 # When would you use the geometric mean? When would you use the geometric mean?

This is my second blog on mean, mode and median in just a week – surely that’s far from average. However, I’ve just been reading about the differences between the arithmetic mean and the geometric mean and would love it if there are maths nerds out there who can tell me when it’s appropriate to use each.

Last week I wrote my why mean, mode and median matter blog, then on theme the following hit in my in-box as part of Paul Lewis’s Radio 4 Moneybox email on Friday…

The Government has come up with a novel explanation about why the CPI is a better prices index than the RPI…the CPI takes account of people trading down… So if Nescafe goes up in value, people trade down and buy a supermarket’s own brand. That is why the CPI shows a lower rise than the RPI. At least, that’s how I read what they said."

"Exactly how it does this is a bit of a mystery to me as it seems to be contained in the difference between the arithmetic (or normal mean ML) and the geometric mean. The geometric mean is defined as…well, it’s quite long so look it up. But suffice it to say that the geometric mean of 3 and 5 and 19 is 6.580844. Certainly that difference between geometric and arithmetic mean is the ‘formula effect’, which leads to CPI being about 0.75 percentage points below RPI given the same data."

So I went onto that bastion of often, though not always, correct information – Wikipedia, to look up the geometric mean, and indeed it’s complex. Essentially you multiply all the numbers together then take the root of it (where the factor you take the root of depends on the number of numbers).

Right, so I’m with that and with a calculator I could work it out – yet there’s still the point of when you use it, again back to Wiki…

"Applications in the social sciences

Although the geometric mean has been relatively rare in computing social statistics, in 2010 the United Nations Human Development Index switched to this mode of calculation, on the grounds that it better reflected the non-substitutable nature of the statistics being compiled and compared:

The geometric mean reduces the level of substitutability between dimensions [being compared] and at the same time ensures that a 1 percent decline in say life expectancy at birth has the same impact on the HDI as a 1 percent decline in education or income. Thus, as a basis for comparisons of achievements, this method is also more respectful of the intrinsic differences across the dimensions than a simple average."

Anyone out there any the wiser… when would you see it as appropriate to use?

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