and the arithmetic mean in the above
formula is defined as
Expanding the deviation
scores x, 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.
Mean of Standard Scores
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.
Variance of Standard Scores
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
Examples
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 in the
standard score form.
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.
Transform
Obtained Scores to Standard Scores
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
Comparability and Translations
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.
Logic behind the
Translations
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.
Reversal of Transformations
Subtract/Add
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
Divide/Multiply
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
Combine
Two Operations
The translation of standard scores back
to 'obtained' scores can be done directly by combining the
two operations above into one, as
New Distribution with a
Selected Mean and Standard Deviation
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.
Reason for
the Translations
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.
Examples
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
Obtained
Scores Level
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.
Deviation
Scores Level
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.
Standard
Scores Level
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.
Data
Visualization: Linear Relationship
Graph the relationship between the original data X and the
linear transformed data z.
New Means
and Standard Deviations
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.
T Scores
(L Scores)
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 (or L 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.
GRE
or GMAT
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.
Elevation,
Scatter, and Shape
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.
Summary
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.
