Standard Scores

Suppose you took a math test and learned that your score was a 47. How would you know if that was a “good” [Page 940]score? One of the things you might want to know is the range of scores that was possible for the test. Most people would consider a score of 47 on a test with a maximum possible score of 50 better than a 47 on a test with a maximum possible score of 100. A different approach would be to see how your score compared to that of others. For a test on which scores of 0–80 were possible, you might feel better about a score of 47 if you learned that the average score was 42 than if it was 62. Standard scores (sometimes referred to as scaled scores) provide a means of interpreting a score in terms of its distance to the average score and takes into account the overall variability or spread of the set of scores.

More specifically, standard scores indicate the location of a score relative to the mean of all scores in standard deviation units. Mean (M) is the mathematical term for the numerical average found by adding all of the scores (often represented as X) and dividing by the number of scores. It is one of three measures of central tendency used in describing sets of scores. Standard deviation (SD) is ameasure of the spread or variability of a set of scores based on the distance between each score and the mean of the set of scores: The greater the SD, the more spread out the scores are.

Although standard scores take many forms, they are all based on z scores. A z score is a standard score with a mean of 0 and a standard deviation of 1. Z scores are found by subtracting the mean from the score and dividing by the standard deviation:

For example, for a set of scores with a mean of 60 and a standard deviation of 8, the z score for a score of 60 would be 0, and the z score for a score of 76 would be

For this same set of scores, the z score for a score of 48 would be

Note that if the score is greater than the mean, the value of z will be positive. Conversely, if the score is less than the mean, z will be negative. If the score is equal to the mean, z will equal 0. In addition, the greater the difference between the score and the mean, the larger the absolute value of z will be.

Other standard scores can be derived from z scores by changing the value of the mean and standard deviation to some other desired values. This is done on most norm-referenced standardized tests, in which people's scores are interpreted in comparison to those of a norming sample of people believed to be representative of those who will take the test. Transformed or “dressed up” standard scores may be used because few people would want to hear that their (or their child's) score on an intelligence test or other standardized test was 0 or 1, let alone a negative number. For example, some intelligence tests use standard scores for which the mean has been transformed or reset to 100 and the standard deviation has been transformed to 15. Thus, a person whose score was equal to the mean of the norming sample would receive a score of100. Another whose score was 1 standard deviation above the mean would receive a score of 115.

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