Very often I read that the distribution of IQ-test results is normal. However, the IQ-scale is left-censored, i.e. test results cannot be lower than zero. The normal distribution, however, is usually not bounded, neither on the left nor right side.
Thus, is it correct to say that left-censored data like IQ-test results is normally distributed?
(Not sure if the term 'censored' is used correctly in this question)
According to Wikipedia, IQ test results are scaled so that the mean corresponds to a score of 100 points and the standard deviation corresponds to 15 points.
Assuming a normal distribution, the proportion of scores that are more than 3 standard deviations below the mean (less than 55) is $\Phi(-3) \approx 0.001349898$. The proportion of scores that are more than 6 standard deviations below the mean (less than 10) is $\Phi(-6) \approx 9.865876e-10$.
As you can see from these numerical examples, the tails of the normal decay so fast that the censoring in the left tail doesn't really matter.
You're right that a distribution of values that can't be below zero cannot be truly normal. But no real data are truly normal. Usually the point of describing data as normal is to acknowledge that the data are close enough to a normal distribution so that the results of traditional parametric tests (e.g. t tests) will be reasonably correct. Or that we can more-or-less assume some useful properties of the distribution, such as that it isn't too skewed and about 99% of the values will fall within three standard deviations of the mean.
The other two answers are entirely correct, but just to addd some info: IQ could most properly be described as a truncated normal distribution, where X is normally distributed with the usual mean and variance if it lies on some interval $(a, b)$. Given the range of human IQ scores, the truncation is of very little substantive relevance.
I am not sure if left-censored is a 'proper' description. That term implies that values under 0 can exist but are not observed due to some censoring process. Because of the way we define and score the IQ scale, I think that censoring is an inapt description. However, the measurement system probably has some floor effect, i.e. it can't measure IQs below some level. In that sense, it may be correct to say that there is censoring.
