Overview

Averages summarize data but behave differently under outliers and transformations. Know which average is being used and what it preserves.

Contest problems often hide which average is appropriate. Recognize when the mean is inappropriate and when the harmonic mean is required.

Key Ideas

• Mean is sensitive to outliers; median is not.
• Mode captures frequency, not magnitude.
• Harmonic mean is suited to rates and reciprocals.
• For weighted data, use weighted mean: $\bar{x} = \frac{\sum w_ix_i}{\sum w_i}$.
• Median depends on ordering, not arithmetic.

Definitions

Let the data be $a_1,\ldots,a_n$.

• Mean: $\frac{a_1+\cdots+a_n}{n}$.
• Median: middle value (or average of two middle values if $n$ is even).
• Mode: most frequent value(s).
• Harmonic mean: $\frac{n}{\sum \frac{1}{a_i}}$ (for positive values).

Worked Example

The average speed for a 60-mile trip is 40 mph going and 60 mph returning. What is the overall average speed?

Let each leg be 60 miles. Time out: $60/40 = 1.5$ hours. Time back: $60/60 = 1$ hour. Total distance is 120 miles, total time is 2.5 hours, so average speed is $120/2.5 = 48$ mph. This is the harmonic mean of 40 and 60.

More Examples

Example 1: Median with Even Count

Data: $2, 5, 7, 9$. Median is $(5+7)/2 = 6$.

Example 2: Weighted Mean

A test has 70% homework and 30% exam. Scores are 80 and 90. The overall score is $0.7\cdot 80 + 0.3\cdot 90 = 83$.

Example 3: Mode

Data: $1,2,2,3,3,3,4$. Mode is $3$.

Strategy Checklist

• Is the quantity a rate? If yes, consider harmonic mean.
• Are there weights? Use weighted mean.
• Is the data skewed? Median may be more meaningful than mean.

Common Pitfalls

• Averaging speeds by simple mean instead of harmonic mean.
• Confusing median with mean in skewed data.
• Forgetting to sort before computing the median.

Practice Problems