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Question 708:



No answer provided yet.The z-score as stated will describe how a data-point lies in relation to the population mean based on the difference and standard deviation. The t-score, often applied in psychometrics (personality tests etc) converts the z-score to a positive number (since a z-score can take on any value from -infinity to +infinity).

The t-score distribution has a mean of 50 and standard deviation of 10 versus the z-score which is based on the standard normal distribution, has a mean of 0 and standard deviation of 1.

The t-score formula is t = 10(z)+50, where z is the z-score described above. For example, using a z-score of -.7386 (the undesirable negative value) we get t = 10(-0.7386) +50 = 42.614 (the desirable positive value).

With a z-score one can also lookup the value using a standard normal table to find the percent of area covered (e.g. a z-score of 1 covers 84% of the area under the normal curve). This percent can then be used as a percentile ranking, which take on a more familiar 0 to 100 boundary. It is important to note though that the relationship is not linear--these percentiles are taken from squeezing area under a curve.  The procedure involves the same steps as the z-score = (rawscore-mean)/standard deviation, just looking up the area from a normal table or a z-score to percentile calculator.

The term Norm-referenced scores means percentile scores are in reference to a large set of students or test takers who have established a baseline for the test. For example, with the SAT standardized test, every few years the gather a large sample of users to establish how difficult or easy questions are in generating the overall SAT score. The average score for this group is set ( or normed) to zero. When someone later then gets a raw score of 600 on the verbal portion of the test and this is in the 90th percentile it means they scored better than 90 percent of the baseline (normed) set of students.

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