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Chapter 2: Problem 105

Golfers' gains During the 2010 Professional Golfers Association (PGA) season, 90 golfers earned at least \(\$ 1\) million in tournament prize money. Of those, 5 earned at least \(\$ 4\) million, 11 earned between \(\$ 3\) million and \(\$ 4\) million, 21 earned between \(\$ 2\) million and \(\$ 3\) million, and 53 earned between \(\$ 1\) million and \(\$ 2\) million. a. Would the data for all 90 golfers be symmetric, skewed to the left, or skewed to the right? b. Two measures of central tendency of the golfers' winnings were \(\$ 2,090,012\) and \(\$ 1,646,853 .\) Which do you think is the mean and which is the median?

Short Answer

Expert verified
a. The data is skewed to the right. b. \(\$2,090,012\) is the mean; \(\$1,646,853\) is the median.

Step by step solution

01

Understanding the distribution of earnings

First, examine the distribution. 53 golfers earned between \(\\(1\) million and \(\\)2\) million, which is the largest group. As earnings increase, the frequency of golfers decreases, with only 5 earning \(\$4\) million or more. This indicates that the majority earn towards the lower end, with a few earning significantly more. Hence, the distribution tails off to the right, showing a few golfers with very high earnings.
02

Determining distribution skewness

In distribution terms, more golfers are concentrated towards the lower end (\(\\(1\) million to \(\\)2\) million), with fewer golfers at the higher earnings end. Since there are a few high earners (tail on the right), the distribution is skewed to the right.
03

Identifying measures of central tendency

The two measures of central tendency provided are \(\\(2,090,012\) and \(\\)1,646,853\). Since the distribution is right-skewed, the mean will be affected by the higher values and will typically be larger than the median. Therefore, \(\\(2,090,012\) is likely the mean and \(\\)1,646,853\) the median.

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Key Concepts

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

Skewness
When looking at data such as the earnings of professional golfers, it's essential to determine how the data is distributed. Skewness tells us about the asymmetry of data. In our exercise, we observe that most of the golfers earn between $1 million and $2 million, with fewer golfers earning much more. This creates a tail that stretches out towards higher earnings. When a distribution has a tail on the right, like this one, it is referred to as being right-skewed.

To recognize skewness:
  • Right-skewed: The tail is on the right side, indicating a few high values.
  • Left-skewed: The tail is on the left side, indicating a few low values.
  • Symmetric: Both sides of the distribution are mirror images.
In summary, skewness helps us understand the balance of data, and in this case, most golfers earn less, with some earning a lot more.
Mean and Median
The concepts of mean and median are fundamental when dealing with data. They help us determine the average and the middle point.
  • The **mean** is the sum of all values divided by the number of values. It's susceptible to extremely high or low values, which can skew it.
  • The **median** is the middle value of a data set that has been ordered from smallest to largest. It is less affected by outliers and skewed data.
In the context of golfers' earnings, the mean ( $2,090,012) is larger than the median ( $1,646,853) because high earners increase the mean value more significantly than they affect the median. This discrepancy often happens in a right-skewed distribution, and it's important to distinguish between these two measures of central tendency when analyzing real-life data.
Distribution Analysis
Analyzing a distribution gives insight into how data is spread across different values. For the PGA golfers' earnings, most players are clustered in one earning range with a gradual decline as earnings increase. Understanding this allows us to make informed comments on the spread of data.
  • **Concentration**: Here, golfers are mainly concentrated between $1 million and $2 million.
  • **Tail**: The presence of few golfers with high earnings extends the data to the right, creating a skewed tail.
By breaking down the distribution, analysts can uncover the nature of data, allowing for informed interpretations. In this case, it reveals a typical right-skewed distribution often seen in income and earnings data, where a few high earners significantly affect the distribution's shape. This understanding is crucial for accurate data analysis and reporting.

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Most popular questions from this chapter

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Categorical/quantitative difference a. Explain the difference between categorical and quantitative variables. b. Give an example of each.

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