68.27 % of the distribution lies within one standard deviation of the mean.
what percent of the distribution lies within one standard deviation of the mean?
68.27 %
but for the gaussian random walk, this is just the standard deviation of the translation distance's distribution after "n" steps. hence, if μ is equal to zero, and since the root mean square(rms) translation distance is one standard deviation, there is 68.27% probability that the rms translation distance after "n" steps will fall between ± σformula_49. likewise, there is 50% probability that the translation distance after "n" steps will fall between ± 0.6745σformula_49.





























section::::gaussian random walk.:anomalous diffusion.
random walk
if a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, "μ" ± "σ", where "μ" is the arithmetic mean), about 95 percent are within two standard deviations ("μ" ± 2"σ"), and about 99.7 percent lie within three standard deviations ("μ" ± 3"σ"). this is known as the "68-95-99.7 rule", or "the empirical rule".





























for various values of "z", the percentage of values expected to lie in and outside the symmetric interval, ci = (−"zσ", "zσ"), are as follows:





























section::::relationship between standard deviation and mean.
standard deviation