Problem 19 - Algebra 2

Question

The Millersburg school board is negotiating a pay raise with the teacher's union. Three of the administrators have salaries of \(\$ 90,000\) each. However, a majority of the teachers have salaries of about \(\$ 45,000\) per year. You are a member of the school board and would like to show that the current salaries are reasonable. Would you quote the mean, median, or mode as the “average” salary to justify your claim? Explain.

Step-by-Step Solution

Answer

Quote the median or mode to show that the typical salary is $45,000.

Step 1: Understand the Salary Distribution

The salary distribution includes three administrators with salaries of $90,000 each, and a majority of teachers with salaries around $45,000. This indicates a skewed distribution due to the much higher salaries of the administrators compared to the teachers.

Step 2: Analyze the Mean Salary

The mean salary is calculated by summing all the salaries and dividing by the number of data points. Due to the presence of salaries significantly higher than others, the mean will be lifted upwards by the higher administrator salaries, potentially not accurately representing the majority of the teachers' salaries as reasonable.

Step 3: Consider the Median Salary

The median is the middle value when all salaries are ordered from least to greatest. In a distribution where the majority of salaries are at $45,000, the median will likely be close to this amount, making it a good measure of the center that reflects what most teachers earn, showing that the typical teacher salary is reasonable.

Step 4: Evaluate the Mode of the Salaries

The mode is the most frequently occurring salary. Since most teachers earn about $45,000, the mode will be $45,000. This provides a clear picture of the most common salary, emphasizing that while a small number of administrators earn much more, the typical salary is $45,000.

Step 5: Determine the Best Measure of Central Tendency

Given that the mean is skewed by the high salaries of a few individuals, the median and mode are better representations of the typical teacher salary. They will provide a more realistic view of what the majority earns, supporting the argument that teacher salaries are reasonable.

Key Concepts

Mean

The mean, often referred to as the average, is calculated by adding up all the numbers in a data set and dividing by the number of values. In the context of salary distribution for Millersburg school, the mean takes the total sum of the salaries of both teachers and administrators and divides by the number of individuals. Here's the issue: the mean can be heavily influenced by extreme values, like the much higher salaries of administrators.

Imagine if only a few individuals make significantly more money than the majority. This scenario will cause the mean to be higher than what most teachers actually earn. Therefore, while calculating the mean gives us a numerical average, it may not accurately reflect the typical salary for most teachers in a skewed salary distribution.

Median

The median is another measure of central tendency, which is the middle value when all data points are ordered from the smallest to the largest. Its primary strength in a salary distribution like Millersburg’s is its resistance to skew from outliers.

Unlike the mean, the median remains unaffected by extremely high or low values. This means that if you arranged all the teacher and administrator salaries in order, the median would be around the salary of most teachers, rather than being pulled toward the higher end by administrator salaries. Hence, the median would be a more accurate reflection of the typical teacher salary in this scenario.

• Arrange all salaries in order
• Identify the middle salary value
• Resistant to extremely high or low salaries

Mode

The mode is the value that appears most frequently in a data set. It shows the most common salary within the group. In Millersburg, where most teachers have similar salaries, the mode provides a straightforward picture of what most people earn.

If the teachers' salaries are primarily around $45,000, the mode becomes a clear indicator of the typical salary, minimally influenced by outliers. The mode is especially useful in categorical data or when the data has a lot of repetition, as it shows the "peak" of the distribution - in this case, what most teachers actually earn.

• Identifies the most frequent salary
• Unaffected by outlier salaries
• Clear representation in categorical or repetitive data

Salary Distribution

A salary distribution breaks down the range and frequency of salaries in an organization. In Millersburg, we see two distinct groups: a handful of administrators with high salaries and a larger group of teachers with salaries around $45,000.

This setup results in a distribution that is not evenly spread or normally distributed, often raising questions about the fairness and representation of salaries within the organization. Understanding this distribution is crucial in choosing the right measure of central tendency that accurately reflects the majority's earnings.

Skewed Distribution

In statistical terms, a skewed distribution occurs when a few values are drastically different from the rest. In Millersburg's salary case, the administrators’ high salaries skew the distribution to the right. When a distribution is skewed, the mean is affected the most, while median and mode remain more stable.

In a right-skewed distribution like this, the mean salary will be higher because it incorporates the larger administrator salaries. Thus, the median and mode become more reliable indicators of a typical teacher’s salary, providing a more realistic view of what most teachers earn.

• Right-skewed due to high salaries
• Mean higher than median and mode
• Median and mode provide better central measures