The Standard Normal Distribution and Z-scores

The standard normal distribution tells us how scores are organized among a population and how these scores can be distributed compared to the rest of the population based upon two parameters: mean and standard deviation. It is a normal distribution of standardized values called z-scores. A z-score is measured in units of the standard deviation. For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean.

The calculation is: x = μ + (z)σ = 5 + (3)(2) = 11

where μ is the population mean, and σ is the standard deviation.

The z-score is 3. The mean for the standard normal distribution is 0 and the standard deviation is 1.

The transformation z=(x−μ)/σ produces the distribution Z~ N(0,1). Thus, when comparing scores, we use this transformation to standardize the range of scores and determine their placement relative to other scores.

Also note that z-scores are either negative or positive. A positive z-score means that the value is higher than the mean. A negative z-score says that the individual scored below the mean.