Z Scores,Standard Scores,and Composite Test Scores Explained

Need for Composite Scores

One way to get an overall perspective is to create a composite score. This is easily done in some circumstances; for example, one may reasonably add or average language, science, math, geography, and history marks to get a single composite score in school examinations. Simple addition or averaging of marks is possible because all subjects are treated equally, all subjects are marked from 0 to 100, and all subjects have higher marks indicating better performance. Simple addition or averaging is not possible for cognitive tasks because different tasks have different everyday importance, because different tasks have different minimum and maximum scores, because in some tasks lower scores indicate better performance, and in other tasks, higher scores indicate better performance, and because some tasks yield scores measured in units of time, others yield scores measured as the number of correct responses, and so on. As examples, verbal memory may be more important than visual memory because of its application to everyday life; tests of processing speed are measured in units of time and lower scores indicate better performance; and tests of memory are measured in the number of units correctly recalled and higher scores indicate better performance. So, simple addition or averaging of scores is not possible.

Computing Z Scores

One solution is to first convert the original (raw) scores for each cognitive task into a new score that is described in the same unit for all tasks. The new scores can then be added or averaged to form a composite score. Conversion of the raw scores into Z scores is one such approach. To do this (for example) for the verbal memory test in the cognitive battery, I would need to perform the following actions with the verbal memory raw scores.

Z scores are similarly calculated for each patient for each of the remaining four cognitive tasks. This is done separately for the baseline and endpoint data.

Understanding Z Scores

If we look at the formula for the Z score, we will immediately realize that the Z score tells us how far above or below the mean an individual’s score is, expressed in units of SD. So, if the M(SD) is 18(4) for the verbal memory scores in the pooled sample, a patient with a verbal memory score of 20 has a Z score of (20−18)/4, or 0.5. That is, the patient’s verbal memory score is half an SD above the mean of the sample. Another patient whose raw score is 12 would have a Z score of (12−18)/4, or −1.5; that is, one and a half SDs below the sample mean.

Interpreting and Using the Z Scores

The raw scores were in different units in the different cognitive tasks. Z scores are all in the same unit, that is, SD. The Z score distribution has a mean of 0 and an SD of 1. Z scores are useful because they allow data to be interpreted or used in many ways, as the following examples show:

The Z score tells us at a glance how the patient has performed relative to the rest of the sample, something that is not evident from the inspection of a raw score.

Z Scores and Composite Scores

When Z scores are added or averaged as described above, each cognitive task receives equal weightage. It is possible to create composite scores in which some tasks are given higher weightage than others, based on preset values for weights. For example, it can a priori be decided that, because verbal tasks are more relevant in everyday life than visual tasks, the verbal memory task should receive twice the weight that the visual memory task receives when computing the composite score. Weights can also be determined and assigned through statistical methods.^{3, 4}

Once the composite score has been calculated for each patient, the M(SD) composite score can be calculated for CR and TV groups separately at the study baseline and at the study endpoint and then processed using usual statistical methods; this can be done whether the composite score is a total or an average of the Z scores of the individual cognitive tasks.

Standard Scores

Some people find it hard to understand Z scores, especially when values are negative (readers are reminded that Z scores have a mean of 0, and that Z values that are negative indicate scores that are below the mean of the sample). This difficulty can be resolved by converting Z scores into other standard scores. The Z score is one example of a standard score; using simple formulae, Z scores can be converted to other standard scores that have only positive values and other specific properties. An example is the T score which has M=50 and SD=10. Stanine and sten scores are based on the same principle. Stanine (standard nine) scores range from 1 to 9, with a mean of 5 and an SD of 2; sten (standard ten) scores range from 1 to 10, with a mean of 5.5 and an SD of 2. Stanine and sten scores are used in some psychological tests. IQ scores are also standardized; they have a mean of 100 and an SD of 15. Readers may note that Z transformation and other methods of standardization do not change the ranking of the original data.

Computing Z Scores: Reprise

The Z score for an individual measurement can be calculated using the mean and standard deviation of a sample, or of a pooled sample, or of the population, depending on the context in which the Z score requires to be derived and used. If the sample comprises a single group, such as a class of students, the Z scores are based on the M(SD) of that group. If there are two groups, such as in the study described in this article, Z scores should be calculated based on the M(SD) of the pooled sample. However, when Z scores are interpreted for a single individual on a test for which population norms are available, the population mean and standard deviation are used rather than the sample M(SD).

As an aside, for the study described in this article, why are the Z scores computed for the pooled sample; why cannot the Z scores be computed for each group separately, and then M(SD) Z scores compared between groups? The answer ought to be obvious. Z scores may create new values but do not change the ranks (order) of the raw scores within a group; and these new values have an M(SD) of 0(1) because, as stated earlier, this is a property of Z scores. So, if Z scores are computed separately for CR and TV groups, the M(SD) of the z scores for each group will be 0(1), making comparisons between groups illogical. However, if the CR and TV groups are pooled, the order of the raw scores will change; whereas the M(SD) of the Z scores for the pooled group will be 0(1), the Z scores for the CR and TV group patients will depend on the new order, and the M(SD) of the Z scores for the CR and TV groups will no longer each be 0(1). So, the M(SD) Z scores thus created can now be validly compared between the CR and TV groups. Pooling of groups is done in certain nonparametric tests, as well. For example, in the Mann–Whitney and Kruskal–Wallis tests, the groups are pooled, individual values are ranked, and then the ranks are compared across groups.

Parting Notes

Knowledgeable readers may recognize that the standardized mean difference that is a measure of pooled effect size in meta-analysis and the (standardized) beta coefficient in regression analysis are both based on principles similar to those discussed in this article. A discussion on these, however, is out of the scope of the present article.