# Zack takes the SAT and his best friend Nick takes the ACT. Zack’s SAT math score is 680, and Nick’s ACT math score is 27. SAT math scores in the county are normally distributed, with a mean of 500 and a standard deviation of 100. ACT math scores in the county are also normally distributed, with a mean of 18 and a standard deviation of 6. Assuming that both tests measure the same kind of ability, who has scored better?

Jan 5, 2015

Zack has scored better than Nick.

To compare the two scores, you should convert their scores to a $Z$-score to determine who has the higher $Z$-score within a normal probability distribution. $Z$ is the standard normal distribution, and as long as a statistic is observed in normal distribution, you can map the statistic onto the $Z$ distribution in order to compare the statistic to other statistics inside the normal distribution.

To convert a random variable (the person's score) into a $Z$-score, you use the following formula:

$Z = \frac{X - \mu}{\sigma}$

$Z$ is the resulting Z-score you want to find, in order to compare the two students scores. X is your random variable you are scoring (in this case, the student's score). $\mu$ is the mean in the probability distribution. $\sigma$ is the standard deviation in the probability distribution.

So, first plug in the corresponding numbers for Zack:

$\frac{680 - 500}{100} = 1.8$

Then, plug in the corresponding numbers for Nick:

$\frac{27 - 18}{6} = 1.5$

As you can see, $1.8 > 1.5$, there for Zack outperformed Nick.

Source: Wikipedia