Final answer:
A z-score represents how many standard deviations a value is from the mean of a distribution. Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean. The z-score is calculated using the formula z = (x - μ) / σ.
Step-by-step explanation:
Understanding Z-Scores
A z-score is a statistical measure that describes a value's relationship to the mean of a group of values, 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. A z-score of 1.0 signifies that a value is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative z-score indicating it is below the mean.
In the context of the z-score examples provided, a z-score of +2 or -2 would represent a value that is two standard deviations from the mean. Similarly, a z-score of +3 or -3 represents a value that is three standard deviations from the mean. These are based on the empirical rule, which is also known as the 68-95-99.7 rule, reflecting the percentage of values within one, two, and three standard deviations from the mean, respectively.
To calculate a z-score for a specific value, the formula is z = (x - μ) / σ, where x is the value being standardized, μ is the mean of the population, and σ is the standard deviation of the population.