Choosing a Measure of Center Under Skew
A statistics class analyzes the salaries of 11 employees at a small company: one owner earns $400,000; ten employees earn between $32,000 and $48,000. The mean salary is about $80,000; the median is $40,000. A student, Priya, reports "the typical employee earns about $80,000" and the class is debating whether that statement is fair. The teacher wants students to connect the shape of the distribution to the choice of summary statistic and to reason about which measure better represents a "typical" value here. A second group proposes reporting the mode instead.
1. Explain, using the specific numbers, why the mean and median diverge so sharply, and evaluate Priya's claim.
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The distribution is strongly right-skewed: ten of eleven salaries cluster in a narrow band ($32k–$48k) while a single value ($400k) sits far to the right.
- The median ($40k) is the 6th ordered value, so it depends only on position-the outlier's magnitude is irrelevant to it. It sits inside the cluster.
- The mean adds all values and divides by 11. The $400k owner contributes $400,000/11 ≈ $36,364 to the mean by itself, pulling it far above the cluster. That single value is why the mean ($80k) roughly doubles the median.
Priya's claim is misleading: $80,000 is not earned by anyone-no employee is near it. The mean is a legitimate statistic but it is not a good description of a "typical" salary when one extreme value dominates the sum. The median better represents the typical employee here because it is resistant to the outlier.
2. Evaluate the second group's proposal to report the mode, and state a general rule linking skew to choice of center.
Read the worked answer
The mode is a poor choice here. With salaries spread across $32k–$48k, the mode is whichever narrow value repeats most-likely an artifact of how salaries were rounded, and it discards the magnitude information of all other values. It is not robust in a useful sense; it is unstable and can be non-representative or even non-unique. For continuous-ish data like salaries, the mode is rarely the best "typical" summary.
General rule:
- Roughly symmetric data with no extreme outliers → mean and median are close; the mean is preferred because it uses all the data.
- Skewed data or data with outliers → the median is preferred because it resists extreme values.
- Use the mode mainly for categorical data or when the most frequent value is itself the question of interest.
So: report the median ($40,000) as the typical salary, and if the mean is reported, pair it with the median or a note about the skew so the $400k owner's effect is transparent.