Approximate geometric means

This post relates to finding the mean of two numbers. If I asked what the mean of 3 and 27 most people would say it was 15 but this is simply one version of the mean, the arithmetic mean (AM), where we look for a number that is an equal distance from the two through addition. The other type is the geometric mean (GM) where we look for a number that you get to by multiplying the smaller number by and then multiplying that number again to get the higher number. In our example it would be 9 as 3×3=9 and 9×3=27. Now for most purposes this isn’t going to help us but there are a few situations where this method is much more useful. For example if we wanted to find the mean of 20,000 and 400 the AM would give us 10,200 but does this really give us an accurate picture of things? Our mean is 25.5 times larger than our smaller number but our larger is only 2 times the size. A more appropriate number might be 2828 which is our GM. This number takes the fact that our lower number is so much smaller much more in to account. Below is the exact process to calculate the GM:

How can we find this value though? We have two main options to do this.

AM method. For this we want to find the AM of the power and the coefficient. This has slightly different processes for even and odd powers.

Even power: What is the AGM of 2000 and 200,000? Firstly we want to turn both numbers into standard form so we have 2×10³ and 2×10⁵. Our best guess for the AGM is then the AM of the coefficients x10 to the power of the AM of the powers, or:

2. Odd powers: What is the AGM of 2000 and 20,000? Again, like above we want to turn the numbers into standard form, 2×10³ and 2×10⁴.

If you’re wandering how we got a value for root 10 then you may want to have a look at the post on Taylor expansions.

Taylor expansion method from the beginning. For example the AGM of 9,000 and 80,000, again turn them into standard form, 9×10³ and 8×10⁴. We then use the process below: