Which mean should you use?
Each mean answers a different question. The table below summarizes when each is the appropriate summary.
| Mean | Use when values combine by | Typical applications |
|---|---|---|
| Arithmetic | Addition | Test scores, heights, temperatures, general averages |
| Geometric | Multiplication | Growth rates, investment returns, ratios, index numbers |
| Harmonic | Reciprocals (rates) | Average speed over equal distances, price-earnings ratios, parallel resistances |
- The geometric and harmonic means are defined only for all-positive values. This calculator omits them when the list contains zero or negative numbers.
- For any all-positive data set, arithmetic mean >= geometric mean >= harmonic mean, with equality only when all values are equal (the AM-GM-HM inequality).
- Averaging percentage growth rates with the arithmetic mean overstates cumulative growth; the geometric mean of the growth factors gives the correct equivalent constant rate.
What are the arithmetic, geometric and harmonic means?
The arithmetic mean is the sum of the values divided by their count. It is the appropriate average for quantities that combine by addition, such as heights, test scores or temperatures, and it is what most people mean by 'the average'.
The geometric mean is the n-th root of the product of n positive values. It is the appropriate average for quantities that combine by multiplication, such as growth rates and ratios: the geometric mean of yearly growth factors gives the constant rate that would produce the same overall growth. It is defined only for all-positive data, because a zero collapses the product to zero and negative values can make the root non-real.
The harmonic mean is the count divided by the sum of the reciprocals of the values. It is the appropriate average for rates over a fixed numerator, the classic case being average speed over equal distances. Like the geometric mean, it requires all values to be positive. For any all-positive data set, the three means obey the inequality: arithmetic >= geometric >= harmonic, with equality only when all values are identical.
How to use this mean calculator
- Enter your numbers separated by commas, for example: 4, 8, 15, 16, 23, 42.
- Read the arithmetic mean, which is always shown.
- If every value is positive, read the geometric and harmonic means as well; they are not defined when the list contains zero or negative values, so they are omitted in that case.
- Choose the mean that matches your problem: arithmetic for additive quantities, geometric for growth rates and ratios, harmonic for rates such as speeds over equal distances.
Formulas for the three means
Arithmetic mean: add the n values and divide by n. Geometric mean: multiply the n values and take the n-th root (computed in practice as the exponential of the mean of the logarithms). Harmonic mean: divide n by the sum of the reciprocals.
Worked example with the values 1, 4, 16. Arithmetic: (1 + 4 + 16) / 3 = 21 / 3 = 7. Geometric: the product is 1 x 4 x 16 = 64, and the cube root of 64 is 4. Harmonic: 3 / (1/1 + 1/4 + 1/16) = 3 / (21/16) = 48/21, approximately 2.285714. Note that 7 >= 4 >= 2.285714, illustrating the arithmetic-geometric-harmonic inequality.
Speed example for the harmonic mean: driving the same distance at 30 km/h and then at 60 km/h gives an average speed of 2 / (1/30 + 1/60) = 40 km/h, not the arithmetic mean of 45 km/h, because more time is spent at the slower speed.
Common mistakes
- Using the arithmetic mean for growth rates: a +50% year followed by a -50% year averages to 0% arithmetically, yet the investment is down 25% overall; the geometric mean of the factors 1.5 and 0.5 (about 0.866) reflects the true average change.
- Averaging speeds over equal distances with the arithmetic mean — equal-distance legs call for the harmonic mean.
- Applying the geometric or harmonic mean to data containing zeros or negative values, where they are undefined.
- Expecting the three means to agree — they coincide only when every value in the list is identical.
よくある質問
What is the difference between the arithmetic and geometric mean?
The arithmetic mean adds the values and divides by the count; the geometric mean multiplies the values and takes the n-th root. The arithmetic mean suits quantities that combine by addition, while the geometric mean suits quantities that combine by multiplication, such as growth factors. For the values 1, 4 and 16, the arithmetic mean is 7 and the geometric mean is 4.
When should I use the harmonic mean?
Use the harmonic mean to average rates when the numerator is fixed — the classic case is average speed over equal distances. Driving the same distance at 30 km/h and 60 km/h averages 2 / (1/30 + 1/60) = 40 km/h, not 45 km/h, because more time is spent at the slower speed. The harmonic mean also appears in averaging price-earnings ratios and combining parallel electrical resistances.
Why does this calculator sometimes show only the arithmetic mean?
The geometric and harmonic means are defined only when every value in the list is positive. A zero makes the geometric mean collapse to zero and the harmonic mean undefined (division by zero), and negative values can make the n-th root of the product non-real. When the list contains zero or negative values, only the arithmetic mean is reported.
Why is the geometric mean right for investment returns?
Investment returns compound multiplicatively. If a portfolio grows by factors of 1.5 (up 50%) and 0.5 (down 50%) in successive years, the overall factor is 1.5 x 0.5 = 0.75. The geometric mean of the factors, sqrt(0.75), approximately 0.866, is the constant yearly factor producing the same result — an average loss of about 13.4% per year, which the arithmetic mean of 0% completely misses.
Is the arithmetic mean always the largest of the three means?
For data sets of positive numbers, yes: the arithmetic mean is greater than or equal to the geometric mean, which is greater than or equal to the harmonic mean. Equality holds only when all values are identical. This is the AM-GM-HM inequality, a standard result in classical algebra.
参考文献
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.1: Measures of Location. nist.gov.
- Weisstein, Eric W. "Pythagorean Means." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Hardy GH, Littlewood JE, Polya G. Inequalities. Cambridge University Press, 1934 (AM-GM-HM inequality).