Final answer:
A z-score more than two standard deviations from the mean is typically considered an outlier, exemplified by a score 3 standard deviations above the mean being an anomaly, according to the 68-95-99.7 empirical rule.
Step-by-step explanation:
When a z-score is used to identify anomalies, a value that falls more than two standard deviations away from the mean in a normal distribution is typically considered an outlier. The theory underlying this approach is often referred to as the empirical rule or the 68-95-99.7 rule, which states that for a normal distribution, approximately 68% of data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations from the mean.
An example of an outlier in this context could be a score of 11 in a distribution with a mean of 5, if the score is 3 standard deviations above the mean. Considering this empirical rule, a z-score of +3 or -3 would represent values that are very unusual and likely to be outliers, increasing the chances of a true anomaly in the dataset.