IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If a certain person has an IQ of 131, what percentage of the population has a lower IQ?

Question

kirbykirby · Answer

Let $X$ be the IQ score. If a person has $X=131$, then the question asks what percentage (or what's the probability) that the population has a lower score. This is effectively asking for $P(X<131)$.

Since the scores are normally distributed (i.e. $X$ follows a normal distribution), you can standardize $X$ to obtain a standard normal distribution, and then use a standard normal table to find out the probability. Remember you standardize by doing:
$$\frac{X-\mu}{\sigma} $$, where $\mu$ is the mean and $\sigma$ is the standard deviation. So:

$$P(X<131)=P\left( \frac{X-100}{15} < \frac{131-100}{15}\right)=P(Z<2.07) $$