z-Score

The z-score is a statistical transformation that specifies how far a particular value lies from the mean of a normal distribution in terms of standard deviations, z-scores are particularly helpful in comparing observations that come from different populations and from distributions with different means, standard deviations, or both. A z-score has meaning only if it is calculated for observations that are part of a normal distribution.

z-scores are sometimes referred to as standard scores. When the values of a normal distribution are transformed into z-scores, the transformed distribution is said to be “standardized” such that the new distribution has a mean equal to 0 and a standard deviation equal to 1.

The z-score for any observation is calculated by subtracting the population mean from the value of the observation and dividing the difference by the population standard deviation, or z = (x − μ)/σ. Positive z-scores mean that the observation in question is greater than the mean; negative z-scores mean that it is less than the mean. For instance, an observation with a z-score of 1.0 would mean that the observation is exactly one standard deviation above the mean of the distribution. An observation with a z-score equal to −0.5 would fall one-half of one standard deviation below the distribution's mean. An observation with a z-score equal to 0 would be equal to the mean of the distribution.

As an example, a researcher looking at middle school students' test scores might benefit from using z-scores as a way of comparing the relative performance of a seventh-grade student on a seventh-grade test to an eighth-grade student on an eighth-grade test. In this example, the researcher knows that the scores for the entire population of seventh graders and for the entire population of eighth graders are normally distributed. The average number of correct answers (out of 100 multiple choice questions) for the population of seventh graders on the seventh-grade test is 65 with a standard deviation of 10. The average score (out of 100 multiple choice questions) for the population of eighth graders on the eighth-grade test is 72 with a standard deviation of 12.

The seventh- and eighth-grade students of interest to this researcher scored 70 correct and 75 correct, respectively. Transforming each raw score into a z-score would be an appropriate way to determine which student scored better relative to his or her own population (cohort). The z-score for the seventh-grade student would be (70 = 65)/10, or 0.5, meaning that he or she scored 0.5 standard deviation above the average for seventh-grade students. The z-score for the eighth-grade student would be (75 = 72)/12, or 0.25, meaning that he or she scored 0.25 standard deviation above the average for eighth-grade students. Relative to his or her peers, the seventh-grade student performed better than the eighth-grade student, despite the eighth grader's higher raw score total.