|
|
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 |