Sokal & Rohlf (1995) and Griffiths et al. Norman & Streiner (2008), Gravetter & Wallnau (2006) and Heiman (2001) all provide reasonable coverage of Z-scores. One clear misuse, which is fortunately rare, is the belief that standardizing measurements will also normalize them. The latter practice can be advantageous, although other methods of analysis (such as covariance analysis) are often preferable. We give examples of their use for standardizing variables prior to analysis, and for removing the effect of an explanatory variable by standardizing the data to the mean value of each level of that variable. Z-scores are used much less in other disciplines. Another misuse may be to use the relatively sophisticated Z-score in a famine emergency, when mid upper arm circumference may be the more appropriate diagnostic tool. One misuse of Z-scores is to use the cut-offs of +2 and +3 to assess obesity - the body mass index is more appropriate for this. Whilst that may usually be the case, it should not be automatically assumed. ![]() The validity of nutritional Z-scores depends on the validity of the current reference values irrespective of ethnic group. Mean Z-scores are used to evaluate the nutritional state of populations relative to the reference population. Cut-off scores of -2 and -3 are used to identify children suffering from malnutrition. Weight for age, height for age, and weight for height Z-scores are computed using international reference data intended to reflect human growth patterns under optimal conditions. In biostatistics probably the commonest use of Z-scores is in the analysis of human nutritional data, especially for children. Sometimes the distribution of the whole sample is examined, in which case the Z-scores will not have a mean of zero and a standard deviation of one - what is of interest is the extent to which their distribution differs from the reference population. In this situation the Z-scores are used to identify those individuals in the sample falling below a specified Z-score. In some applications (such as weight-for-age in nutritional studies), the Z-scores are not based upon the known population mean and standard deviation, but on an external reference population. In other words converting data to Z-scores does not normalize the distribution of that data! ![]() If however, the original distribution is skewed, then the Z-score distribution willĪlso be skewed. You can then make assumptions about the proportion of observations below or above specific Z-values. If the original distribution is normal, then the Z-score distribution will be normal, and you will be dealing with a standard normal distribution. The shape of a Z-score distribution will be identical to the original distribution of the raw measurements. If your Z-score distribution is based on the population mean and population standard deviation, then the mean and the standard deviation of the Z-score distribution will only approximate to zero and one if the sample is random. ![]() If your Z-score distribution is based on the sample mean and sample standard deviation, then the mean and standard deviation of the Z-score distribution will equal zero and one respectively. Commonly a known reference population mean and standard deviation are used. A Z-score is calculated by subtracting the mean value from the value of the observation, and dividing by the standard deviation. The sign of the Z-score (+ or - ) indicates whether the score is above (+) or below ( - ) the mean. A Z-score serves to specify the precise location of each observation within a distribution. In other words it merely re-scales, or standardizes, your data. ![]() The resulting misuse is, shall we say, predictable.Ī Z-score (or standard score) represents how many standard deviations a given measurement deviates from the mean. Statistics courses, especially for biologists, assume formulae = understanding and teach how to do statistics, but largely ignore what those procedures assume, and how their results mislead when those assumptions are unreasonable.
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