Step by Step
1
The mean
Mean = sum of all values ÷ n. Because every value contributes to the sum, a single extreme outlier can dramatically pull the mean toward it.
2
The median
Median = the middle value when the data is sorted (or the average of the two middle values for an even-sized dataset). Since it only depends on the position of values, not their exact magnitude, it's resistant to outliers.
3
When to use the median
For skewed distributions or datasets containing significant outliers, the median gives a more representative measure of "typical" center, since it isn't distorted by extreme values.
4
When either measure works
For roughly symmetric distributions without major outliers, the mean and median will be close to each other, and either can be reported.
Applied Walkthrough
1
A small company has salaries of $40k, $45k, $50k, $55k, and the CEO's $500k. Calculate the mean: (40+45+50+55+500)/5 = 690/5 = $138k.
2
This mean of $138k doesn't represent a "typical" salary at all — it's been dramatically pulled upward by the CEO's outlier salary.
3
Now calculate the median: sorting the data (40, 45, 50, 55, 500), the middle value is $50k.
4
The median of $50k is a far more representative measure of the "typical" salary at this company, since it isn't distorted by the single extreme outlier — this is exactly why skewed datasets should be summarized using the median rather than the mean.
Exam Application
Exams test whether you know that the mean is sensitive to outliers while the median is resistant, and whether you can correctly choose the appropriate measure of center for a skewed or outlier-containing dataset.
⚠ Common Trap
The most common trap is defaulting to the mean automatically, without checking whether the data is skewed or contains outliers — in those cases, the median is the more appropriate and representative measure of center.
✓ Quick Self-Check
1. How is the mean calculated?
Sum of all values divided by n.
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2. How is the median found?
The middle value when the data is sorted (or the average of the two middle values for an even-sized dataset).
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3. Which measure of center is more resistant to outliers — mean or median?
Median.
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4. In the salary example, why was the median a better representation of "typical" salary than the mean?
Because the mean was dramatically pulled upward by the CEO's outlier salary, while the median remained representative of the bulk of the data.
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5. For a skewed distribution, which measure of center should typically be reported?
The median.
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