Measures of Center: Mean and Median
Compute and interpret the mean and median. Understand why the median is resistant to outliers and skew while the mean is not. Choose the appropriate measure for a given distribution shape.
A lesson of AP Statistics, in Exploring One-Variable Data, about 10 minutes.
By the end of this lesson, you will be able to calculate and interpret the mean and median, explain when each is the better measure, and understand how outliers and skewness affect each.
The Mean
$$\bar{x} = \frac{\sum x_i}{n}$$
The mean is the arithmetic average — the balance point of the distribution. It uses every value in the calculation.
Interpreting the mean: "The average [variable] for this sample was [value] [units]."
Example: Data: 5, 8, 9, 11, 17
$\bar{x} = \frac{5 + 8 + 9 + 11 + 17}{5} = \frac{50}{5} = 10$
Key property: the mean is sensitive to outliers. A single extreme value can pull the mean far from where most of the data lies.
The Median
The median is the middle value of ordered data.
Procedure:
- Order the data from least to greatest
- If n is odd: median = the middle value
- If n is even: median = average of the two middle values
Example (odd n): 5, 8, 9, 11, 17 → median = 9
Example (even n): 5, 8, 11, 17 → median = $\frac{8+11}{2} = 9.5$
Key property: the median is resistant to outliers. It depends only on the middle values, not the extremes.
Mean vs. Median: Which to Use?
| Situation | Use |
|---|---|
| Symmetric distribution, no outliers | Mean (more mathematically useful) |
| Skewed distribution | Median |
| Outliers present | Median |
| Need to compute variance or SD | Mean is required |
Effect of skewness on the mean:
- Right-skewed: the long tail pulls the mean above the median (mean > median)
- Left-skewed: the long tail pulls the mean below the median (mean < median)
- Symmetric: mean ≈ median
Classic example: household incomes are right-skewed (Bill Gates is in someone's neighborhood). The median income is more representative than the mean income.