Final answer:
Calculating z-scores assumes a normal distribution of data. Z-scores are standardized values based on how many standard deviations a score is from the mean, allowing comparisons across different datasets. This concept is central to statistical methods like hypothesis testing.
Step-by-step explanation:
To calculate z-scores, we make the assumption that the dataset follows a normal distribution. A z-score is used to describe the exact location of a score within a distribution and is given by the formula z=(x-μ)/σ, where x is the score, μ is the mean, and σ is the standard deviation. In the case of a standard normal distribution, which has a mean of 0 and a standard deviation of 1, z-scores allow us to determine how many standard deviations a score is from the mean.
For example, if we have a data set with a mean of 5 and a standard deviation of 2, and we want to determine the z-score for a score of 11, we use the formula: z = (11 - 5) / 2 = 3. This means that the score of 11 is 3 standard deviations above the mean.
The empirical rule, which corresponds to approximately 68%, 95%, and 99.7% of the values falling within ±1, ±2, or ±3 standard deviations from the mean, respectively, helps to visualize this concept. The z-score is essential when performing statistical tests such as hypothesis testing using the Student's t-distribution, where the assumption of data being normally distributed is also important.