Question

Exercise 21 looked at the running times of movies released in \(2005 .\) The standard deviation of these running times is \(19.6\) minutes, and the quartiles are \(Q_{1}=97\) minutes and \(Q_{3}=119\) minutes. a) Write a sentence or two describing the spread in running times based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain.

Short answer

Quartiles imply a 22-minute spread in central 50%, while standard deviation is 19.6 minutes overall. Quartiles are robust against outliers, but standard deviation might mislead if data is skewed.

Step-by-step solution

Understanding Quartiles and Spread

The quartiles divide the data into four equal parts. Here, we have \(Q_1 = 97\) minutes and \(Q_3 = 119\) minutes, which implies that 50% of movie running times are between these values. The interquartile range (IQR) is the range from \(Q_1\) to \(Q_3\), which is \(119 - 97 = 22\) minutes. This indicates that the central spread of the running times is 22 minutes.

Understanding Standard Deviation and Spread

The standard deviation gives a measure of the average deviation of each running time from the mean. The standard deviation is 19.6 minutes, suggesting that, on average, movie running times deviate by 19.6 minutes from the mean. This provides information on the overall spread around the average.

Concerns with Using Quartiles

Using quartiles and the interquartile range (IQR) is robust to outliers, but it doesn't reflect the potential variability within the upper and lower 25% of the data. If the data has significant skew or outliers in those regions, quartiles may not accurately reflect the total spread.

Concerns with Using Standard Deviation

Standard deviation is sensitive to outliers and skewed data. If the data is not normally distributed or has extreme values, the standard deviation might give a misleading impression of variability. It's essential to know the data distribution when interpreting standard deviation.

Key concepts

Quartiles

The term "quartiles" divides a data set into four equal sections. They help describe the distribution's shape and spread. In any given data set, the first quartile \(Q_1\) is the median of the lower half, and the third quartile \(Q_3\) is the median of the upper half. For the running times of movies in 2005:

• \(Q_1 = 97\) minutes: This means that 25% of the movies have running times less than or equal to 97 minutes.
• \(Q_3 = 119\) minutes: This indicates that 75% of the movies have running times less than or equal to 119 minutes.

Using quartiles, we find that the middle 50% of movies have running times between 97 and 119 minutes. The Interquartile Range (IQR) is the difference between \(Q_3\) and \(Q_1\), which is:\[IQR = Q_3 - Q_1 = 119 - 97 = 22 \text{ minutes}\]This value represents the spread of the middle half of the data. Quartiles and IQR are useful because they resist the influence of outliers. However, they can overlook the spread of extreme values.

Standard Deviation

Standard deviation is a key measure in statistics that reflects how much variation there is from the mean. For the movie running times in 2005, the standard deviation is 19.6 minutes. This suggests the following:

• On average, each movie's running time is about 19.6 minutes away from the average running time.
• It provides a sense of how spread out the movie times are around the mean.

A large standard deviation indicates a wide spread of times, while a smaller one suggests they are closer to the mean. However, one must be cautious as standard deviation can be distorted by outliers or skewed data. When interpreting standard deviation, it's crucial to have insight into the data's normality, as significant skewness or unusual values can influence the measure.

Spread Measurement

Spread measurement in statistics helps illustrate the dispersion or variability in a data set. Both quartiles and standard deviation are methods to measure spread, each with its own strengths:

• **Interquartile Range (IQR)**: This is the range between \(Q_1\) and \(Q_3\), focusing on the middle 50% of the data. It's resistant to outliers but tells less about extremes.
• **Standard Deviation**: It provides a sense of average deviation from the mean, offering a complete view of variability but can be swayed by outliers or skewness.

When choosing a spread measurement method, consider the data distribution. If data contains outliers, IQR might be preferable due to its robustness. Conversely, if data is normally distributed without extreme values, standard deviation can offer a more nuanced view of spread. Understanding the nature and distribution of your data is paramount to selecting the right measurement tool.