Question

As professional sports teams become more and more profitable, the salaries paid to the players have also increased. In fact, many sports superstars are paid huge salaries. If you were asked to describe the distribution of players' salaries for several different professional sports, what measure of center would you choose? Why?

Short answer

Answer: The median is the most appropriate measure of center for describing the distribution of professional players' salaries. This is because the median is resistant to outliers and provides a more accurate representation of the central tendency in the distribution, unlike the mean, which is sensitive to outliers, or the mode, which might be difficult to determine given the diversity of salaries.

Step-by-step solution

Understand the different measures of center

There are three main measures of center: mean, median, and mode. 1. Mean: The average of all the data points. It is calculated by adding up all the values and then dividing by the number of values. 2. Median: The midpoint value of a dataset when it is ordered from least to greatest. 3. Mode: The value that appears most often in a dataset.

Consider characteristics of the salaries data

The exercise states that sports superstars are paid huge salaries. Therefore, we know that there is likely to be a skew in the distribution and some outliers (the superstars) since their salaries will be significantly higher than the rest of the players.

Choose the best measure of center based on the data characteristics

Based on the characteristics of the salary data, choosing the median as the measure of center would be the most appropriate in this situation. This is because the median is resistant to outliers, meaning that it will not change significantly with the presence of a few very high or low values. As a result, it will provide a more accurate representation of the central tendency in the distribution of players' salaries.

Explain why the other measures of center are not suitable

The mean would not be a suitable choice because it is sensitive to outliers, and due to the presence of sports superstars with significantly higher salaries, the mean would be dragged upwards and would not provide an accurate representation of the central tendency of the data. The mode would not be a suitable choice either since there is a high probability that many players earn different salaries, making the mode difficult to determine or having several modes, which wouldn’t effectively represent the distribution of players' salaries. In conclusion, the median is the best measure of center for this exercise, as it is robust to outliers and provides an accurate representation of the central tendency of the professional players' salaries distribution.

Key concepts

Mean

The mean, often referred to as the average, is a foundational concept in statistics. To calculate the mean, you sum up all the values in a dataset and then divide by the count of the values. In mathematical terms, if you have a dataset with values \( x_1, x_2, ..., x_n \) the mean \( \bar{x} \) is calculated as \( \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i \).

Despite its widespread use, its main drawback is sensitivity to extreme values, or 'outliers'. In distributions with outliers, such as a sports team's salaries where a few superstars may earn exponentially more than their teammates, the mean can become an overestimation of what most players earn. It represents the 'balance point' of the dataset, but in skewed distributions, this balance can be misleading as it doesn't necessarily represent the typical value in the dataset.

Median

The median is the value that divides a dataset in half, with an equal number of values above and below it. To find the median, you first need to arrange the data in ascending order and then identify the middle value. If there is an even number of observations, the median is the average of the two middle values. It's symbolically expressed as the middle value of \( x_{(n+1)/2} \) in an ordered set, or \( \frac{x_{n/2} + x_{(n/2)+1}}{2} \) if the dataset has an even number of values.

Unlike the mean, the median is unaffected by outliers or skewed distributions, making it a more robust measure of central tendency in such cases. It's particularly useful when describing data such as income or property value, where high-end outliers can misrepresent the average.

Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or even no mode at all if no data point repeats. It is particularly useful in understanding categorical data or data that are likely to cluster around certain values.

For instance, in a shoe store, the most common shoe size sold would be the mode of the shoe size dataset. However, when it comes to salaries, especially in professional sports where individual contracts vary widely, determining a mode can be impractical or may not add meaningful insight into the central tendency of the dataset.

Skewed Distribution

In a skewed distribution, data points are not symmetrically distributed around the mean. There are two types of skewness: right-skewed (positive skew) and left-skewed (negative skew). In right-skewed distributions, the tail on the right side (higher values) is longer, indicating that there are a number of outliers pulling the distribution in that direction.

Using measures of central tendency such as the mean in a skewed distribution can be misleading, as these measures might not accurately reflect the dataset's typical value. It is crucial to identify skewness as it affects the interpretation of the data and the choice of the appropriate measure of central tendency.

Outliers

Outliers are data points that differ significantly from other observations in a dataset; they can be exceptionally high or low. Outliers may occur due to variability in the measurement or possibly due to experimental error. They are important to recognize as they can greatly affect the mean and skew the analysis of data.

Statistical tools, including the median and interquartile range, are often used to get a sense of the typical values in a dataset that includes outliers. Outliers require careful consideration; they should not be immediately disregarded as they may contain valuable information about the dataset, such as the presence of superstars in sports salary data.