Power index in the measurement of the variance from a Gaussian field

The rms fluctuation (variance) sigma of a cosmic held alpha(x) is an import ant measure to quantify the initial fluctuation of the universe and is usua lly determined by the formula sigma(2) = [alpha(x)(2)]. We investigate the necessity of using this specific formula, under the assumption that the ini tial fluctuation is random-Gaussian-distributed. We calculate the expected finite-volume effect on sigma obtained from a general formula [\alpha\(m)]. We find that although the finite-volume effect is minimal at the conventio nal choice m = 2, it is almost insensitive to m around m = 1 similar to 3. Therefore we can reduce the relative contribution of tail parts, which migh t be considerably contaminated by other effects (such as measurement errors ), at a very small sacrifice of the finite-volume effect.