Types of Mean
The "mean" is a measure of central tendency. There are several types of mean, each useful in different contexts.
Arithmetic Mean
AM = (x₁ + x₂ + ... + xₙ) / n
Geometric Mean
GM = ⁿ√(x₁ × x₂ × ... × xₙ)
Harmonic Mean
HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
For any dataset: HM ≤ GM ≤ AM (AM-GM-HM inequality)
Three Types of Mean Compared
| Mean Type | Formula | Best Used For |
|---|---|---|
| Arithmetic Mean | (x₁+x₂+...+xₙ)/n | Symmetric data, test scores, temperatures |
| Geometric Mean | ⁿ√(x₁×x₂×...×xₙ) | Investment returns, growth rates, ratios |
| Harmonic Mean | n / (1/x₁+1/x₂+...+1/xₙ) | Speeds, rates, frequencies |
Worked Examples — All Three Means
Dataset: 2, 8, 4, 16, 32
Arithmetic Mean = (2+8+4+16+32)/5 = 62/5 = 12.4
Geometric Mean = ⁵√(2×8×4×16×32) = ⁵√(32768) = 8.0
Harmonic Mean = 5/(1/2+1/8+1/4+1/16+1/32) = 5/0.9688 ≈ 5.16
Geometric Mean = ⁵√(2×8×4×16×32) = ⁵√(32768) = 8.0
Harmonic Mean = 5/(1/2+1/8+1/4+1/16+1/32) = 5/0.9688 ≈ 5.16
The AM-GM-HM Inequality
For any positive dataset, these means always satisfy:
HM ≤ GM ≤ AM
Equality holds only when all values are identical.
Example above: 5.16 ≤ 8.0 ≤ 12.4 ✓
Equality holds only when all values are identical.
Example above: 5.16 ≤ 8.0 ≤ 12.4 ✓
When to Use Each Mean
- Arithmetic mean: Most common situations — exam scores, temperatures, heights. Sensitive to outliers.
- Geometric mean: Compound interest, population growth, investment portfolio returns. Use when values multiply together over time.
- Harmonic mean: Average speeds (when same distance travelled at different speeds). Example: Drive 60 km/h one way, 40 km/h return — average speed = harmonic mean = 2/(1/60+1/40) = 48 km/h, NOT 50.
Arithmetic Mean in the Real World
The arithmetic mean is used in countless everyday and professional contexts:
| Application | What is averaged | Why useful |
|---|---|---|
| Academic GPA | Grade points per credit hour | Summarises academic performance |
| Weather reports | Daily temperatures | Monthly/annual climate summaries |
| Sports statistics | Points/goals per game | Compares player/team performance |
| Stock market | Daily closing prices | Moving averages for trend analysis |
| Quality control | Product measurements | Detecting drift from target specs |
Moving Averages
A moving average calculates the mean of the most recent n data points as new data comes in. It is widely used in finance, weather forecasting, and signal processing to smooth out noise and reveal trends:
7-day moving average of temperatures:
Days 1–7: (18+21+23+19+22+25+20)/7 = 21.1°C
Days 2–8: (21+23+19+22+25+20+24)/7 = 22.0°C
Days 1–7: (18+21+23+19+22+25+20)/7 = 21.1°C
Days 2–8: (21+23+19+22+25+20+24)/7 = 22.0°C