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The great utility of the z-score

Relationship between percentile and RSD - residual standard deviation There is great utility in using thez-score, i.e. the difference between observed (y) and predicted (Y) value divided by the residual standard deviation (RSD) about the mean predicted value:

z-score = (y - Y)/RSD

The great advantage is that the z-score can be used for any index; if the deviations (y - Y) are normally distributed, regardless of whether we are dealing with the hemoglobin concentration, standing height, serum creatinine concentration or FEV1, the z-score discloses how rare or common the observation is. For example, if the z-score is smaller than -1.64 then the observation occurs in only 5% of the reference population. A z-score > +1.96 is encountered in only 2½ per cent of all healthy subjects.

In the reference population, the trajectory from the lowest to the highest z-score encompasses 0 to 100% of all subjects. If we depict the cumulative percentage of the population up to a certain z-score (Y-axis) as a function of the z-score (X-axis), an S-shaped curve is obtained: this then depicts graphically the relationship between percentile and SDS in the reference population.

Z-scores for Caucasians and African AmericansIn overt pathology observed values fall outside the lowest (0%) and highest (100%) percentiles of a healthy population. On that account percentiles are not very practical in pathological cases, but the Z-score looses none of its usefulness. If the Z-score of the FEV1 = -3, for example, this signifies that the FEV1 is far below the 2½ percentile in a healthy population. Particularly if the subject presented with respiratory symptoms chances are small that we are dealing with a ‘normal’ FEV1.

On the right z-scores for FEV1 for healthy African Americans according to the Global Lung Function Initiative have been superimposed on those for healthy Caucasians. The overlap is almost perfect. Hence, even though the absolute values for FEV1 between the two ethnic groups differ by about 14%, using the z-score removes differences between ethnic groups so that the interpretation of test results is identical for any ethnic group. Thus, a z-score of -1.64 implies that the measurement is at the lower 5th percentile of whatever ethnic group. In fact, the z-score is independent of age, height, sex and ethnic group, and thus unbiased by any of these variables. This is not the case for percent of predicted.

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