Competitive Exams: Standardized Distribution Scores, or Z-Scores
Actual scores from a distribution are commonly known as a “raw scores.” These are expressed in terms of empirical units like dollars, years, tons, etc. We might say “The Smith family's income is $29, 418.” To compare a raw score to the mean, we might say something like “The mean household income in the US is $2, 232 above the Smith family's income.” This difference is an absolute deviation of 2, 232 emirical units (dollars, in this example) from the mean. When we are given an absolute deviation from the mean, expressed in terms of empirical units, it is difficult to tell if the difference is “large” or “small” compared to other members of the data set. In the above example, are there many families that make less money than the Smith family, or only a few? We were not given enough information to decide. We get more information about deviation from the mean when we use the standard deviation measure presented earlier in this tutorial. Raw scores expressed in empirical units can be converted to “standardized” scores, called z-scores. The z-score is a measure of how many units of standard deviation the raw score is from the mean. Thus, the z-score is a relative measure instead of an absolute measure. This is because every individual in the dataset affects value for the standard deviation. Raw scores are converted to standardized z-scores by the following equations: Population z-score Sample z-score where is the population mean, is the sample mean, is the population standard deviation, s is the sample standard deviation, and x is the raw score being converted.
For example, if the mean of a sample of I. Q. Scores is 100 and the standard deviation is 15, then an I. Q. Of 128 would correspond to: = (128 − 100)/15 = 1.87
For the same distribution, a score of 90 would correspond to: z = (90 − 100)/15 = -0.67
A positive z-score indicates that the corresponding raw score is above the mean. A negative z-score represents a raw score that is below the mean. A raw score equal to the mean has a z-score of zero (it is zero standard deviations away). Z-scores allow for control across different units of measure. For example, an income that is 25, 000 units above the mean might sound very high for someone accustomed to thinking in terms of US dollars, but if the unit is much smaller (such as Italian Lires or Greek Drachmas), the raw score might be only slightly above average. Z-scores provide a standardized description of departures from the mean that control for differences in size of empirical units. When a dataset conforms to a “normal” distribution, each z-score corresponds exactly to known, specific percentile score. If a researcher can assume that a given empirical distribution approximates the normal distribution, then he or she can assume that the data's z-scores approximate the z-scores of the normal distribution as well. In this case, z-scores can map the raw scores to their percentile scores in the data. As an example, suppose the mean of a set of incomes is $60, 200, the standard deviation is $5, 500, and the distribution of the data values approximates the normal distribution. Then an income of $69, 275 is calculated to have a z-score of 1.65. For a normal distribution, a z-score of 1.65 always corresponds to the 95th percentile. Thus, we can assume that $69, 275 is the 95th percentile score in the empirical data, meaning that 95% of the scores lie at or below $69, 275. The normal distribution is a precisly defined, theoretical distribution. Empirical distributions are not likely to conform perfectly to the normal distribution. If the data distribution is unlike the normal distribution, then z-scores do not translate to percentiles in the “normal” way. However, to the extent that an empirical distribution approximates the normal distribution, z-scores do translate to percentiles in a reliable way.