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Ranking driving performance of professional golfers.A group of researchers developed a new method forranking the total driving performance of golfers on theProfessional Golf Association (PGA) tour (The Sport Journal,Winter 2007). The method requires knowinga golfer鈥檚 average driving distance (yards) and drivingaccuracy (percent of drives that land in the fairway).The values of these two variables are used to compute adriving performance index. Data for 40 PGA golfers (asranked by the new method for a recent tour year) aresaved in the accompanying file. The first five and last fiveobservations are listed in the table below.

a.Find the mean, median, and mode for the 40 driving performance index values.

b.Interpret each of the measures of central tendency, part a.

c.Use the results, part a,to make a statement about the type of skewness in the distribution of driving performance indexes. Support your statement with a graph.

Step by step solution

01

Finding the mean, median, and mode for the 40 driving performance index values

The mean is calculated as follows:

$$\begin{gathered} \mathbf{Mean}\mathbf{=}\frac{\mathbf{Sum}\mathbf{}\mathbf{of}\mathbf{}\mathbf{all}\mathbf{}\mathbf{driving}\mathbf{}\mathbf{performance}\mathbf{}\mathbf{index}}{\mathbf{No}\mathbf{.}\mathbf{of}\mathbf{}\mathbf{performing}\mathbf{}\mathbf{index}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{77}\mathbf{.}\mathbf{07}}{\mathbf{40}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{93} \end{gathered}$$

Therefore, mean = 1.93

The median is the centermost value of the data set. Since the data set is even, we will calculate the median using the 2 central values.

$$\begin{gathered} \mathbf{Median}\mathbf{=}\frac{{\mathbf{20}}^{\mathbf{th}}\mathbf{+}{\mathbf{21}}^{\mathbf{st}}\mathbf{value}}{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{1}\mathbf{.}\mathbf{75}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{76}}{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{3}\mathbf{.}\mathbf{51}}{\mathbf{2}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{755} \end{gathered}$$

Therefore, the median is 1.755.

The value repeated maximum times is 1.4. Therefore 1.4 is the mode.

02

Interpreting the measures of central tendency, part a

As per the mean, the average driving performance of golfers ranks around 1.93. While 50% of golfers鈥� performance is below 1.755 and the performance of the rest 50% is above 1.755. And the most common performance index is 1.4.

03

Determining the skewness in the distribution of driving performance indexes

The distribution of driving performance indexes is rightward skewed because the mean is greater than the median. The extreme values on the right tail caused the mean to shift to the right.

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