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Chapter 3: Problem 52

In 1994, Major League Baseball (MLB) players went on strike. At the time, the average salary was \(\$ 1,049,589\), and the median salary was \(\$ 337,500\). If you were representing the owners, which summary would you use to convince the public that a strike was not needed? If you were a player, which would you use? Why was there such a large discrepancy between the mean and median salaries? Explain. (Source: www.usatoday.com)

Short Answer

Expert verified
As an owner, you would use the mean (average) salary to argue against the need for a strike, highlighting that the average salary is over a million dollars. As a player, you would use the median salary, pointing out that half of the players earn less than \$ 337,500. The large discrepancy between the mean and median results from a skewed distribution of salaries, where a small number of superstars earn much more than the majority of players.

Step by step solution

01

Understand the Mean and Median

The mean (average) is found by adding all numbers in the data set and then dividing by the number of values in the set, while the median is the 'middle' value in the data set. The middle point is found by arranging all data points and picking out the one in the middle (or if there are two middle points, taking the mean of those two numbers).
02

Analyze the Mean Salary

The average salary of MLB players in 1994 was \(\$ 1,049,589\). Owners can use this value to argue against the strike as it portrays a high salary. This might make it seem like the players are earning plenty of money.
03

Analyze the Median Salary

The median salary in 1994 was \(\$ 337,500\), much lower than the average. If you were a player, you would cite the median salary in arguing that a strike was needed. This indicates that at least half the players earn less than \(\$ 337,500\) a year.
04

The Discrepancy between Mean and Median

The discrepancy between the mean and median is caused by the distribution of values within the data set. If the salaries were evenly distributed, the mean and median would be closer together. In this case, the existence of a few incredibly high salaries (from superstar players) 'skews' the distribution, driving up the average while leaving the median relatively low. This indicates an uneven salary distribution, with a few players earning much more than the rest.

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Key Concepts

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

Statistical Analysis
When engaging in statistical analysis, the goal is to extract meaningful information from data to make informed decisions. In the context of salaries, statistical measures such as mean and median provide insights into the overall compensation trends.

For instance, the average or mean salary is calculated by summing all individual salaries and dividing by the number of players. It gives an overall sense of the salary scale but can be influenced by extreme values.

On the other hand, the median salary is the middle value when all salaries are lined up from lowest to highest. It represents the 'central' salary and is not distorted by outliers. This makes the median a more reliable indicator when the data includes exceptionally high or low numbers.
Notably, thorough statistical analysis allows for understanding the nature of the data, highlighting disparities that might not be apparent at first glance. Understanding both mean and median salaries offers a more comprehensive view of the earnings landscape.
Mean and Median Interpretation
The interpretation of mean and median has real-world implications, particularly when discussing salaries. If representing MLB owners in 1994, highlighting the mean salary of \( \$ 1,049,589 \) suggests that players, on average, earned substantial wages, perhaps dissuading public sympathy for the strike.

The median salary of \( \$ 337,500 \) tells a different story. Players could argue that this figure is more representative of what most players make, thereby justifying the need for better pay. The median provides a more realistic picture of the financial situation of the 'typical' player, especially when the data is skewed by a few large salaries.

Choosing the Right Measure

Selecting which measure to spotlight鈥攎ean or median鈥攄epends on the argument one wishes to present, as each offers a distinct perspective on the same data set.
Data Set Distribution
The distribution of a data set is crucial as it affects statistical measures and the conclusions drawn from them. The fact that the mean and median MLB salaries in 1994 were significantly different signals an uneven distribution with a right-skewed shape.

A right-skewed distribution occurs when a data set contains a few much higher values than the rest, which pulls the mean towards these higher numbers. This is typical in salary distributions where top earners vastly outpace the average worker's income.

Impact of Skewness

This skewness inflates the mean, potentially painting an unrealistic picture of everyday earnings. Therefore, understanding the shape of data distribution helps in choosing the right measure for accurate representation, as it indicates the presence of outliers that can distort the average.

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Most popular questions from this chapter

Babies born weighing 2500 grams (about \(5.5\) pounds) or less are called low- birthweight babies, and this condition sometimes indicates health problems for the infant. The mean birth weight for U.S.-born children is about 3462 grams (about \(7.6\) pounds). The mean birth weight for babies born one month early is 2622 grams. Suppose both standard deviations are 500 grams. Also assume that the distribution of birth weights is roughly unimodal and symmetric. (Source: www.babycenter.com) a. Find the standardized score \((z\) -score), relative to all U.S. births, for a baby with a birth weight of 2500 grams. b. Find the standardized score for a birth weight of 2500 grams for a child born one month early, using 2622 as the mean. c. For which group is a birth weight of 2500 grams more common? Explain what that implies. Unusual z-scores are far from \(0 .\)

The tables below show the 2015 unemployment rates for states in the northeastern and midwestern regions of the United States. Compare the unemployment rates for the two regions, commenting on the typical unemployment rate of each region and then comparing the amount of variation in the unemployment rate for each region. (Source: 2017 World Almanac and Book of Facts) Northeast States \(\begin{array}{llllllll}5.6 & 5 & 3.4 & 5.6 & 5.3 & 5.1 & 5 & 3.7\end{array}\) Midwest States \(\begin{array}{llllllllll}5.9 & 4.8 & 3.7 & 4.2 & 5.4 & 3.7 & 5 & 3 & 2.7 & 4.9\end{array}\) \(\begin{array}{ll}3.1 & 4.6\end{array}\)

Assume that men's heights have a distribution that is symmetric and unimodal, with a mean of 69 inches and a standard deviation of 3 inches. a. What men's height corresponds to a \(z\) -score of \(2.00\) ? b. What men's height corresponds to a z-score of \(-1.50 ?\)

Data at this text's website show the number of central public libraries in each of the 50 states and the District of Columbia. A summary of the data is shown in the following table. Should the maximum and minimum values of this data set be considered potential outliers? Why or why not? You can check your answer by using technology to make a boxplot using fences to identify potential outliers. (Source: Institute of Museum and Library Services) $$ \begin{aligned} &\text { Summary statistics }\\\ &\begin{array}{lcccccccc} \text { Column } & \text { n } & \text { Mean } & \text { Std. dev. } & \text { Median } & \text { Min } & \text { Max } & \text { Q1 } & \text { Q3 } \\ \text { Central } & 51 & 175.76471 & 170.37319 & 112 & 1 & 756 & 63 & 237 \\ \text { Public } & & & & & & & \\ \text { Libraries } & & & & & & & & \end{array} \end{aligned} $$

In 2017 a pollution index was calculated for a sample of cities in the eastern states using data on air and water pollution. Assume the distribution of pollution indices is unimodal and symmetric. The mean of the distribution was \(35.9\) points with a standard deviation of \(11.6\) points. (Source: numbeo. com) see Guidance page \(142 .\) a. What percentage of eastern cities would you expect to have a pollution index between \(12.7\) and \(59.1\) points? b. What percentage of eastern cities would you expect to have a pollution index between \(24.3\) and \(47.5\) points? c. The pollution index for New York, in 2017 was \(58.7\) points. Based on this distribution, was this unusually high? Explain.
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