 ### Description

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