Final answer:
A z-score of approximately 1.74 for an SAT score of 720 indicates that the score is 1.74 standard deviations above the mean. A SAT math score of 692.5 represents 1.5 standard deviations above the mean. Z-scores provide a way to compare two scores from different tests with different means and standard deviations.
Step-by-step explanation:
The z-score is a statistical measure that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. To calculate the z-score of an SAT math score of 720, you subtract the mean of the SAT math section scores from the individual's score and then divide by the standard deviation. Using the provided data with a mean (µ) of 520 and standard deviation (o) of 115:
z = (720 - 520) / 115 ≈ 1.74
This z-score means that a score of 720 is approximately 1.74 standard deviations above the mean SAT math score.
To find a SAT math score that is 1.5 standard deviations above the mean, we would perform the following calculation:
Score = µ + (z * o) = 520 + (1.5 * 115) ≈ 692.5
A score of approximately 692.5 means the individual's performance is 1.5 standard deviations above the mean SAT math score.
Comparing two different test scores using z-scores allows for a fair comparison regardless of the differing scales or distribution of the tests. To determine who scored better between a student who got a 700 on the SAT math test and another who scored a 30 on the ACT math test, we calculate the respective z-scores using the means and standard deviations of their tests. Higher z-scores indicate better performance relative to each test's mean.