Question

Distribution of Blocked Shots in the NBA The variable Blocks in the dataset NBAPlayers2015 includes information on the number of blocked shots during the season for each of the 182 players in the dataset. (a) Use technology to find the mean and the standard deviation of the number of blocked shots. (b) Use technology to find the five number summary for the same variable. (c) Which set of summary statistics, those from part (a) or part (b), is more resistant to outliers and more appropriate if the data are heavily skewed? (d) Use technology to create a graph of the data in Blocks and describe the shape of the distribution. (e) Is it appropriate to use the \(95 \%\) rule with these data? Why or why not?

Short answer

Mean and standard deviation can be obtained for the number of blocked shots. A five number summary can be also calculated. The five number summary (part b) is more resistant to outliers and more appropriate if the data are heavily skewed. A histogram can be created to view the data distribution. The suitability of the \(95 \%\) rule will depend on whether the data is normally distributed or not.

Step-by-step solution

Find the Mean and Standard Deviation

Use the 'mean' and 'standard deviation' function in the spreadsheet or statistical software to calculate the mean and standard deviation of the data in the 'Blocks' column. The mean is the average of the data, while the standard deviation measures the amount of variation or dispersion of a set of values. Both are measures of central tendency.

Calculate the Five Number Summary

The five number summary consists of the minimum, first quartile (25th percentile), median (second quartile or 50th percentile), third quartile (75th percentile), and maximum. These can be calculated using the 'min', 'quartile', 'median' and 'max' functions in your spreadsheet or statistical software.

Analyze Resistance to Outliers

Part (a) uses the mean and standard deviation, which are affected by outliers. If an extremely large or small value is added, the mean and standard deviation will change significantly. Part (b), however, uses median and quartiles. These values are more resistant to outliers because they depend on the position of the data, not their value. Therefore, part (b) is more appropriate if the data is heavily skewed or has outliers.

Create a Graph

Creating a graph will help visualize the distribution of the dataset. A suitable type for this dataset is a histogram. The horizontal axis will represent blocks and the vertical axis will represent the number of players with a particular number of blocks.

Analyze the 95% Rule

The \(95 \%\) rule, or the empirical rule, states that for a normal distribution, \(95 \%\) of the data will fall within 2 standard deviations of the mean. If our data set is normally distributed, the rule can be applied. However, if the data is heavily skewed or has substantial outliers, the rule may not hold true. This can be determined by analyzing the shape of the created graph and comparing it with a normal distribution curve.