The number of standard deviations required to include a given percent of a population that has a normal distribution can be looked up in a table or calculated.
To determine the number of standard deviations to include X percent of population using a table:
(1) Convert the percent to a decimal fraction.
(2) Divide the fraction by 2 (to reflect symmetry about the mean).
(3) Look up the number of standard deviations to give that value in a table.
Alternatively the values can be calculated. The following equations were fit to the normal distribution data in JMP.
number of standard deviations =
= (A * ((area)^3))) + (B * ((areas)^2))) + (C * (area)) + D
where:
• Area is the decimal fraction at the end of step 2 above.
Fraction Range
A
B
C
D
0 to 0.3643
0
2.3194752
2.1057859
0.0106404
0.3643 to 0.4893
745.29186
-892.7877
361.33164
-48.09123
0.4893 to 0.4987
0
6414.5363
-6266.883
1532.9654
Above 3 standard deviations the area under the normal distribution curve ranges from 0.997 to 1.000.
Area Under Normal Distribution Curve
Number of SD
0.999
3.27
1.000
3.90
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