T Scores

T scores are area-normalized scores standardized to have a mean of 50 and a standard deviation of 10. They are like normally distributed z scores except that they have been subjected to a linear transformation that changes the mean of the distribution from zero to 50 and changes the standard deviation from1.0 to 10. As a result, T scores are always positive, usually ranging from about 20 to 80, although more extreme values are possible. Ordinarily, T scores are reported without a decimal point; that is, they are rounded to whole numbers (48.4 becomes 48, 48.6 becomes 49).

T scores were first described by McCall and named in honor of three famous psychologists of the era whose last names began with T. Since that time, T scores have provided the metric for a number of psychological and educational tests. Although McCall defined T scores as being normalized, thereby giving them many of the advantages of normal-curve z scores without some of the disadvantages, subsequent users have often failed to incorporate the normalization step and have simply transformed ordinary z scores into a scale with a mean of 50 and a standard deviation (SD) of 10 and called them T scores. Thus, when T scores are mentioned as a metric for a test, it is important to determine whether the normalization step has been taken.

Properly constructed T scores cannot be computed directly from the raw scores unless the raw scores are themselves normally distributed. This is because any linear transformation, such as computing z scores, does not change the shape of the original distribution. If the raw score distribution is positively skewed, the z scores and any transformations of those z scores will also be positively skewed to the same degree.

The determination of area-normalized scores requires the use of a table of the normal distribution such as is found in almost any statistics book and some measurement texts, and it always involves as a first step the determination of normalized z scores. The process is as follows: Make a frequency distribution and cumulative frequency distribution of the raw scores.

Determine the percentile rank of each raw score. The formula is

where

• PRj is the percentile rank of raw score j,
• cfj−1 is the cumulative frequency for the next lower raw score,
• .5 fj is half of the frequency for raw score j,
• N is the total number of cases.

Once the percentile ranks have been determined, look up the z score that has the corresponding percentile rank.

• a.
  For percentile ranks less than 50, this will be the negative z score, which has an area beyond itself equal to the percentile rank. For example, if the percentile rank of a raw score is 16, the z score that has 16% of the distribution below it is −1.00.
• b.
  For percentile ranks greater than 50, subtract 50 from the PR and look up the z score that has the remaining percentage between itself and the mean. For example, if the percentile rank of a raw score is 84, we would look up the z score that has 34% of the distribution between itself and the mean. This z score is + 1.00. (Occasionally, you may find a table of the cumulative normal distribution in which the relationship between percentile ranks and z scores can be read directly.)
• c.

  You now have a set of z scores that has been forced into a normal distribution by being assigned values corresponding to the percentile ranks of the normal distribution. To obtain T scores, multiply each z score by 10, then add 50. That

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