Question

As professional sports teams become a more and more lucrative business for their owners, the salaries paid to the players have also increased. In fact, sports superstars are paid astronomical salaries for their talents. If you were asked by a sports management firm to describe the distribution of players" salaries in several different categories of professional sports, what measure of center would you choose? Why?

Short answer

Answer: The median is the most appropriate measure of center to describe the distribution of players' salaries in different categories of professional sports.

Step-by-step solution

Understanding the measures of center

The mean (average) is calculated by dividing the sum of the values by the number of values. This is probably the most widely known measure of center, but it can be easily influenced by extreme values known as outliers. In the context of professional sports salaries, outliers are common, as some exceptional players might earn significantly more than the majority of others. This means that the mean might not always be an accurate representation of the central tendency of the data set. The median is the middle value of a dataset in a sorted order. If the dataset has an odd number of values, the median would be the middle value. If there is an even number of values, the median would be the average of the two middle values. The median is less influenced by outliers than the mean, which makes it more suitable for situations with a skewed distributions like the sports salaries. The mode is the value that occurs most frequently in the dataset. For salaries, this might not be the most useful measure of center, since salaries can differ by even one cent, leading to no mode or multiple modes. Therefore, the mode is less relevant in this case.

Choosing the appropriate measure of center

Given the potential for extreme outlier salaries in professional sports, as well as the continuous nature of salaries where exact matches are rare, the median is the most appropriate choice as a measure of center for describing the distribution of salaries. The median represents the value that separates the highest 50% of salaries from the lowest 50% of salaries, and is therefore less influenced by extreme values, providing a better description of the overall distribution of sports salaries.

Conclusion

In this case, to describe the distribution of players' salaries in several different categories of professional sports, it is recommended to use the median as the measure of center. This choice takes into account that salaries are prone to extreme values (outliers), and it's less influenced by those values when compared to the mean or mode.

Key concepts

Mean

The mean is often what people think of when they hear "average." It’s a calculation that shows the typical value in a data set by adding up all the numbers and dividing by how many numbers there are. For example, if you wanted to find the mean salary of a group of players, you would sum up all their salaries and divide that sum by the number of players.
However, the mean has a big drawback: it's sensitive to outliers. Outliers are values much higher or lower than the rest of the data. In professional sports, star players might have salaries that are much higher than others. These high salaries can dramatically increase the mean, giving a misleading picture of the typical salary.
Choosing the mean as the measure of center can sometimes lead to misinterpretation in such cases. It may not accurately reflect what most players earn.

• Simple to calculate: Add all numbers, then divide.
• Sensitive to outliers: Might not represent typical values due to extreme values.
• Useful: When the data has no extreme values, the mean gives a good sense of the average.

Median

The median is the middle value in a data set listed in order. When we find the median, we line up all the values from lowest to highest and find the number that divides the data into two equal halves. If there is an odd number of items, the median is the exact middle one; if even, it’s the average of the two middle numbers.
The beauty of the median is its resistance to outliers. In a list of salaries, a superstar’s salary won’t affect the median much. It strikes a balance by showing where the middle of the pack lies.
For professional sports salaries, this feature makes the median more reliable than the mean. With salaries ranging widely due to a few high earners, the median can paint a clearer picture of what a typical salary might look like.

• Resistant to outliers: Largely unaffected by extreme values.
• Reflective of typical values: Gives a better sense of the center in skewed data.
• Ideal for skewed distributions: Like those of most salaries.

Mode

The mode is the value that occurs most frequently in a data set. It tells us which value appears more often than the rest. When data is grouped into categories, like clothing sizes, the mode can be very helpful.
However, when it comes to professional salaries, the mode might not be very beneficial. Salaries are unique and can differ by even small amounts like a cent, making it possible to have no mode or many modes.
This is why the mode isn't the most suitable measure when discussing sports salaries. While it can show which salary appears most often, it doesn’t say a lot about the overall distribution of salaries.

• Useful for categorical data: Excellent when items naturally fall into categories.
• Less informative with unique values: Limited use in detailed numerical data such as salaries.
• Multiple modes possible: Can have more than one mode if numerous values occur with the same frequency.