If you have a variable X that is normally distributed with mean m and variance s2 then the z-score, Z = (X - m)/s.
Z has a standard Normal distribution.
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A z table is used to calculate the probability of choosing something that is normally distributed. In order to use it, first a z score is needed. A z score is the number of standard distributions a value is away from the mean of the data. In order to find the z score, take the value of the datum, subtract the mean, then divide by the standard deviation. The result is a z score. Look up the z score on the table to find the probability of getting anything equal to or lesser than the value you chose.
Z = (x-mu)/sigma. So, for your example, any x value can be transformed to Z-score by the formula Z = (x-100)/20.
it doesn't exist.
If the Z Score of a test is equal to zero then the raw score of the test is equal to the mean. Z Score = (Raw Score - Mean Score) / Standard Deviation
To find what z-score represents the 80th percentile, simply solve for 0.8 = F(z), where F(x) is the standard normal cumulative distribution function. Solving gives us: z = 0.842