Transformations

Deviation and Standard Scores  With the variable X [1 2 3 4 5] on the vector display, you may experiment with its transformations under the Transformations menu. Clicking on the Deviations from the Mean command, variable X can be transformed into deviation scores x [-2 -1 0 1 2] with the mean of zero. Clicking on the Standard Z Scores command, variable X can be transformed into standard scores z [-1.414   -.707   0   .707   1.414] with the mean of zero and variance (and standard deviation) equal to one. By squaring the standard scores (Operations, Exponentiate Variables) and observing the mean of the squared standard scores, you can demonstrate that the variance (and standard deviation) of the standard scores is one.

 Entering different variables, you can demonstrate that this property is independent of any particular data.

Deviations from the Mean and from the Median  Click on the Univariate under the Data menu and enter a skewed variable where the mean and median (C, centralwerth) differ. Click on the Descriptive Statistics menu and select the Sum, the Median, and the Mean. On the Transformations menu select the Deviations from the Mean and the Deviations from the Median. Compute absolute values (Functions, Absolute Values) for these two types of the deviation scores. For clarity, click on the variable names and rename the variables X, |x|, and |c|.

You may observe the demonstration of the theoretical postulate that when the mean and the median differ, the sum of the absolute distances from the median (14.0) is smaller than the sum of the absolute distances from the mean (15.6).

Linear vs. Area Transformations Comparison of linearly transformed standard scores (Transformations, Standard z Scores) with the area transformed standard scores (Transformations, Normalized z Scores)

shows, that while the means and variances of both the linearly and area transformed skewed scores equal to zero and one, only the area transformations changes the skewness of a skewed variable to zero.

Linear Transformations  Test scores can be linearly transformed to scores with a "desirable" mean and standard deviation. Examples of such scores are L-Scores with the mean of 50 and the standard deviation 10, C-Scores with the mean of 100 and the standard deviation 15, or D-Scores with the mean of 500 and the standard deviation 100. Select (Structures, Random)

and click the Uniformly Distributed Integer Numbers command.

Select Deck of Cards, specify the number of rows and columns of the random data to generate (for the example 100,1), and click the Accept command to generate a random variable with the mean 26.23 and the standard deviation 14.869. Select (Transformations, Linear Transformations) to linearly transfer the generated random variable to a variable with the mean of 50 and the standard deviation of 10

and click the Append command. Rename the variables X and L.

Experiment with translations of the variable X to variables with the mean and standard deviation of your choice.

Demonstration that the maximum value of the standard score z is the square root of n-1

Entering numbers with one an extremely high number illustrates the fact that in the case of real data, the maximum value of z is not infinity, but the square root of n-1, i.e., 2.00 for the example.

Also, notice that if the limit of the non-extreme values is negligible or the non-extreme values equal to zero, the standard deviation of the distribution equals the square root of the n-1 (or about the square root of the n for the large samples, or the square root of the n for the infinitely large samples of the theoretical model).

These observations were used when we solved the problem of the indeterminate proportions within the Thurston's pair comparisons method. (In Krus, D. J. & Kennedy, P. H. (1977) Normal scaling of dominance matrices: The domain-referenced model. Educational and Psychological Measurement, 37, 189-193.)