I have seen scores from this test circulate on twitter and gcs:

Looking closely, there seems to be a bit of a problem here. If you average out the scores in picture 1, you get a score of 127.7, and if you average out the scores in picture 2, you get 131. However, the Full Scale IQ score is below both of these averages. This is impossible, as the full scale score should always be above the average of the subtests.

Let me illustrate this with a simulation. Lets say I have 3 vectors b, c, and d.

c and d correlate at 0.63

b and c correlate at 0.43

b and d correlate at 0.62

In every single case, the FSIQ score was more extreme than the average score of each subtest. This is obvious if you look at the first 19 rows:

Empirically speaking, it looks like this is also true outside of simulations. Some researchers assigned the openpsychometrics FSIQ test to 88 college students, 72 of which also took the WASI-II. On average, the students scored about 5 points less on the FSIQ test than the WASI-II.

Emil also sent me this document, proving that the average of the subtests should always be less extreme than the full scale score.

edit: apparently if you refresh itâ€¦ the score increases

In the example scores, I was the 123 score. The person who told me said their score increased to above 130. Still, my point still stands about it not being possible for the average of the subtests to be larger than the full scale score.

## Openpsychometrics FSIQ test underestimates IQ scores by about 5 points

It scored me about 1.5 points above my avg of the three group factors. Could just be that verbal is weighted more heavily than the two other subtests (although I scored pretty badly on the verbal one so the same should apply to me. I let cookies expire and closed the tab so I also can't see what happens when I refresh).

In general this is not always the case. Here I provide an abstract argument:

Let A and B both be uncorrelated tests with mean 0 and SD 1. Then we devise some test C which is merely the summed score of A and B. Then C has mean 0 and SD 1.414.

Now say that your score on each test A and B is +1. Your summed score on C is 2, which, normalized to C's SD, is itself a z-score of 1.414, significantly higher than your score on A and B - and this is *fine.* Each subtest adds more information about you as a candidate. After all, it's not likely for most people to achieve +1 for both subtests A and B; the chance for this is only 2.5%. So even though A and B may not be that strongly "C-loaded," you're still somewhat far out on the overall C-distribution.

But in terms of this specific test? Oh, who knows. Forget all these tests, you're as smart as I say you are. (*Everyone* is as smart as I say they are.)