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Z Transformation

Z transformation is the process of standardization that allows for comparison of scores from disparate distributions. Using a distribution mean and standard deviation, z transformations convert separate distributions into a standardized distribution, allowing for the comparison of dissimilar metrics. The standardized distribution is made up of z scores, hence the term z transformation. Z scores are a special type of standard score in which each unit represents one standard deviation from the mean; z scores always have a distribution mean of 0 and a standard deviation of 1. This entry details a variety of issues central to understanding z transformation. First, standardization and z scores are explained. The formula for z transformation is provided and discussed. Second, an example of comparison across different metrics is provided to demonstrate how z transformation makes such comparisons possible. Two scores are compared and other applications of z scores are previewed.

Standardization and Z Scores

The first step in z transformation is to convert the original units to a standard unit. Standardizing is done by subtracting the distribution mean from the original value and then dividing by the standard deviation. The new value is the z score or standardized value, and the formula is

z = x − μ σ ,

where x is an observation, µ is the mean of the distribution, and σ is the standard deviation. A z score describes how many standard deviations an observation moves away from the mean, as well as the direction in which it moves. Positive scores indicate the observation is larger than the mean, and negative scores indicate the observations are smaller than the mean. Z scores always have a distribution mean of 0 and a standard deviation of 1, so the numerical value represents the number of standard deviations between x and µ. A z score of –1 describes a data point that is exactly 1 standard deviation below the mean, whereas a positive 1 is by definition 1 standard deviation above the mean.

Applying Z Scores

Although the definition of z transformation may sound technical, conceptually z transformation and z scores themselves are fairly simple and extremely useful. For example, with measures of central tendency, any two scores can be meaningfully compared. Imagine you want to get into a competitive program in videogame design, but the program is in Russia and you need to demonstrate your proficiency in Russian. You took two tests, the Russian Language Proficiency (RLP) and the Test of Former Soviet Languages (TFSL). Your videogame design program only requires one score, and you want to send your best performance. You scored a 30 on the RLP and a 650 on the TFSL. Which score do you send? Given the different metrics and hence different distributions, the choice is not obvious. However, z transformations will allow you to convert to a common metric to make the comparison simple.

In this example, the comparison appears difficult because of the different properties of the two distributions in terms of central tendency and variability. We know the two test scores in their original metrics, but we also know the mean and standard deviation of the two distributions. For the RLP, the mean is 20, and the standard deviation is 8. For the TFSL, the mean is 550, and the standard deviation is 150. That’s all the information we need. Making use of the z score formula, we know the RLP mean is 20, with a standard deviation of 8. Just work the formula: 30–20 = 10, so 10 is 1.25 standard deviations above the mean (10/8). Now we do the same for the TFSL, which has a mean of 550, with a standard deviation of 120. The score 650 minus the mean 550 is 100; 100/120 = .83 standard deviation above the mean. You should send the RLP score if you want to make the best impression.

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