Power Index in the Measurement of the Variance from a Gaussian Field

The rms fluctuation (variance) σ of a cosmic field α(x) is an important measure to quantify the initial fluctuation of the universe and is usually determined by the formula σ²=<α(x)²>. We investigate the necessity of using this specific formula, under the assumption that the initial fluctuation is random-Gaussian-distributed. We calculate the expected finite-volume effect on σ obtained from a general formula <|α|^m>. We find that although the finite-volume effect is minimal at the conventional choice m=2, it is almost insensitive to m around m=1~3. Therefore we can reduce the relative contribution of tail parts, which might be considerably contaminated by other effects (such as measurement errors), at a very small sacrifice of the finite-volume effect.