Standard Scores

A z Scoreis an actual or raw score converted intostandard deviation (SD) units -- hence the term standard score -- sothat it can be placed on the normal distribution curve. A z score indicates how much ascore deviates from the mean of the distribution. Simply knowing a z scoreoffers no information about the raw score, but it indicates how well a persondid compared to other test-takers in the norm group.

The units of a z score are from -3 SD to +3 SD, and 0 equals the mean.Therefore, positive z scores exceed the mean, while negative z scores are lessthan the mean. Knowing the simple formula for calculating a z score provides abetter understanding of its relationship to the distribution: z score =(raw score - mean)/SD.

For example, if a test has a mean of 35 and a SD of 5, a score of 20 hasa z score of -3 ((20 - 35) / 5 = -3) or 3 SD below the mean.

z Scores are beneficial to educators because they allow comparisons to bemade between tests with different distribution characteristics, i.e. mean andSD. For example, if a different version of the test described above was given,and the mean dropped from 35 to 30, and the SD increased from 5 to 10, then ascore of 20 on this version of the test has a z score of -1 ((20 - 30) / 10 =-1)) or 1 SD less than the mean. Therefore, a score of 20 on this version ofthe test is relatively better than it was on the earlier version.

There are standard scores other than the z score. As evidenced above, zscores are often negative and may contain decimal places. To eliminate thesecharacteristics, z scores often are converted to T scores. This isaccomplished using the simple formula: T score = 10(z score) + 50. Forexample, a z score of -2.5 becomes a T score of 25. Therefore, the T scorescale from 20 (-3 SD) to 80 (+3 SD) has a mean of 50 and a SD (standarddeviation) of 10. The T score has two major advantages: it is always a wholenumber, and it is never a negative number. The range of T scores is from 1 to100 with a mean of 50.

The Standard Aptitude Test (SAT) Example: The distribution of rawscores, and thus the mean and the standard deviation, on the SAT vary from yearto year depending on the difficulty of the test and the ability of theexaminees. Also, there are several editions of the test in use at any one time. In order for the scores to be comparable between tests, the raw scores areconverted to a standard score scale.

The Standard Aptitude Test (SAT) is scored on a standard scale very similarto the T score scale mentioned above, except the scale is from 200 (-3 SD) to800 (+3 SD). The mean always equals 500 and the SD always equals 100,regardless of the test edition.

If one edition of the SAT as described above has a raw mean score of 650 and aSD of 50, then a raw score of 700 is 1 SD above the mean (650 + 50 = 700). Therefore, the raw score of 700 would be converted to a standard score of 600. A standard score of 600 has the same meaning from test to test. It is always 1SD above the mean.

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