Areas Under Normal Distributions Prerequisites Distributions, Central Tendency, Variability, Introduction to Normal Distributions Learning Objectives State the proportion of a normal distribution within 1 and within 2 standard deviations of the mean Use the calculator "Calculate Area for a given X" Use the calculator "Calculate X for a given Area." Areas under portions of a normal distribution can be computed by using calculus. Since this is a non-mathematical treatment of statistics, we will rely on computer programs and tables to determine these areas. Figure 1 shows a normal distribution with a mean of 50 and a standard deviation of 10. The shaded area between 40 and 60 contains 68% of the distribution.   The normal distribution shown in Figure 1 is a specific examples of the general rule that 68% of the area of any normal distribution is within one standard deviation of the mean. Figure 2 shows a normal distribution with a mean of 75 and a standard deviation of 10. The shaded area contains 95% of the area and extends from 55.4 to 94.6. For all normal distributions, 95% of the area is within 1.96 standard deviations of the mean. For quick approximations, it is sometimes useful to round off and use 2 rather than 1.96 as the number of standard deviations you need to extend from the mean so as to include 95% of the area.