Mean Plus Two Standard Deviations. Under this rule 68 of the data falls within one standard deviation 95 percent within two standard deviations and 99 7 within three standard deviations from the mean. It will always be between 0 and 25 outside two standard deviations from the mean and for continuous unimodal densities i think the bounds are 0 to 11 1 1 9 outside two standard deviations from the mean.
Hence one can interpret the value of the standard deviation by reference to the normal curve. To compute the probability that an observation is within two standard deviations of the mean small differences due to rounding. It will always be between 0 and 25 outside two standard deviations from the mean and for continuous unimodal densities i think the bounds are 0 to 11 1 1 9 outside two standard deviations from the mean.
Pr μ 2 σ x μ 2 σ φ 2 φ 2 0 9772 1 0 9772 0 9545 displaystyle pr mu 2 sigma leq x leq mu 2 sigma phi 2 phi 2 approx 0 9772 1 0 9772 approx 0 9545.
An estimate of the standard deviation for n 100 data taken to be approximately normal follows from the heuristic that 95 of the area under the normal curve lies roughly two standard deviations to either side of the mean so that with 95 probability the total range of values r represents four standard deviations so that s r 4. To compute the probability that an observation is within two standard deviations of the mean small differences due to rounding. We see that the majority of observations are within one standard deviation of the mean and nearly all within two standard deviations of the mean. We can do the same with any number of other distributions.
