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Krus, D. J., & Kennedy, P.H. (1977) Lost: McCall's T scores: why? Educational and Psychological Measurement, 37, 257-261.

Lost: McCall's T scores: Why?

David J. Krus and Patricia H. Kennedy
University of Minnesota

 

Summary. -Description of an algorithm for the area transformations of test scores and comparison of area and linear transformations.

 

In the early twenties, William McCall proposed a new scale to be named in honor of " Thorndike-Terman, or, for brevity, a T”’ (McCall, 1922, p. 299). In a personal letter to the authors, William McCall described the development of the T-scores as follows:

 

"While I was a student of Thorndike, I was led to believe that every trait measured has to be normal. I got suspicious and asked my brother who never heard of the normal curve to make a mark on the ground for the intelligence of every man for miles down the road. The result -- a normal curve! A few days later to entertain my six year old niece, we cut down a good-sized bush and measured the length of the leaves on it. A normal curve, of course. Not at all!  A strong trimodal curve. Being too early to repeat the experiment, I have tried to get some student interested enough to repeat the study to no avail. I'll probably go grieving to the grave without learning whether trimodality is characteristic of leaves on bushes! As you can see, the T score is not wholly free from difficulties. Moreover, the T unit proved to be too technical for U.S. students of education and even more so to Chinese teachers during my days in China, so I found it necessary to invent a simpler unit, called the G score..."

 

An elaboration of Pearson’s (1906) concept of mentaces, the T scale was probably the first articulated description of the area transformation of test scores. As similar to linear transformations, area transformations adjust the mean and standard deviation of the distribution into convenient units. But unlike linear transformations, area transformations into the normal distribution also standardize the third moment about the mean.

 

Advantages of area transformations are obvious. Out of the infinite number of possible empirical distributions of test scores, the normal distribution is most frequently assumed and approximated. It is also most frequently studied, in considerably greater detail than other possible test score distributions. Normalization thus allows the application of knowledge concerning properties of standard normal distribution toward the interpretation of the obtained scores.

 

Despite the advantages of traditionally defined T scores as area transformations to a normal distribution with a mean of 50 and standard deviation of 10, a recent trend toward the redefinition of T scores can be observed. Thus, e.g., Chase (1976, pp. 89-90) has defined T scores as

 

and has commented that “this l0z value we add algebraically into our T-score mean of 50. The result will be a T-score value for the raw score under consideration.” In a similar vein, Wright (1976, pp. 153-154) has defined T scores as

 

and has cautioned that “only if the two distributions are known to have the same mean, standard deviation, and shape can percentiles and standard scores be used for comparing scores between as well as within the distributions.”

 

Since the natural occurrence of score distributions with the same shapes is relatively rare, the utility of thus defined T scores for inter-distributional comparisons would be low indeed. Interestingly enough, it was for this very purpose that T scores were defined in the first place (McCall, 1922, p. 305):

 

“thus the T scale method was developed not only to provide a more satisfactory reference point and unit of measurement, but also to provide a method of combining scoring units which yields a genuine scale”

 

Given the obvious advantages of area transformed T scores, it is hard to understand why they are being avoided. The most likely reason behind these attempts for linear redefinition of T scale is that contrary to McCall’s assurance, the computation of T scores is quite tedious. The detailed description of the whole procedure is well presented elsewhere (McCall, 1939; Guilford, 1965; Magnusson, 1967). Briefly, it consists of computation of cumulative frequencies of test scores and their adjustments with respect to class interval means. The cumulative frequencies for the class interval means are converted to proportions which in turn must be interfaced with proportions of the unit area of the standard normal distribution. Only at this point a linear transformation to a desired mean and standard deviation unit is performed.

 

Area Transformations

 

Skewness and kurtosis describe departures from normality in the distributions of variables. There are several transformations for changing the distributions of variables into distributions closer to normal distribution. These transformations vary with respect they are able to accomplish this goal. One of the most efficient transformations in this respect is the area transformation.

 

Split Point Scores into Adjacent Intervals

Let us describe area transformation of scores on variable X [1 2 3 6 7] shown as black squares. Each score (1.0) is split into two components (.50 +.50) shown as black diamonds. The theoretical reason for this split is that each score of the variable X is only a point estimate of the each score's interval. The first interval stretches from minus infinity to 1.5, the second interval is located between 1.5 and 2.5, the third interval is 2.5 - 3.5, the fourth 3.5 - 5.5, the fourth 5.5 - 6.5, and the fifth interval stretches from 6.5 to plus infinity. 

 

 

The numerical algorithm for the area transformation consists of several steps: In the first numerical line of the diagram below, the scores were split into .50 - .50 parts. In the second numerical line the split scores were reassembled within each score's interval.

 

 

Cumulate the Scores and Convert them into Proportions

In the third numerical line, the scores within each interval were cumulated. In the fourth numerical line, the cumulated scores were converted into proportions by dividing the cumulative frequencies by the total n of cases, for the example, by 5.0.

 

Convert Proportions to Corresponding z Scores

The resulting proportions in the fourth numerical line were then converted to their corresponding z-scores, shown at the bottom of the diagram. Note that proportions equal to 1.00 have to be deleted, since they correspond to infinitely large values.

 

 

For this conversion, we have to use the table of the z scores and their corresponding areas. 

 

 

Area Transformations vs. Linear Transformations

The linearly transformed standard scores were obtained from deviation scores by dividing by the standard deviation. Area transformed standard scores were read from a statistical table, associating standard scores with their corresponding areas under the normal distribution.

 

 

 

Note that the skewness of the original variable X and of the linearly transformed standard scores is .24. The skewness of the area transformed distributions is zero. The area transformation also alters kurtosis, but does not transform platykurtic or leptokurtic distributions into mesokurtic distributions.

 

Area Transformed T Scores 

As a last step, the standard scores were transformed into T scores, this time using the linear transformation

 

where  are area transformed z scores, as shown in the following table.

 

 

Comparison of the Initial Distribution and the Area Transformed T Scores

In the following diagram, top distribution is the initial distribution of the obtained scores; the bottom distribution is the distribution of area transformed T scores. Notice that the normalizing the distribution made the scores of the new distribution evenly spaced. The skewness of the original variable X, as computed in the previous sections was .24. The skewness of the area-transformed distributions is zero. Area transformations also alter kurtosis, but do not change platykurtic or leptokurtic distributions into mesokurtic distributions.

 

 

Discussion

 

The advantage of using the McCall area transformations is substantial. Prior to the area transformation, however, the skewness of the scores to be transformed should be tested for statistical significance. Critical values for the coefficients of skewness and kurtosis can be found in Egon S. Pearson's (1930) tables.

 

If skewness is not statistically significant, the departure of the distribution from normality can be considered due to random factors, and the distribution can be normalized.  In general, area transformations are a better method of the normalization of data than other methods, as, e.g., the square root method, or the often used arc sine transformation. Area transformations can potentially save substantial effort that is often associated with rewriting a test instrument.

 

If the coefficient of skewness is statistically significant, other avenues leading to normality might be explored, such as rewriting test items to compensate either for 'low ceiling' or 'high floor' effects. Area transformations on markedly skewed distributions should not be done, as the may obscure potentially important factors contributing to the distribution's departure from normality.

 

 REFERENCES

Chase, C. I. Elementary statistical procedures. New York: McGraw-Hill, 1976.

Guilford, J. P. Fundamental statistics in psychology and education. New York: McGraw-Hill, 1965.

Magnusson, D. Test theory. Reading, Mass.: Addison-Wesley, 1967.

McCall, W. A. How to measure in education. New York: Macmillan, 1922.

McCall, W. A. Measurement. New York: Macmillan, 1939. Pearson, E. S. A further development of tests of normality. Biometrika, 1930, 22, 239249.

Pearson, E. S. (1930) Statistical significance tables for skewness and kurtosis. Biometrika, 22, 239-249

Pearson, K. On the relationship of intelligence to size and shape of the head, and to other physical and mental characters. Biometrika, 1906, 5, 105146.

Wright, R. L. Understanding statistics. New York: Harcourt, Brace Jovanovich, 1976.