SAT versus ACT Eleanor scores $680$ on the SAT Mathematics test. The distribution of SAT scores is

symmetric and single-peaked, with mean $500$and standard deviation $100$. Gerald takes the American College Testing (ACT) Mathematics test and scores 27. ACT scores also follow a symmetric, singlepeaked distribution—but with mean $18$ and standard deviation $6$. Find the standardized scores for both students. Assuming that both tests measure the same kind of ability, who has the higher score?

The average score on the SAT Mathematics test is $500$. Eleanor received a $680$ on the SAT Mathematics test. The SAT Mathematics test has a standard deviation of $100$. Gerald received a $27$on the ACT Mathematics test. The average ACT Mathematics test score is $18$ points. The ACT Mathematics exam score has a standard deviation of $6$ points.

The inflection points are the points where the curve shifts from being concave up to being concave down. These inflection points are always one standard deviation distant from the mean on a normal density curve.

Eis's standardized score is shown below.

$$\begin{gathered} z\left(Eleanor\right)=\frac{x-\mu }{\sigma } \\ =\frac{680-500}{100} \\ =1.80 \end{gathered}$$

The standardized score for G can be found in the table below.

$$\begin{gathered} z\left(Gerald\right)=\frac{x-\mu }{\sigma } \\ =\frac{27-18}{6} \\ =1.50 \end{gathered}$$

E's score of $680$was $1.80$standard deviations higher than the mean, whereas G's score of$27$was $1.50$standard deviations higher. Therefore, the standardized score for Eis $180$ and the standardized score for G is $150$