Estimation and the Approximate Geometric Mean

When estimating a value, it is often easier to start with an upper and lower bound on that value. Once you have an upper and lower bound, you can pick a representative point estimate in that interval. The first and most obvious candidate is the (arithmetic) mean of the upper and lower bounds, but this is only valid if the upper and lower bounds are close together, or have the same order of magnitude. If the upper and lower bounds span multiple orders of magnitude, then it is better to use the geometric mean.

An Example

Suppose your bounds are 1 and 100. The arithmetic mean of 1 and 100 is 50.5. While that seems well and good, we can look at this point estimate from a less appealing perspective. The estimate 50.5 is about 50 times bigger than our lower bound, and about 2 times smaller than out upper bound. If we use the geometric mean instead, we get a point estimate of 10, which is ten times bigger than our lower bound, and 10 times smaller than our upper bound. In terms of scale, our point estimate of 10 is right in the middle of our two bounds, 1 and 100.

Approximating the Geometric Mean

Given two numbers, x and y, their geometric mean is \sqrt{xy}. Square roots are generally difficult to compute by hand, but there is an approximation. If we rewrite x and y in scientific notation, as a coefficient times a power of ten, then we can write the approximate geometric as the mean of the coefficients and the mean of the exponents.

Let x = 4 \times 10^{2}, and y = 7 \times 10^{6}. Then \sqrt{xy} \approx 5.5 \times 10^{4}. This result is within 4% of the actual value.

If the sum of the exponents is odd, then the mean of the exponents will end in fractional .5 part. If we recall that 10^{0.5} = \sqrt{10} \approx 3, then we can drop the .5 in the exponent, and just multiply the coefficient by 3.

Let x = 2 \times 10^{3}, and y = 3 \times 10^{2}. Then \sqrt{xy} \approx 2.5 \times 10^{2.5} \approx 7.5 \times 10^{2}. This result is within 3% of the actual value.

Simplifying Products

The \sqrt{10} \approx 3 is also a very useful tool when simplifying products with many terms. If you rewrite a terms in scientific notation, then you can round the coefficients to one, three, and ten.

Suppose you want the product of 121, 318, and 940. Then you can quickly write (1.2)(3.2)(9.4)(10^{2^{3}}) \approx (1)(3)(10)(10^{6}) \approx 3 \times 10^{7}. This result is within 18% of the actual value.


  1. Mahajan. Street Fighting Mathematics
  2. Weinstein, Adam. Gue??timation

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