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📚 Online Statistics Education
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1 Introduction2 Graphing Distributions3 Summarizing Distributions4 Describing Bivariate Data5 Probability6 Research Design7 Normal Distribution8 Advanced Graphs9 Sampling Distributions10 Estimation11 Logic of Hypothesis Testing12 Tests of Means13 Power14 Regression15 Analysis of Variance16 Transformations17 Chi Square18 Distribution Free Tests19 Effect Size
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7.3 Areas of Normal Distributions

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.

Normal curve with a mean of 50 and a standard deviation of 10 on an axis from 10 to 90. The region between 40 and 60 — one standard deviation either side of the mean — is shaded blue and contains 68% of the area.
Figure 1. Normal distribution with a mean of 50 and standard deviation of 10. 68% of the area is within one standard deviation (10) of the mean (50).

Figure 2 shows a normal distribution with a mean of 100 and a standard deviation of 20. As in Figure 1, 68% of the distribution is within one standard deviation of the mean.

Normal curve with a mean of 100 and a standard deviation of 20 on an axis from 20 to 180. The region between 80 and 120 is shaded blue: again exactly one standard deviation either side of the mean, and again 68% of the area, even though the numbers on the axis are quite different from Figure 1.
Figure 2. Normal distribution with a mean of 100 and standard deviation of 20. 68% of the area is within one standard deviation (20) of the mean (100).

The normal distributions shown in Figures 1 and 2 are specific examples of the general rule that 68% of the area of any normal distribution is within one standard deviation of the mean.

Figure 3 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.

Normal curve with a mean of 75 and a standard deviation of 10 on an axis from 35 to 115. The shaded blue region runs from about 55.4 to 94.6 — 1.96 standard deviations either side of the mean — and contains 95% of the area.
Figure 3. A normal distribution with a mean of 75 and a standard deviation of 10. 95% of the area is within 1.96 standard deviations of the mean.

The normal calculator can be used to calculate areas under the normal distribution. For example, you can use it to find the proportion of a normal distribution with a mean of 90 and a standard deviation of 12 that is above 110. Set the mean to 90 and the standard deviation to 12. Then enter "110" in the box to the right of the radio button "Above." At the bottom of the display you will see that the shaded area is 0.0478. See if you can use the calculator to find that the area between 115 and 120 is 0.0124.

Screenshot of the normal-area calculator. The plot shows a normal curve on an axis labelled 54 to 126, with the thin upper tail beyond about 110 shaded grey. The parameters read Mean 90, SD 12, with the Above radio button selected and 110 entered. The Results panel reads Area (probability) = 0.0478.
Figure 4. Display from calculator showing the area above 110.

Say you wanted to find the score corresponding to the 75th percentile of a normal distribution with a mean of 90 and a standard deviation of 12. Using the inverse normal calculator, you enter the parameters as shown in Figure 5 and find that the area below 98.09 is 0.75.

Screenshot of the inverse normal calculator. Area is set to 0.75, Mean to 90, SD to 12, and the answer appears in the Below field as 98.09; the plot shades everything left of about 98 on an axis from 54 to 126 — the 75th percentile of that distribution.
Figure 5. Display from normal calculator showing that the 75th percentile is 98.09.

Normal and Inverse Normal Calculator

Adapted from Online Statistics Education: A Multimedia Course of Study (onlinestatbook.com), Project Leader: David M. Lane, Rice University. Developed with NSF support. The original work is in the public domain; it is cited here at the authors' request. Changes were made: reformatted as an accessible XYZ web edition with native MathML. License: Public-Domain.