How does the harmonic mean method work?

Answer 1

The mean is a measure of central tendency, and the three most common ways to calculate the mean are the arithmetic mean (sum the values and divide by the number of values, n); the geometric mean (multiply all the values and raise them to a power of 1/n, where n is again the number of values); and the harmonic mean, where you divide n (the number of values) by the sum of the reciprocals.

Using the examples where there are three values, 3, 5, and 7, the arithmetic mean would be: (3+5+7)/3 = 15/3 = 5; the geometric mean would be (3*5*7)^1/3 = cubed root of (3*5*7) = cubed root of 105 = 4.72; and the harmonic mean would be 3/(1/3 +1/5 +1/7) = 3/0.676 = 4.44. The geometric mean is best when taking the average of values that are normalized to a reference point (relative values). The harmonic mean is best when small values have more meaning than large value, such as in population genetics when a small number of individuals can put the population in a genetic bottleneck, where alleles are lost from the population, whereas large numbers of individuals do not cause this type of impact. The arithmetic mean is used most commonly, and applies under most circumstances as a good measure of central tendency.