The z-score, also known as the z-value, represents the number of standard deviations that a given value x is from the mean of a data set. The following formula can be used to convert any x-value into its corresponding z-score.

Here, μ represents the mean and σ the standard deviation of the distribution. The z-value corresponding to a sample mean x is called a z-statistic and is calculated using a similar formula.

In this formula, s is the standard deviation of the sample, n is the sample size, and μ is the population mean.

Example

Consider a distribution with mean 12 and standard deviation 2.5. The z-score corresponding to x=11.5 is computed as follows. z = 11.5-12/2.5 ⇔ z = -0.2 Consequently, 11.5 is 0.2 standard deviations to the left of the mean. The z-scores can be used to standardize a normal distribution. Then, for a random z-value of a standard normal distribution, the Standard Normal Table can be used to determine the corresponding area under the curve.