Final answer:
Z-scores are statistical measurements used to describe a value's position relative to the mean of a set of values, standardizing scores to enable comparison across different normal distributions. The standard normal distribution is used with a mean of 0 and a standard deviation of 1. The empirical rule helps to contextualize the z-scores within the normal distribution.
Step-by-step explanation:
Understanding Z-Scores in Statistics
Z-scores are a statistical measurement that describe a value's relationship to the mean of a group of values. A z-score is calculated by subtracting the mean from the value and then dividing the result by the standard deviation. This process of standardizing allows researchers to compare positions of scores between two or more different normal distributions. The standardization process converts different normal distributions to a standard normal distribution with a mean of 0 and a standard deviation of 1, known as Z ~ N(0, 1).
For example, if you have two normal distributions of weight gains, say X ~ N(5, 6) and Y ~ N(2, 1), and you want to compare weight gains x=17 from the first and y=4 from the second, you would use z-scores to standardize these values. This makes it possible to see that both scores are two standard deviations to the right of their respective means, even though the raw scores and scales are different.
It is also useful to reference the empirical rule, which indicates that roughly 68% of values within a normal distribution fall between z-scores of -1 and 1, about 95% between z-scores of -2 and 2, and about 99.7% between z-scores of -3 and 3. Understanding and utilizing z-scores is crucial for making valid comparisons across different datasets in statistics.