A real estate agent reports the mean home price in a neighborhood. Most houses cost $250,000–$350,000, but one mansion sold for $5,000,000. Why is the mean potentially misleading here?
AThe mean is always lower than the median in housing data, making it systematically inaccurate
BThe mean is pulled upward by the extreme outlier, making the neighborhood appear more expensive than a typical home — the median would better represent what a typical buyer would pay
CThe agent should use the mode, because the most frequent price is always the most accurate measure
DThe mean is mathematically incorrect whenever outliers are present
This is the mean's fundamental weakness: a single extreme value (outlier) can pull it far from what most people would call 'typical.' The $5,000,000 mansion might push the mean to $400,000 even if every other house sold for $280,000. The median — the middle value when data is sorted — is unaffected by how extreme the outlier is, making it a much better measure of a 'typical' home price. This is exactly why median household income is reported rather than mean household income.
Question 2 Multiple Choice
A student is asked for the median of the data set: 3, 7, 9, 12. She answers '9 because it's in the middle.' What did she do wrong?
AShe forgot to sort the data before finding the median
BShe should have used the mean instead of the median
CWith an even number of values, the median is the average of the two middle values: (7 + 9) / 2 = 8
DShe identified the wrong middle value; it should be 7
When a data set has an even number of values, there is no single middle value. The correct median is the mean of the two values closest to the center. For 3, 7, 9, 12 (already sorted), the two middle values are 7 and 9, so the median is (7+9)/2 = 8. Selecting either middle value without averaging is one of the most common median errors. Note: the data here was already sorted; in practice, always sort before locating the middle.
Question 3 True / False
A data set can have no mode if all values are different, or multiple modes if several values appear with equal highest frequency.
TTrue
FFalse
Answer: True
The mode is the most frequently occurring value, but this requires that some value actually repeats. If all values are unique (e.g., 3, 7, 11, 14), the data set has no mode. If two values each appear the same number of times and more than any other (e.g., 4, 4, 7, 7, 9), the data set is bimodal. Unlike mean and median, a data set is not guaranteed to have exactly one mode — and forcing one where none exists is a common error.
Question 4 True / False
In any data set, the mean and median will typically be close to each other because both measure central tendency.
TTrue
FFalse
Answer: False
Mean and median can differ substantially when data is skewed or contains outliers. In a right-skewed distribution (a few very high values, like salaries with some billionaires), the mean is pulled toward the tail while the median stays near the bulk of the data. The mean of {1, 2, 3, 4, 100} is 22, while the median is 3 — a dramatic difference. It is precisely because mean and median can diverge that choosing the right measure matters. Treating them as interchangeable is the error this topic is designed to correct.
Question 5 Short Answer
A company's employee salary data is right-skewed because the CEO earns $5 million while most employees earn $50,000–$80,000. Which measure of central tendency best represents a 'typical' employee's salary, and why?
Think about your answer, then reveal below.
Model answer: The median best represents a typical employee's salary. The mean would be dragged upward by the CEO's extreme salary, making the 'average' appear far higher than what most employees actually earn. The median is resistant to outliers — it reports the salary of the middle-ranked employee, which accurately reflects what a typical person at the company earns. The mode might also be useful if there is a common salary tier, but the median is the standard choice when data contains extreme values.
This is why median is used for official income statistics (median household income, median wage) rather than mean. A few extremely high earners inflate the mean without changing what most households actually experience. Using the right measure is not just a math skill — it is a critical thinking skill about what 'typical' really means in a given context.