Z Score

Definition

A z score is the deviation of a raw score from the mean and expressed in terms of standard deviation units. The mean of z scores is always 0 and the standard deviation is always 1. z scores above the mean have a positive value, whereas z scores below the mean have a negative value. z scores are interpreted using the z-distribution table, and most statistical textbooks include these tables for reference.

Calculation

The deviation score is the difference of any score (X) from the mean (M). When this deviation score is divided by the standard deviation (SD), z score of any raw score can be calculated. The following formula is used to calculate z scores (Z):

$$Z=\frac{\mathrm{X}-M}{SD}$$

The following example illustrates the calculation of a z score of a raw score using a communication-related example. Suppose that a group of communication researchers measure the organizational identification of newcomers using an organizational identification scale. One newcomer scores 20 in the identification scale. The sample mean for this score is 25 and the standard deviation is 2. Using the z-score calculation formula, [Page 1900]one can deduct the mean score of 25 from the individual’s score of 20 and divide it by the standard deviation of 2. The result represents the z score of this individual’s raw score on the organizational identification scale. Using this formula, communication researchers can transform raw scores into z scores manually, yet several statistical analysis packages have the function of z transformations for easier automated calculations.

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