Mean Plus Two Standard Deviations. The area within plus and minus two standard deviations of the mean constitutes about 95 percent of the area under the curve see figure 2. Because standard deviation is a measure of variability about the mean this is shown as the mean plus or minus one or two standard deviations.
Consider A Normal Distribution With Mean 33 And Standard Deviation 9 What Is The Probability A Value Sele Normal Distribution Statistics Math Probability Math from www.pinterest.com
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. Because standard deviation is a measure of variability about the mean this is shown as the mean plus or minus one or two standard deviations. 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.
Hence one can interpret the value of the standard deviation by reference to the normal curve.
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. 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. 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. The area within plus and minus two standard deviations of the mean constitutes about 95 percent of the area under the curve see figure 2.
