Gaussian Integrals

For psychometric purposes, the Gaussian distribution is given by:

and the corresponding Gaussian integral by

For a = 0,

.

Let , or and .

Then = = = = .

To average all the scores on the right-hand side of the bell curve, we write:

= = .

This can be rewritten:

==.= = 12.767 for s = 16.

Thus, the "average" above-average IQ = 112.767 112.8.

Averaging IQ's Above 120

To normalize the average IQ's above 120, we must integrate

to get the normalization constant.

As before, we set , and . Then at x = 20, x' = = 0.8838. As before,

= = = 0.10565.

=

=

=

=

= 27.665

The complete and final formula then becomes: =

where "cutoff" = , and s c = threshold s.

—½¢

Table of normalizing constants:

 s c Area 0.1 0.46017 0.2 0.42074 0.25 0.40129 0.3 0.38209 0.4 0.34458 0.5 0.30854 0.6 0.27425 0.7 0,24196 0.75 0.22663 0.8 0.21186 0.9 0.18406 1.0 0.15866 1.1 0.13567 1.2 0.11507 1.25 0.10565 1.3 0.09680 1.4 0.08076 1.5 0.06681 1.6 0.05408 1.7 0.04457 1.75 0.04056 1.8 0.03593 1.9 0.02872 2.0 0.02275 2.1 0.01786 2.2 0.01390 2.25 0.01222 2.3 0.01072 2.4 0.00820 2.5 0.00621 2.6 0.00466 2.7 0.00347 2.75 0.00298 2.8 0.00256 2.9 0.00159 3.0 0.00135 3.25 0.00058 3.5 0.00023 3.75 0.00009 4.00 0.00003