Measures of Center

Mean (\(\bar{x}\))

Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Interpretation: balance point of the data; if data were balanced on a fulcrum at \(\bar{x}\), it would balance

Properties:

• Uses every value in dataset
• Not resistant: one extreme value shifts mean significantly
• Pulled toward outliers and skewed tail
• For symmetric distributions: mean ≈ median

When to use: symmetric, unimodal data with no extreme outliers

Median

Definition: middle value when data ordered from smallest to largest

• If n is odd: middle value is at position \(\frac{n+1}{2}\)
• If n is even: average of two middle values at positions \(\frac{n}{2}\) and \(\frac{n+1}{2}\)

Properties:

• Resistant: unaffected by extreme outliers
• Divides data in half: 50% below, 50% above (50th percentile)
• For skewed data: median is more representative than mean

When to use: skewed distributions, data with outliers

Mode

Definition: value that appears most frequently

Properties:

• Only measure for categorical data
• Can have multiple modes (bimodal, multimodal)
• Not useful if all values appear once (no mode)

When to use: categorical data; for quantitative data, usually less informative

Effect of Outliers

Example: Dataset: 10, 12, 14, 16, 18 (mean = 14, median = 14)

Add outlier 50:

• New dataset: 10, 12, 14, 16, 18, 50
• New mean = \(\frac{10+12+14+16+18+50}{6} = \frac{120}{6} = 20\)
• New median = \(\frac{14+16}{2} = 15\)

Conclusion: mean shifted from 14 to 20 (43% change); median only shifted from 14 to 15 (7% change)

Worked Example

Scenario: Customer wait times (minutes) at service desk: 3, 5, 5, 7, 8, 10, 11, 45

Mean: \(\bar{x} = \frac{3+5+5+7+8+10+11+45}{8} = \frac{94}{8} = 11.75\) minutes

Median: Ordered list has 8 values. Middle two are 7 and 8. Median = \(\frac{7+8}{2} = 7.5\) minutes

Mode: 5 (appears twice; all others appear once)

Outlier effect: The 45-minute wait is an outlier. Mean (11.75) is pulled up; median (7.5) remains representative.

Decision: Report median (7.5 min) to customers. Investigate why 45 happened.

Decision Rule: When to Use Each

Common Mistakes

1. Calculating mean incorrectly: divide by n, not n−1
2. Confusing median and mode: median is middle value; mode is most frequent
3. Ignoring outliers: use median when extreme values present
4. Using mean with categorical data: not meaningful

AP Exam Tip

When asked "which measure is best?" answer:

1. Identify if data is skewed or has outliers (look at boxplot/histogram/stemplot)
2. If yes: use median ("resistant to outliers")
3. If no (symmetric): either mean or median is fine (mention both are close)
4. Cite the shape or presence of outliers in your justification

Example response: "Use median (8 minutes) because the distribution is right-skewed with an outlier at 45 minutes, and the median is resistant to extreme values."