Example of Calculating a z Score

Match each variable from the formula for computing a z-score to its correct definition.

In a psychological study, a researcher uses the formula $z = \frac{X - M}{SD}$ (where $X$ is the raw score, $M$ is the mean, and $SD$ is the standard deviation) to standardize data. What is the primary function of dividing the difference ($X - M$) by the standard deviation ($SD$) within this formula?

In a psychological study on cognitive processing speeds, researchers use the $z$ score formula ($z = \frac{X - M}{SD}$) to standardize scores for comparison across different testing conditions. Arrange the following participant scenarios in order of their resulting $z$ score magnitude (absolute value), from the highest magnitude (top) to the lowest magnitude (bottom).

A peer reviewer claims that a researcher must have made an error in applying the formula $z = \frac{X - M}{SD}$ because the resulting $z$ score was negative, even though the distribution's standard deviation ($SD$) was a positive value.

A researcher is developing a new standardized psychological assessment and needs to establish the norming parameters such that a raw score of $65$ is transformed into a standardized $z$ score of exactly $+2.5$. Which of the following sets of technical specifications for the distribution's mean ($M$) and standard deviation ($SD$) must the researcher implement to achieve this design goal?

In the formula for computing a $z$ score, $z = \frac{X - M}{SD}$, the numerator is calculated by subtracting the individual raw score ($X$) from the mean of the distribution ($M$).

In a psychological study on reaction times, the sample mean ($M$) is 450 milliseconds and the standard deviation ($SD$) is 40 milliseconds. A participant has a raw reaction time ($X$) of 410 milliseconds. Using the formula $z = \frac{X - M}{SD}$, this participant's $z$ score is _____.

A psychological researcher standardizes memory recall scores across different groups. Match each participant's raw score ($X$) and group parameters (mean $M$ and standard deviation $SD$) to their correct standardized $z$ score.

A researcher standardizes a set of test scores using the z-score formula. If the standard deviation ($SD$) of the distribution decreases while a participant's raw score ($X$) and the mean ($M$) remain constant (where $X > M$), the value of the resulting $z$ score will _____.

A researcher claims that a score of $X = 82$ on a test with mean $M = 70$ and standard deviation $SD = 4$ is more extreme than a score of $X = 35$ on a test with mean $M = 50$ and standard deviation $SD = 5$. Arrange the steps in the correct order to evaluate this claim using standard scores.

State the mathematical formula used to compute a $z$ score, and clearly define each of the four variables represented in this formula based on the distribution parameters.

Explain how the relationship between Participant A's raw score ($X$) and the distribution's mean ($M$) affects the numerator of the formula, and explain what the sign of the resulting $z$ score indicates about Participant A's performance relative to the group mean.

A participant in a psychological study has a raw score of $X = 40$. The mean of the distribution is $M = 50$ and the standard deviation is $SD = 5$. Apply the $z$ score formula to calculate this participant's $z$ score, showing the intermediate steps of your calculation.