If you are a mono-g-ist, you can use the estimate of g (IQ) to get an idea of what is the typical range of achievement scores for a child with that IQ. Not every child with the same IQ will have the same achievement scores. Not even mono-g-ists believe that. Also, it is simply not true that achievement cannot be higher than IQ. Equally false is the assumption that if achievement is higher than IQ, then the IQ is wrong. These misconceptions are based on two premises: one true, the other false. If potential is the range of all possible outcomes, it is logically true that people cannot exceed their potentials. The false premise is that IQ and achievement tests are measured on the “potential scale.” By analogy, if I say, “This thermometer reads -10 degrees. I know from my understanding of physics that Brownian motion never stops and thus no temperature dips below zero. Therefore, this thermometer is incorrect.” My premise is true, if the thermometer is on the Kelvin scale. However, it is on the Celsius scale and so there is no reason to believe that something is amiss. IQ and achievement simply are not measured on the “potential scale.” They are measured with standard scores, which are transformed deviations from a population mean. Because of this about half of all people have academic achievement scores that are higher than their own IQ. There is nothing wrong with this.
 And not every child with the same achievement scores will have the same IQ.
This post is an excerpt from:
Schneider, W. J. (2013). Principles of assessment of aptitude and achievement. In D. Saklofske, C. Reynolds, & V. Schwean (Eds.), Oxford handbook of psychological assessment of children and adolescents (pp. 286–330). New York: Oxford