Commenter pumpkinhead asks:

What would be the general formula for working the composite IQ from 3 or more sub-tests?

Let’s say your verbal IQ is in the top 1% (135), your spatial IQ is in the top 1% (135) and your working memory IQ is in the top 1% (135).  One might think your composite IQ is also in the top 1% (IQ 135) since that’s the average of all three,  however this ignores the fact that it’s much harder to score 135 in three completely different tests than it is to score 135 on just one, so someone who is in the top 1% in all three tests is perhaps in the top 0.3% overall (IQ 141).

Because IQ is simply the number of standard deviations you score above the mean, multiplied by 15 and added to 100, in order to calculate your composite score on multiple tests, you need to know the mean and standard deviation of the composite score.  Knowing the mean is fairly straightforward.  Since the mean IQ by definition is 100, the mean of a composite of three tests is just triple the mean of each sub-test, which would be 300.

The standard deviation of the composite is trickier, because unless the correlation between subtests is perfect, you can not simply triple it.

Instead you supposedly use the following formula:

Composite σ = SQUARE ROOT OF:  σ12 + σ22 + σ32 + 2r12σ1σ2 + 2r13σ1σ3 + 2r23σ2σ3

σ refers to standard deviation, r refers to the correlation (between any pair of the three subtests in its subscript) and subscripts refer to tests 1, 2 and 3 respectively, so for example r12 refers to the correlation between subtests 1 and 2 and σrefersto the standard deviation of subtest 3.  This formula can be extrapolated to any number of tests.

So applying this formula to verbal + spatial  + working memory IQ and assuming a correlation of 0.67 between verbal and spatial ,  0.57 between verbal and working memory, and 0.5 between spatial and working memory:

Composite σ =  SQUARE ROOT OF: 152 + 152 + 152 + 2(0.67)(15)(15) + 2(0.57)(15)(15) + 2(0.5)(15)(15)

Composite σ = SQUARE ROOT OF: 225 + 225 + 225 + 301.5 + 256.5 + 225

Composite σ = SQUARE ROOT OF: 1458

Composite σ = 38

So someone who score IQ 135 on all three subtests has a total score of 405 (135 * 3 = 405) which is 105 points higher than the average person who has a total score of 300 (IQ 100 * 3 = 300) and with a composite SD of 38,  they are 2.76 standard deviations above average (105/38 = 2.76),  which equates to an IQ of 141 (because 2.76 * 15 = 41 and 41 + 100 = 141)

Of course I have serious doubts about this composite σ  formula but I’ll save that for another day.

 

 

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