In conducting meta-analysi, we come across to the article that have reported geometric mean. Is it possible to convert geometric mean to arithmetic mean ?
No, it is not possible. The only thing you can say for sure is that the geometric mean is equal to or less than the arithmetic mean.
Arithmetic and geometric mean have their own purpose: static conditions and accordingly dynamic conditions (growth/velocities). Yet their is an interesting relation: the logarithm of the geometric mean is the arithmetic mean of the logarithms, whereat You may choose an arbitrary base of the logarithms. That's so good for big n near the Universe N. Calculating arithmetic means is more comfortable in logarithmic scaled figures.
Unfortunately business people do not mention these rules from their math coaches. Therefore You find shocking transformations and stupid means in reports of director's meetings.
You can easily check financial reports by the inequality geometric mea
There is a precise formula relating the geometric mean to the arithmetic mean involving the central moments of the distribution. This can be approximated in many cases by geometric mean = arithmetic mean - variance/2 but the exact formula is impacted by higher-order moments like skew and kurtosis. I derive this in the Appendix of the paper below
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