Averages are tools for summarizing data, but the ordinary arithmetic mean is not always the right choice. When values multiply over time, such as investment growth or population change, the geometric mean gives a better typical rate. When values are rates for the same fixed quantity, such as speeds over equal distances, the harmonic mean gives a better typical rate.
Choosing the correct mean prevents misleading conclusions from data that look simple at first.
Key Facts
- Arithmetic mean: A = (x1 + x2 + ... + xn) / n
- Geometric mean for positive values: G = (x1 x2 ... xn)^(1/n)
- Harmonic mean for positive values: H = n / (1/x1 + 1/x2 + ... + 1/xn)
- For positive unequal numbers: H < G < A
- Use geometric mean for multiplicative change: overall growth factor = (1 + r1)(1 + r2)...(1 + rn)
- Use harmonic mean for averaging rates over equal amounts of the denominator, such as equal distances: average speed = total distance / total time
Vocabulary
- Arithmetic mean
- The sum of the data values divided by the number of values.
- Geometric mean
- The nth root of the product of n positive values, often used for multiplicative growth.
- Harmonic mean
- The reciprocal of the arithmetic mean of reciprocals, often used for averaging rates over equal quantities.
- Growth factor
- A multiplier that shows how much a quantity changes, such as 1.08 for an 8 percent increase.
- Rate
- A comparison of two quantities with different units, such as miles per hour or dollars per item.
Common Mistakes to Avoid
- Using the arithmetic mean for percent growth rates, which is wrong because growth compounds by multiplication rather than addition.
- Averaging equal-distance speeds with the arithmetic mean, which is wrong because slower speeds take more time and should have greater effect on the overall average.
- Taking the geometric mean of negative values, which is wrong in many real data contexts because the standard geometric mean requires positive values.
- Forgetting to convert percents into growth factors, which is wrong because 20 percent growth and 10 percent loss should be calculated as 1.20 and 0.90 before multiplying.
Practice Questions
- 1 An investment grows by 10 percent in the first year and 30 percent in the second year. What is the average annual growth rate using the geometric mean?
- 2 A cyclist rides 12 km uphill at 8 km/h and 12 km downhill at 24 km/h. What is the average speed for the full trip?
- 3 A student wants to summarize test scores of 70, 80, and 90, while another wants to summarize growth multipliers of 1.10, 0.95, and 1.20. Explain which mean each student should use and why.