MOD 3.3 | VID 2 | Standardized Test Z-Score Calculations and Comparisons

ACT Test Score Distribution:
- Mean score ( $\mu$ ): $21.1$
- Standard deviation ( $\sigma$ ): $5.2$
- Distribution shape: Approximately bell-shaped.
SAT Mathematics Test Score Distribution:
- Mean score ( $\mu$ ): $514$
- Standard deviation ( $\sigma$ ): $117$
- Distribution shape: Approximately bell-shaped.

Conceptual Meaning of a Z-Score:
- The z-score of a data value measures the exact number of standard deviations that value lies away from its population mean.
Formula for Calculating a Z-Score from a Raw Data Value:
- Given a value $x$ from a population with mean $\mu$ and standard deviation $\sigma$ :
- $z = \frac{x - \mu}{\sigma}$
Formula for Calculating a Raw Data Value from a Z-Score:
- Given a population with mean $\mu$ and standard deviation $\sigma$ , a value $x$ corresponding to a given z-score $z$ is:
- $x = \mu + z \times \sigma$

Z-Score Calculation for an ACT Score of $27$ :
- Population mean ( $\mu$ ): $21.1$
- Population standard deviation ( $\sigma$ ): $5.2$
- Raw score ( $x$ ): $27$
- $z = \frac{27 - 21.1}{5.2}$
- $z = 1.13$
- Interpretation: An ACT score of $27$ is $1.13$ standard deviations above its population mean.
Z-Score Calculation for an SAT Math Score of $650$ :
- Population mean ( $\mu$ ): $514$
- Population standard deviation ( $\sigma$ ): $117$
- Raw score ( $x$ ): $650$
- $z = \frac{650 - 514}{117}$
- $z = 1.16$
- Interpretation: An SAT math score of $650$ is $1.16$ standard deviations above its population mean.
Relative Performance Comparison:
- The SAT math score of $650$ yields a z-score of $1.16$ .
- The ACT score of $27$ yields a z-score of $1.13$ .
- Because $1.16 > 1.13$ , the SAT math score of $650$ is higher relative to its population of scores than the ACT score of $27$ .