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If you have a variable X distributed with mean m and standard deviation s, then the z-score is (x - m)/s.
If X is normally distributed, or is the mean of a random sample then Z has a Standard Normal distribution: that is, a Gaussian distribution with mean 0 and variance 1. The probability density function of Z is tabulated so that you can check the probability of observing a value as much or more extreme.
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Another name for a standardized value of an observation in a normal distribution?
zscore
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