Question

How is a Z score related to a raw score?

a) A Z score is the same as a raw score.
b) A Z score is a measure of how many standard deviations a raw score is from the mean.
c) A Z score is the average of all raw scores in a data set.
d) A Z score represents the total range of raw scores in a dataset.

Answer 1

A Z score is a measure of how many standard deviations a raw score is from the mean. Thus, the correct answer is option b) A Z score is a measure of how many standard deviations a raw score is from the mean.

Step-by-step explanation:

A Z score, or standard score, is a statistical measure that quantifies a raw score's relationship to the mean of a group of scores, indicating how many standard deviations an individual score is above or below the mean. Mathematically, the formula for calculating the Z score (Z) is given by:

`\[Z = \frac{(X - \bar{X})}{\sigma}\]`

where:

-X is the raw score,

-
`\(\bar{X}\)` is the mean of the dataset,

-
`\(\sigma\)` is the standard deviation.

The numerator
`\((X - \bar{X})\)` represents the raw score's deviation from the mean, and dividing by \(\sigma\) standardizes this deviation. If Z is positive, the raw score is above the mean; if negative, it's below.

For instance, if a raw score X is 75, the mean
`\(\bar{X}\)` is 70, and the standard deviation
`\(\sigma\)` is 5, the Z score would be:

`\[Z = ((75 - 70))/(5) = 1\]`

This Z score of 1 indicates that the raw score is 1 standard deviation above the mean.

In summary, a Z score provides a standardized measure that facilitates comparisons across different datasets with varying means and standard deviations. It allows researchers and analysts to understand where a specific raw score stands relative to the average, making it a crucial tool in statistics and data analysis. Therefore, the correct answer is option b) A Z score is a measure of how many standard deviations a raw score is from the mean.