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When summarizing normalized data (for example, percentage data), one must use the geometric mean instead of the arithmetic mean. Thus, instead of using the arithmetic standard deviation, one shall use the geometric standard deviation.

When using arithmetic 'descriptors', we can describe the results as mean $\pm$ standard deviation. However, when dealing with geometric 'descriptors', we must describe them as the range from (the geometric mean divided by the geometric standard deviation factor) to (the geometric mean multiplied by the geometric standard deviation factor), since one cannot add/subtract "geometric standard deviation factor" to/from geometric mean.


My question is: What is the correct notation for representing this operator?
I'm looking for the notation of what is equivalent to $\pm$ for the case of geometric mean and geometric standard deviation.

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By definition $\mu_g=\left(\prod_{i=1}^nx_i\right)^{1/n}$, $\sigma_g=\exp\left(\sqrt{\frac{\sum_{i=1}^n(\log x_i-\log\mu_g)^2}{n}}\right)$, so if you take the logarithm of both you get arithmetic mean and standard deviation for $\log X$. If further $\log X$ is approximately normal you could have something like a confidence interval $\log\mu_g\pm z\times\log\sigma_g$. You can then exponentiate both side to get $\mu_g\times/\div\sigma_g^z$.

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  • $\begingroup$ I understand that. However, is there a compact notation for express geometric mean 'times div' geometric SD? $\endgroup$ Apr 15, 2019 at 8:13
  • $\begingroup$ @IagoCarvalho I am not sure. I don't think there is any... $\endgroup$
    – Steven Xu
    Apr 16, 2019 at 3:23
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You've pretty much answered your own question. You want to express the data as the geometric mean times or divided by the geometric standard deviation. Here are three ways to do so:

  • Enter the multiply x (in the Insert Symbol dialog of Word) followed by a slash: 9.5 ×/ 2.3
  • Enter a superscripted period followed by a slash: 9.5 ./ 2.3
  • Use the Unicode “DIVISION TIMES” symbol (U+22C7) that superimposes the multiply × and divide ÷ symbols: 9.5 ⋇ 2.3. The problem is this symbol looks to me more like an asterisk than a multiply-or-divide symbol.

Instead of reporting the geometric mean and geometric SD, an alternative would be to report the 95% confidence interval of the GeoMean: 4.13 to 21.9.

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