Geometric and harmonic mean calculator
The four classical means always order themselves the same way: harmonic, geometric, arithmetic, quadratic. Each suits a different kind of average.
Geometric mean
5.656 854 249
- Arithmetic mean
- 7.5
- Harmonic mean
- 4.266 666 667
- Quadratic mean (RMS)
- 9.219 544 457
- Inequality holds
- 1
Show the working
- Value 1 = 2
- Value 2 = 4
- Value 3 = 8
- Value 4 = 16
- Arithmetic mean = (2+4+8+16)/4 = 7.5
- Geometric mean = (2*4*8*16)^(1/4) = 5.656 85
- Harmonic mean = 4 / (1/2 + 1/4 + 1/8 + 1/16) = 4.266 67
- Quadratic mean (RMS) = sqrt((2^2+4^2+8^2+16^2)/4) = 9.219 54
- Inequality holds = 4.266 67 <= 5.656 85 && 5.656 85 <= 7.5 && 7.5 <= 9.219 54 ? 1 : 0 = 1
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- arithmetic = sum / n
- geometric = (product)^(1/n)
- harmonic = n / sum(1/x)
- quadratic = sqrt(sum(x^2)/n)
- harmonic <= geometric <= arithmetic <= quadratic, always.
Use the geometric mean for growth rates and the harmonic mean for rates over a fixed distance — average speed on an out-and-back trip is a harmonic mean, not an arithmetic one.