Final answer:
A Z-score is a statistical measurement indicating how far away a data point's score is from the mean, measured in standard deviations. John's Z-score of -0.21, which is higher than Ali's -0.3, implies John is closer to the mean score compared to Ali. The Empirical Rule states that in a standard normal distribution, about 68%, 95%, and 99.7% of values fall within one, two, and three standard deviations from the mean, respectively.
Step-by-step explanation:
Understanding Z-Scores
A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score.
Using the given examples, we can understand Z-scores better. For instance, if John's Z-score is -0.21 and Ali's Z-score is -0.3, it means that John's score is closer to the mean than Ali's, and therefore, John has a better performance indicator than Ali when considering their respective categories. This conclusion uses the principle that higher Z-scores indicate a value above the mean, while lower scores indicate a value below the mean.
The Empirical Rule, also known as the 68-95-99.7 rule, helps us to understand how data is spread in a normal distribution. For a standard normal distribution, with a mean (μ) of 0 and standard deviation (σ) of 1, approximately 68 percent of the values fall within one standard deviation of the mean (between Z-scores of -1 and 1), about 95 percent within two standard deviations (between Z-scores of -2 and 2), and about 99.7 percent within three standard deviations (between Z-scores of -3 and 3).