Approximations for the geometric mean are found by expressing it in terms of a series of central moments (variance, skewness, kurtosis and higher) and eliminating various terms by arguing that they are small for the case concerned. By including those terms which are not small for your application you can improve your estimate.
In summary, the geometric mean is given by,
$$
\left(\prod_{i=1}^n X_i\right)^{1/n}
=
\prod_{i=1}^n X_i^{1/n}
=
\prod_{i=1}^n \exp \left[ \frac{1}{n}\ln X_i \right]
=
\exp \left[ \frac{1}{n} \sum_{i=1}^n \ln X_i \right]
$$
Using the arithmetic mean, $\mu_1$, we define
$$
x_i=\frac{X_i - \mu_1}{\mu_1}
$$
and use the binomial expansion for $\ln (1 + x)$
$$
\ln X_i - \ln \mu_1 = \ln (1+x_i)
=
x_i
- \frac{x_i^2}{2}
+ \frac{x_i^3}{3}
- \frac{x_i^4}{4}
+ \frac{x_i^5}{5}
- \ldots
$$
$$
\frac{1}{n}\sum_{i=1}^n \ln X_i =
\ln \mu_1
- \frac{1}{n}\sum_{i=1}^n \frac{x_i^2}{2}
+ \frac{1}{n}\sum_{i=1}^n\frac{x_i^3}{3}
- \frac{1}{n}\sum_{i=1}^n\frac{x_i^4}{4}
+ \frac{1}{n}\sum_{i=1}^n\frac{x_i^5}{5}
- \ldots
$$
So, without approximation we can write
$$
\left(\prod_{i=1}^n X_i\right)^{1/n}
=
\exp \left[ \frac{1}{n} \sum_{i=1}^n \ln X_i \right]
=
\mu_1 \exp\left[{
-\frac{\mu_2}{2 \mu_1^2}
+\frac{\mu_3}{3 \mu_1^3}
-\frac{\mu_4}{4 \mu_1^4}
+\frac{\mu_5}{5 \mu_1^5}
- \ldots
}\right]
$$
Where the $\mu_i$ are the population$^\dagger$ central moments (arithmetic mean, variance, skewness, kurtosis and higher order)
$$
\mu_1 = \frac{1}{n}
\sum_{i=1}^n X_i
\hspace{1cm}
\mu_{j>1} = \frac{1}{n}
\sum_{i=1}^n(X_i-\mu_1)^j
$$
At this point we may choose how many moments to include and how many terms to retain in the expansion of $\exp x = 1 + x + \frac{x^2}{2!} + \ldots$
For example if we retain only $\mu_2$ and truncate $\exp x = 1 + x$ we arrive at:
$$
\left(\prod_{i=1}^n X_i\right)^{1/n} \simeq
\mu_1
-\frac{\mu_2}{2 \mu_1}
$$
Squaring this and noting that we have implicitly ordered $\left(\frac{\mu_2}{\mu_1}\right)^2$ negligible gives the Latané approximation
$$
\text{Geometric Mean}^2 = \text{Arithmetic Mean}^2 - \text{Population Standard Deviation}^2
$$
Higher order approximations may be found by including higher moments in $\ln (1 + x_i)$ and retaining more terms in the exponential.
$\dagger$ The sample central moments are defined analogously with $\frac{1}{n} \rightarrow \frac{1}{n-1}$. This is Bessel's correction aimed at the reduction of finite population bias.
The reference you probably want is:
Journal of Financial and Quantitative Analysis
1969 / 06 Vol. 4; Iss. 2
Geometric Mean Approximations of Individual Security and Portfolio Performance
William E. Young and Robert H. Trent
screnshot of calculation
