geometric mean and the normal distribution

If you have data that are sampled from a normal distribution, what is the relationship between the arithmetic and geometric means? Would it ever make sense to report the geometric mean instead of the arithmetic mean? (Assume that all the values are positive; no zeros, no negative values)

• Do you mean "arithmetic mean" rather than "arrhythmic mean"? If you really mean "arrhythmic mean", could you explain what one of those is? (A search shows the term occurring at least a few times, for example in a book with all French authors, but it's not a standard term in statistics written in English and would need explanation to be widely understood.) – Glen_b Feb 15 '16 at 23:14
• Would you be happy to have a geometric mean $\sqrt{x_1x_2}$ have imaginary value when $x_1$ and $x_2$ have opposite sign? – Dilip Sarwate Feb 15 '16 at 23:38
• Sorry I meant the arithmetic means – Bassam Feb 16 '16 at 10:02
• Any normal distribution, no matter what $\mu$ & $\sigma^2$, will include $0$ & negative values. It is of course possible that your particular sample includes only strictly positive values, but the population must include negative values. – gung Feb 16 '16 at 15:11
• Why do you want to use a geometric mean? – kjetil b halvorsen Feb 16 '16 at 18:12