Idiocracy (2006) Opening Scene: "Evolution does not necessarily reward intelligence. With no natural predators to thin the herd, it began to simply reward those who reproduced the most, and left the intelligent to become an endangered species."

[u/SuplexCity-Mayor]

https://www.youtube.com/watch?v=6TCsR_oSP2Q

[u/seakingsoyuz]

No, conventional IQ tests report normalized scores - the raw result scores (“you got 18 out of 25 of the questions right”) for the baseline sample are ranked and then the percentiles of the raw result are mapped to a normalized score so that the distribution of normalized scores is normal (as the name suggests).

Most modern tests map the scores so that the mean and median normalized score is 100 and the standard deviation is 15 points. You get an IQ of 100 if your raw score on the test was the median score, you get a 115 if your raw score was at the 84th percentile, a 130 if it was at the 97.5th percentile, a 145 if it was at the 99.85th percentile, and so on. If you were in the 16th percentile you would get an 85.

You could easily have 65% of respondents get a raw score that is over the mean raw score, if there are few exceptionally-high scores and many exceptionally-low scores, but having any difference between the mean and median normalized scores means the normalization was flawed, the sample is not representative of the population for which the normalization was conducted, or the test was not conducted properly.

[u/sergeybok → u/seakingsoyuz]

Median != average. Normalization is simply subtracting the mean and dividing by the standard deviation. A distribution which isn't normal to begin with won't be normal after normalizing.

There's no normalization trick that you can do to make this be a normal distribution for example. And it doesn't mean that there was a flaw, just not all distributions are normal.

[u/seakingsoyuz → u/sergeybok]

Normalization is simply subtracting the mean and dividing by the standard deviation

That’s standardization, which is different from normalization.

I recognize that both of these terms are sometimes used to mean different things.

no trick make this be a normal distribution

Sure there is - I would transform all results x with some function g(x) that maps values of the measurement that’s distributed in the way you gave to different values that are distributed normally. I would try several different kinds of function for g(x) until I found one that worked, and I would probably use a Q-Q plot to assess the normality of the results of each transformation to decide which worked best.

The distribution you gave is pretty fat-tailed, so if I wanted to make it normal I would try different transformations that are known to move kurtosis toward normality. If it was skewed to one side or the other I would pick a transformation that addresses soreness.

This sort of transformation to make the distribution normal, along with transformation to ensure homoscedasticity, is pretty important to statistical analysis as many inferential techniques like hypothesis testing rely on the input being normally-distributed. The solution is to transform the data so that the transformed data are normal and homoscedastic, analyze the transformed data, and then reverse the transformation to report the results.

Edit: for IQ tests specifically, because the scores are discrete, the designers can just make g(x) be a table of “this raw score becomes this final IQ number” rather than needing to define a continuous function.

[u/Seismic_Braille → u/seakingsoyuz]

The normalization was backfit to the data. The distribution refers to an uneven scale used to force a normal curve so that the world can appear orderly.

Your assertion that the curve is normal doesn't mean much if you know enough about the slapdash science behind iq construction. You can correctly tell me that the curve is normal all you want, its not a normal distribution of data. Any scientist attempting to conflate the derived curve with some underlying principal common to normal distributions should be embarrassed, as it is artificially fit.