Researchers performing statistical analysis are often interested how an individual data point fits into the hypothesized distribution of the population from which it comes. Calculated z-scores allow researchers to obtain a number representing where this datum fits in relation to the mean of the distribution; the calculated z-score for a datum is a standardization of that data point. The closer any z-score is to zero, the more it adheres to the hypothesized distribution.
A data point's z-score comes straight from the calculations of the mean and standard deviation for a data sample. The formula for calculating a z-score is Z = (x -- m)/s, where "x" is the datum, "m" is the mean and "s" is the standard deviation. Thus, for any given data point, you can calculate the z-score needing only the mean and standard deviation of the sample.
The critical z-score does not truly belong to a data set. Instead, the critical z plays the role in giving the researcher an idea of how extreme the data point for which the calculated z-score is in the sample. The critical z-score is on whole a benchmark to which the calculated z-score is compared. If the calculated z-score surpasses the critical z-score, it is said that the datum is an extreme value, differing greatly from the mean of the data.
The critical z-score's calculation is less of a calculation and more of a reference algorithm. To find a critical z, you need to decide on a alpha value. The alpha value, a number between 0 and 1, represents how the researcher conceptualizes the "tails," or areas at the extreme edge, of a distribution. The most common alpha value is 0.05, representing the tails as being those outer edges of a distribution that contains 5 percent of the values. After deciding on an alpha value, the researcher can use a Z-table to look up the corresponding critical z-score. That is, the critical z-score is entirely determined by the alpha value.