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 |
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 |