Final answer:
A z-score indicates the number of standard deviations a value is from the dataset's mean. It's calculated with the formula z = (x - μ) / σ. By using z-scores, we can standardize different distributions for comparison.
Step-by-step explanation:
A z-score is calculated using the formula z = (x - μ) / σ, where x is the value in question, μ is the mean of the dataset, and σ is the standard deviation of the dataset. The z-score represents the number of standard deviations that x is away from the mean. If x is above the mean, the z-score is positive; if x is below the mean, the z-score is negative. A z-score of 0 means that x is exactly at the mean. To find a value x given a z-score and dataset parameters, we rearrange the z-score formula to x = μ + zσ. If you are to find the value that is one standard deviation above the mean, you would calculate x = μ + σ. Similarly, to find the value that is two standard deviations below the mean, you would calculate x = μ - 2σ.
Z-scores enable us to convert different distributions into the standard normal distribution for comparison, which has a mean of 0 and a standard deviation of 1. About 95 percent of x values from a normally distributed dataset lie within two standard deviations from the mean. This has implications in various statistical analyses, such as hypothesis testing and data normalization.