The Misleading Average Salary
A small company has 9 employees, including the owner. The owner claims that the average (mean) salary is $45,000 per year, hoping to attract new workers. A job candidate, Jordan, is suspicious because she knows that most employees are paid hourly and earn much less. Jordan asks for the median salary, which the owner reveals is $30,000. Jordan also learns that at least 5 employees earn less than $35,000. She wants to understand how the mean can be so much higher than the median. She decides to construct a possible list of 9 salaries (in thousands of dollars) that satisfies these facts: the mean is 45, the median is 30, and at least 5 salaries are below 35. She also wonders which measure-mean or median-better represents a typical employee's salary, and why. Help Jordan by producing a valid list of 9 salaries and explaining the reasoning behind the difference between the two measures.
1. Construct a list of 9 salaries (in thousands) with mean 45, median 30, and at least 5 salaries below 35. Show that your list satisfies all conditions.
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We need 9 numbers. Median 30 means the 5th number (when sorted) is 30. At least 5 numbers below 35 means the first five numbers (including the median) can be below 35. Let's set the five smallest numbers to be 20, 22, 25, 28, 30. That's 5 numbers, all ≤30, so below 35. The four largest numbers must be at least 30 (to keep median 30) and we need the total sum to be 9*45 = 405. Sum of first five: 20+22+25+28+30 = 125. Remaining sum for four numbers: 405 - 125 = 280. We need four numbers, all ≥30. To make the mean high, let the owner have a very large salary. Let the four numbers be 30, 32, 40, 178? Wait, 30+32+40+178 = 280. But check sorted list: 20,22,25,28,30,30,32,40,178. Median is 5th = 30. Salaries below 35: 20,22,25,28,30,30,32 (seven numbers) all <35. Mean = 405/9=45. Works. However, to avoid duplicate 30s, we can use 20,22,25,28,30,33,35,42,170? Sum: 20+22+25+28+30+33+35+42+170 = 405? Let's compute: 20+22=42, +25=67, +28=95, +30=125, +33=158, +35=193, +42=235, +170=405. Sorted: 20,22,25,28,30,33,35,42,170. Median 5th = 30. Salaries below 35: 20,22,25,28,30,33 (six) <35. Good. So a valid list is 20,22,25,28,30,33,35,42,170 (thousands).
2. Explain why the mean is higher than the median in your list, and what causes the difference.
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The mean is pulled upward by the very large salary of 170 (the owner). The median, being the middle value, ignores the extreme high value. In our list, six of the nine salaries are at or below 33, so a typical employee earns around 28-30. The owner's huge salary adds 170 to the total, raising the mean to 45. The median of 30 reflects that half the employees earn 30 or less. The difference arises because the distribution is skewed right: a few very high values stretch the mean but do not affect the median much. This is why the mean can be misleading when there are outliers.
3. Which measure-mean or median-better represents a typical employee's salary in this company? Justify your answer.
Read the worked answer
The median is a better representation of a typical employee's salary. In this company, 5 out of 9 employees earn less than $35,000, and the median is $30,000, meaning half earn $30,000 or less. The mean of $45,000 is inflated by the owner's exceptionally high salary (e.g., $170,000 in our list). A new worker would likely earn closer to $30,000 than $45,000. The median is resistant to outliers, while the mean is sensitive to extreme values. Therefore, the median gives a more accurate picture of what a typical employee earns. The owner's use of the mean is misleading.