Example of a z Score

Which of the following correctly expresses the formula for computing a z score?

In psychological research, standardizing scores allows for easier comparison between different distributions. Match each mathematical component of the z score formula ($z = \frac{X - M}{SD}$) with its specific conceptual role in this process.

By analyzing the mathematical relationships within the $z$ score formula ($z = \frac{X - M}{SD}$), arrange the following hypothetical participants in order from the highest (most positive) $z$ score to the lowest (most negative) $z$ score.

A researcher concludes that when evaluating which of two participants is more 'extreme' relative to their respective groups, simply comparing the difference between each raw score and its group mean ($X - M$) is a mathematically sound and sufficient method, even if the groups have different standard deviations ($SD$). This conclusion is valid according to the principles of the formula ($z = \frac{X - M}{SD}$).

A psychological researcher is designing a 'Unified Cognitive Profile' to compare students across different assessments. The goal is to create a standardized reporting system where a raw score of $X = 120$ on a Memory Recall test (with a mean of $M = 100$) and a raw score of $X = 55$ on a Mental Rotation test (with a mean of $M = 40$) both represent an identical relative standing of exactly $2.5$ standard deviations above their respective population means. Which set of subtest standard deviations ($SD$) must the researcher establish in the test design to achieve this specific outcome?

In the z score formula, the standard deviation ($SD$) is subtracted from the raw score ($X$) before dividing by the mean ($M$).

A researcher evaluates a participant's performance on a standardized test of processing speed. The participant's raw score ($X$) is 18, the population mean ($M$) is 24, and the standard deviation ($SD$) is 4. Using the formula $z = \frac{X - M}{SD}$, the participant's $z$ score is _____.

Based on the standard formula ($z = \frac{X - M}{SD}$), arrange the procedural steps a researcher must follow to compute a $z$ score for a particular participant's raw score.

A clinical psychologist is reviewing assessment data from four different patients across various standardized measures. Using the formula $z = \frac{X - M}{SD}$, calculate and match each patient's test results to their correct $z$ score.

By mathematically analyzing the structural relationships within the formula $z = \frac{X - M}{SD}$, a researcher can deduce that if the difference between a participant's raw score and the mean ($X - M$) is held constant, an increase in the standard deviation ($SD$) will invariably cause the absolute value of the resulting $z$ score to _____.