Interpreting a Skewed Salary Survey
A small firm publishes salaries for its 11 employees (in thousands of dollars): 42, 45, 47, 48, 50, 52, 55, 58, 62, 70, 210. The 210 is the owner's salary. A recruitment brochure states: "Average salary at our firm is $67,000." A job applicant calculates the median and finds it is much lower. The applicant also notices the brochure does not say which average is used. The firm argues the mean is a legitimate statistic. The applicant wants to decide whether the brochure is misleading and what a fair summary would be.
1. Calculate the mean and median, and explain why they differ so much.
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Mean: sum = 42+45+47+48+50+52+55+58+62+70+210. Grouping: (42+45+47+48+50) = 232; (52+55+58+62+70) = 297; plus 210. Total = 232 + 297 + 210 = 739. Mean = 739 ÷ 11 = 67.18 (thousand) ≈ $67,000. The brochure's figure is correct as a mean.
Median: with 11 values ordered, the median is the 6th value = 52 (thousand) = $52,000.
Why they differ: the distribution is strongly right-skewed. The single value 210 is far above the rest (the next highest is 70). The mean is pulled upward by this outlier because it uses every value's magnitude; the median depends only on position, so one extreme value barely moves it. The gap ($67k vs $52k) is about $15k, almost entirely due to the owner's salary. Most employees earn between $42k and $70k, so $52k better represents a typical worker.
2. Assess whether the brochure statement is misleading, and state what additional information would make it fair.
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The statement is technically true but materially misleading.
- It reports a real statistic (the mean is $67.2k), so it is not a lie.
- But a reader naturally interprets "average salary" as "what I might earn." No ordinary employee earns near $67k; only the owner exceeds it. The mean is inflated by one outlier.
- Omitting which average was used, and omitting the median, exploits the ambiguity of the word "average."
To be fair, the brochure should include:
- The median ($52k), which represents a typical employee.
- The range or the number of staff, so readers see the spread.
- A note that the mean is influenced by senior/owner pay.
A defensible summary: "Median salary $52,000; mean $67,000 (includes owner)." This lets the reader judge for themselves rather than being steered by a single favourable number. The core judgement is that selecting the statistic that flatters the firm, while suppressing the one that describes the typical worker, is misleading even though each number is accurate.