Data sets consist of variables, which can be conceptualized as containing a core enveloped by two layers. The outermost layer is characterized by the arithmetic mean, associated with the obtained scores. Once known, the mean can be removed from the obtained scores by subtraction. By removing the outermost layer, we obtain the deviation scores. Since the arithmetic average was peeled from the obtained data, the mean of the deviation scores is always zero. The statistics most closely associated with the second layer, the deviation scores, is variance. Once computed, it can be removed by dividing deviation scores by the standard deviation, thus removing the second layer. What remains are the standard scores or z-scores, the core.
Computation of Standard Scores
This progression from obtained scores to z-scores is illustrated by the answers to the question I like poetry. First, the mean (3.00) is calculated and subtracted from each of the obtained score. The resulting deviation scores are squared, summed, averaged, and the standard deviation is computed by taking the square root of the variance (2). The deviation scores are then divided by the standard deviation (1.41) to obtain standard scores. These steps are shown as
The formula expressing the operations necessary to obtain z scores is
The denominator of the z-score formula is defined as
The numerator can be obtained from the raw scores as
and the arithmetic mean in the above formula is defined as
Expanding the deviation scores, the z scores are also often written as
The above formula defines standard
scores by using the obtained scores and is equivalent to the definitional
formula expressing standard scores 
as deviation scores divided by their standard
deviation.
Mean and Variance of Standard Scores
The mean and variance of standard scores are computed by formulae, which while preserving their generic structural form substitute the standard scores in lieu of the obtained or deviation scores, as appropriate. The mean of the standard scores thus can be computed as
which always equals zero.
This is demonstrated by substituting the right side of the equivalence to the above equation
and by taking into the consideration the previous proof that the sum of deviation scores always equals zero. Since the mean of standard scores is always zero, subtracting the mean from a set of standard scores will leave them unchanged. Standard scores thus do not have to be transformed before their variance is computed; they already have the properties of deviation scores. Standard scores can be directly squared, summed, and divided by n to compute their variance by using the formula
The above formula is sometimes also written as
It can be formally shown that
the variance of a set of standard scores will always be one. Substituting the
right side of the expression into the above formula results in
Since
then
and
Let us use the z scores associated with the variable X = [1 2 3 4 5] to illustrate the above theoretical findings.
Suppose that we do not know the discussed scores are standard scores. Treating them as any other set of obtained scores, their mean is computed and found to be zero.
Subtracting zero from the obtained scores leaves them unchanged. From this, we can conclude that the 'obtained' scores are either deviation scores or standard scores. When computing the variance by directly squaring the numbers in the first column, we find that the variance equals one. Thus, the set of scores above must be standard. The verification that the mean of a set of scores equals zero and its variance (and standard deviation) equals one is one method to ascertain whether a set of scores is in the standard score form.
Meeting at Café Apollinaire
Peter Weir's motion picture Dead Poets Society based upon a script written by Tom Schulman, takes place in a New England preparatory school during the mid 1950's. Filled with passion, it left lasting impression on many who had seen it. Motivated by Weir's motion picture, our friends Allen, Becky, Cathy, Debra, and Edgar decided to found their own Dead Poets Society. Their meetings take place in a nearby city at the Café Apollinaire. Debra recently returned from Europe. She studied at Sorbonne where she took courses on Baudelaire, Sartre and Ionesco. She also bought a car that had odometer calibrated in both miles and kilometers.
During the meeting, Debra told her friends about the book Bon Jour Tristesse by Françoise Sagan. Debra especially liked the passage when the heroine, noticing friends' tattoo remarks 'something that lasts in this world of impermanence.' That leads her to ponder the substantial and superfluous, immanent and extrinsic, essential and redundant. To illustrate points she makes in rapid succession, she drew on the paper napkin five arbitrary distances measured in miles and in kilometers and translated them from obtained to standard scores.
After liberating distances from arbitrary units imposed upon them by France and England, she commented on the stark beauty of the absolute distances, their parallels of values, simplicities of means, and equivalencies of variances and standard deviations.
Comparability of Standard Scores
The desirable properties of
standard scores with their mean equal to zero and both its variance and the
standard deviation equal to one facilitate their comparability. They also
permit the translations of standard scores into other score distributions with
different means and standard deviations. The means and variances of these new
distributions of scores can be selected at will. The logic behind these
translations is that since the means and variances of the obtained test scores
have been removed by linear transformations, they can be brought back by a
reversal of these transformations. Alterations of means and variances during a
translation back to new 'obtained' scores will not have any effect on the shape
of these scores. Only the scatter and elevation will be changed. Often, it is
advantageous to change the mean and variance into some convenient numbers; by
convenient we mean numbers such as 10, 50, 100, and possibly others.
Since the obtained scores are
transformed to deviation scores by subtracting the mean,
they can be transformed from
the deviation scores back to 'obtained' scores by adding the mean to each
deviation score, as
By the same token if the
standard scores are obtained by dividing the deviation scores by their standard
deviation as
then they can be transformed
back into deviation scores by multiplying the deviation scores by a standard
deviation
The translation of standard
scores back to 'obtained' scores can be done directly by combining the two
operations above into one, as
As demonstrated previously,
the means and standard deviations of standard scores are constants: 0 and 1,
respectively. The means and standard deviations of obtained scores are
incidental to every set of the obtained test scores. Why not translate standard
test scores back to a new distribution of test scores, this time with a mean
and standard deviation defined by some easy to remember and manageable numbers
such as 50 and 10, or, 100 and 15?
The practice of transforming z-scores into new distributions
with conveniently selected means and standard deviations is frequently observed
in the course of developing standardized tests and test batteries. The reason
for translations of standard scores into a new set of scores is, in part,
psychological. About half of the scores when transformed to standard scores
will have negative values. To get a low score on a test is bad enough. Getting
a negative score is more difficult to accept than getting a score that is low.
Translations from distributions of standard scores to new distributions
circumvent the appearance of negative numbers at the lower end of the scale and
facilitate interpretations of results. Suppose you have to interpret
performance on two tests. For the sake of simplicity, consider X to be short
arithmetic test and Y a test of reading ability, illustrated as
You are aware of the fact
that the magnitudes of the obtained scores are dependent on the units of
measurement and thus these tests are not directly comparable. After computing
the means for variables X and Y, subtracting them from each obtained score, you
removed the non-standard means from both variables and obtained thus the
deviation scores. Next, you squared, summed and averaged the deviation scores
to get the variances. Taking the square root of variances you obtained the
standard deviations. Finally, by dividing deviation scores by their standard
deviations, you obtained the standard scores. You checked your computations by
noting that the means of deviation scores equal zero and that their variances
equal those of the obtained scores. You also noticed that the means and
variances of standard scores are equal to zero and one, respectively.
Inspecting the magnitudes of the standard scores on both X and Y variables in
the example, we see no differences; the performance of all five subjects on
both tests was the same.
At the obtained scores level,
the standardized test scores could have been different, but their means and
variances still would be the same. For the purpose of interpretation of the
obtained results, the means and variances of the standard scores are
irrelevant. Would it be possible to transform standard scores to another scores
with new means and standard deviations? For the above example, let us translate
the standard variables into a new distribution, U, with the mean of 10 and
standard deviation equal to 3. Multiplying the standard scores by 3 and adding
10, as shown below, can do this.
These linear transformations
are interesting and perhaps surprising. Translations of
z-scores into scores with mean of 50 and standard
deviation of 10 are called T-scores. Similar translations are used for scores
on many of the admission and qualification tests such as the GRE, GMAT, or
LSAT.
Probably the best-known instrument using the T-score
scale is the Minnesota Multiphasic Personality Inventory (MMPI). Measures
obtained on this instrument are scored with respect to several clinical scales,
e.g., Depression (D) and Schizophrenia (Sc) and T-scores are used to plots
these scales. An example of the plot of the MMPI scales is shown below.
The transformation of obtained scores X into scores
such as GRE or GMAT is as 100 z +500. While the means and standard deviations
of T scores remain constant, the means and standard deviations of most
admission the admission and qualification test scores such as CEEB or GRE
fluctuate from year to year. The norms for these tests were collected in the
past and since the average performance on these tests change over time, so do
the means and variances reported for a particular year.
Caepalic components of
variables
A variable can be conceptualized as having three
caepalic (from L. caepa, onion) components: a core, enveloped by two layers.
The outermost layer is characterized by the arithmetic mean, associated with
the obtained scores. The mean can be removed from the obtained scores by
subtraction. By removing the outermost layer, we obtain the deviation (from the
arithmetic mean) scores. Since the arithmetic mean was peeled off from the
obtained data, the mean of the deviation scores is always zero. The statistics
closely associated with the second layer is variance. It can be removed by
dividing deviation scores by its square root, thus removing the second layer.
What remains is the core; the standard scores with mean equal to zero and
standard variance equal to one, i.e.,
where X signifies the obtained scores, x stands for the
deviation scores
and zx for the standard scores
Another view of linear transformations of obtained
scores into the deviation and standard form conceptualizes the mean as
elevation and the variance as scatter. Thus, it is possible to characterize
obtained scores as containing elevation, scatter, and shape, and deviation
scores as containing scatter and shape. Standard scores contain shape only.
Schematically, this classification is summarized as
The presence of the property is signified by the plus
sign, the minus sign signifies its absence. The linear transformation from
obtained to the deviation scores removes elevation, so only scatter and shape
remains. Transformation from deviation to standard scores removes scatter
(non-standard variance) from scores so that only their shape remains. These
characterizations are the keys to comparability of scores across scales.
The score transformations,
discussed so far, can be written in tabular form as
The means of the obtained
scores can have any value, but the means of deviation and standard scores are
always zero. The variances of the obtained and deviation scores can have any
value, however, the variances of the obtained and deviation scores are the
same. The variances and standard deviations of standard scores are always one.