Problem 20 - 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 the head of the teacher’s union and maintain that a pay raise is in order. Which of the mean, median, or mode would you quote to justify your claim? Explain your reasoning.

Step-by-Step Solution

Answer

Use the median or mode to justify a pay raise, as they reflect typical teacher salaries better than the mean.

Step 1: Understanding the Data

The administrators earn \(\\( 90,000\) each, while most teachers earn \(\\) 45,000\). This indicates a disparity in earnings between administrators and teachers.

Step 2: Identifying Data Distribution

List the salaries involved: several accrue \(\\( 45,000\), and three earn \(\\) 90,000\). The skew is towards higher salaries, as administrators earn more, but with fewer individuals impacted by that high salary.

Step 3: Calculating the Mean

Calculate the mean by dividing the sum of all salaries by the number of individuals. However, since no actual number of teachers is given, we assume many have \(\$ 45,000\), potentially skewing the mean towards administrators' salaries, making the mean appear relatively high.

Step 4: Calculating the Median

The median is the middle value when the salaries are ordered. With many teachers earning \(\\( 45,000\), the middle value will likely fall at or near \(\\) 45,000\), indicating that this is most reflective of the central tendency for teachers.

Step 5: Determining the Mode

The mode is the value that occurs most frequently in the dataset. As many teachers earn \(\\( 45,000\), this makes \(\\) 45,000\) the mode, which accurately represents the teachers' most common salary.

Step 6: Choosing the Measure

To argue for a pay raise, use the mode or median. Mode and median best represent what most teachers earn, emphasizing their low salaries compared to administrators, rather than the mean, which could be skewed by the higher administrative salaries.

Key Concepts

Mean

The mean, often referred to as the average, is calculated by summing all the values in a dataset and dividing by the number of values.
In the context of the Millersburg school board scenario, finding the mean involves adding all the administrators' and teachers' salaries and dividing this total by the number of employees.
A key point to remember with the mean is that it can be heavily influenced by extreme values, also known as outliers.

• In the given problem, each administrator's high salary of \(90,000\) acts as such an outlier.
• This drives the mean up, potentially above what a typical teacher earns.
• Thus, in this context, the mean might not offer an accurate representation of what most teachers earn.

For negotiations, using the mean salary could mislead people to believe that the typical salary is higher than it truly is for the majority of personnel.

Median

The median is the value separating the higher half from the lower half of a dataset when it is ordered.
In simpler terms, it's the middle number in a sorted list.
In the example of the school board, sorting all salaries with several teachers at \(45,000\) and three administrators at \(90,000\) places our focus on the middle value.

• If there is an odd number of total salaries, the median is the exact middle value.
• If there's an even number of salaries, it's the average of the two middle numbers.

Given that most teachers earn \(45,000\), this salary will likely serve as the median, clearly reflecting the central tendency of their pay.
Therefore, the median offers a more accurate depiction of the teachers' wage situation because it is less affected by the high salaries of the administrators.

Mode

The mode is the most frequently occurring value within a dataset.
When it comes to salaries, the mode represents the salary that is earned by the greatest number of people.

• In this context, most teachers earn around \(45,000\).
• Hence, \(45,000\) becomes the mode of the salary dataset.

By focusing on the mode, the representation of the typical salary scenario for the majority of teachers is maintained.
The mode would be a compelling metric to cite for requesting a pay raise, as it underscores the most common salary being significantly lower than the administrators.
Since the mode reflects the salary most frequently earned, it highlights the gap between the majority of teachers and the fewer higher-paid administrators.