The location of the 99.9^th centile

© February 2006 Paul Cooijmans

For societies that select at or above the 99.9^th centile in intelligence, it is relevant to know where it lies on a standard scale like IQ. This is not at once obvious. Here are a few considerations:

The 99.9^th centile is the point at or below which 99.9 % of a population scores. So it depends entirely on the population in question which actual intelligence level it corresponds to. IQ societies normally have the adult US population in mind, or in any case an adult Western population.

More concrete, it is the point that yields 99.9 when the following calculation is done: take the number of people scoring under it; add to that HALF the number of people with exactly that score; divide by the total number of people and multiply by 100.

So it is an exact point, not an interval or class, as many mistakenly think. You score "at or above" the centile, not "in" it. People who speak of "to score in the 99.9^th centile" therewith betray they don't know what a centile is.

Other terms than centile may be used to describe the same point; a few examples (including centile):

• The proportion .999;
• The 999^th millile;
• The 99.9^th centile;
• The 19.98^th vigintile;
• The 9.99^th decile;
• The 3.96^th quartile.

The generic term for these types of scores is "quantile".

Quantiles are the direct scores of IQ tests. The standard scores are obtained by converting the quantiles via the normal distribution to z-scores, IQs and so on. The reason for this conversion is that with quantiles one cannot do computations like addition and multiplication because they are extremely unlinear. Scores with a normal distribution though are assumed to be on a linear scale (interval scale), and on a linear scale one can do those computations. So the reason to use IQs is that computations with IQs are more simple than computations with quantiles. But when doing calculations with IQs, one is really manipulating quantiles in the background.

Some oppose this and say that forcing the scores into a normal distribution does not give outliers a chance to show how far they are lying out (outliers are pulled in to the edges this way). This is theoretically true, but in practice it is a useless insight as it is not yet possible to measure intelligence directly on a standard scale. We can only count how often each score occurs and use the resulting quantiles as norms. IQs are quantiles in disguise.

The 99.9^th centile, according to the normal distribution, lies 3.08 standard deviations above the mean. That is a point, not an interval. To understand how this relates to IQ, one must know IQs are centred intervals, such that for instance IQ 100 is the interval between (including) the following values:

• 99.5000... [infinite 0s] ...1
• 100.4999...

The exact values 99.5, 100.5, 101.5 and so on must ideally be assigned at random or alternately to either the interval above or that below it. In practice, they are rounded up by most software and calculators.

When the SD (standard deviation) of the IQ scale is set at 15, 3.08 SD becomes IQ 146.2. But IQs are whole numbers representing centred intervals, so one must look closer. Here is the contents of the IQs 146 and 147, which are relevant here:

IQ 146

• 145.5000... is the 99.88th centile
• 146 is 99.89
• 146.2 is 99.9
• 146.4999... is 99.9

IQ 147

• 146.5000... is 99.9
• 147 is 99.91
• 147.4999... is 99.92

So most in the interval IQ 146 are below 99.9, a minority in IQ 146 are at or above it, and all in IQ 147 are above it. Since IQs are reported as whole numbers, the pass level in IQ has to be 147 in theory. Only one misses a few in the top of the interval 146 then. There are a few ways to deal with this:

• Arrange the norms of your tests so that IQ 146 ALWAYS is at or above 99.9, so really shift the interval up a bit so that it almost coincides with 147; problems are that norms of other tests may not follow this practice, and that you corrupt the linearity of the scale.
• Express the pass level to a tenth of a point; a problem is that IQs are not reported to a tenth.
• Express the norm as a quantile and use the reported or known quantiles of the tests for admission.

The latter is the best solution to my current insight.