When we speak of “the” mean of two values a and b, everyone generally takes this to mean half the sum of a and b. This is certainly the most common approach, but not necessarily the most appropriate. Consider a product whose price rises by 10% one year and 50% the next. At first glance, we might be tempted to say that the average increase is 30% per year. However, two successive increases of 30% amount to multiplying by 1.32, or 1.69. In other words, this represents a 69% increase. Yet successive increases of 10% and 50% amount to multiplying by 1.1 × 1.5 = 1.65, corresponding to an increase of “only” 65%. The “average increase” is therefore not 30%. So how can we find it? By using the growth factor k, which must satisfy k 2 = 1.65 and is therefore approximately 1.284, which corresponds to an increase of 28.4%. This growth factor k is the geometric mean of 1.1 and 1.5.
A few simple diagrams
------------------
In our example, the arithmetic mean—that is, the usual mean—of 1.1 and 1.5 is (1.1 + 1.5)/2 = 1.3 and is therefore greater than the geometric mean (approximately 1.284). Is this always true? Given two positive numbers x and y, must we always have
The answer is yes: squaring both sides of the inequality is enough to prove it algebraically. But what better way to understand it than with a simple diagram? Draw a line segment of length x + y and construct a semicircle on it. The radius r of this semicircle is the arithmetic mean of x and y. Where is the geometric mean hiding? It is the height h of the right triangle ABC. The ever-useful Pythagorean theorem gives
(*x 2 + *h 2 ) + (*y 2 + *h 2 ) = (x + y) 2.
All that remains is to expand and simplify, giving
