is why understanding and detecting outliers is crucial for an accurate representation of audience opinion.

Outliers in Data

Outliers are data points that significantly differ from other observations. They can arise due to variability in the measurement or possibly indicate experimental error; sometimes, they may also be precisely what's of interest. In statistics, identifying outliers is critical as they can distort overall analysis results, leading to misleading conclusions.

Using the IQR method is one of several ways to detect outliers. This involves multiplying the IQR by a factor (commonly 1.5) and subtracting it from Q1 to find the lower boundary and adding it to Q3 for the upper boundary. Any data falling outside of these set boundaries is considered an outlier. For the HollywoodMovies data, no audience scores were extreme enough to be outliers. However, knowing how to detect outliers is essential for interpreting datasets objectively, especially in dynamically scored platforms like Rotten Tomatoes.