# Can left-censored data be normal distributed?

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)

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 $$\Phi(-6) \approx 9.865876e-10$$.
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.