
ZScores
& TScores
 Overview
The purpose of the statistical techniques discussed
in this chapter and lecture is to convert (transform)
individual scores into a standard form which provide
a more meaningful description of the individual scores
within the distribution.
We discuss two very closely related techniques:
 ZScores: ZScores are a conversion (transformation)
of individual scores into a standard form, where
the transformation is based on knowledge about the
population's mean and standard deviation.
The formula for computing zscores is:
 TScores: TScores (or standardized scores)
are a conversion (transformation) of raw individual
scores into a standard form, where the conversion
is made without knowledge of the population's mean
and standard deviation.
Since we don't know the population's parameters,
we estimate them by using our best guess:
Their corresponding sample statistics. This means
the formula for computing standard scores is:
TScores are not discussed in the book at this
point, but we will need the idea later on.
We discuss these two statistical methods in this
lecture.
If you wish, you can download a
ViSta Applet to further pursue the analyses reported
here (see instructions
for configuring your system to do this).
 ZScores
 Definition:
 ZScores are a transformation of raw scores into
a standard form, where the transformation is based
on knowledge about the population's mean and standard
deviation.
 ZScore Formula:


SAT Math Example
Consider the SATMath variable that we observed
in the survey done on the first day of class.
Sample Statistics, Population Parameters
and Sample Frequency Distribution for SAT Math

Statistics & Parameters 
Sample Frequency Distribution 
Sample Statistics
Samp. Mean = 589.39
Samp. Stand. Dev. = 94.35 

Population Parameters
Pop. Mean = 460
Pop. Stand. Dev. = 100 
Note that red is for males, blue for females.
 Your own SAT Math ZScore
 You can compute your own SAT Math ZScore by using
the following formula:
Q: Why would you want to do this?
A: Because the standardized zscore value
tells you several useful things:
 The sign tells you whether your score
is above or below the population mean.
 The number tells you the number of
standard deviations your score is from (above
or below) the population mean.
 When we get to the next chapter (see page
171) we will be able to easily convert the zscore
into percentiles. (Preview: Z=0 is 50%; Z=1
is 84%; and Z=2 is 97%).
 Here is the frequency histogram for Math ZScores.
Compare this with the original raw score frequency
histogram shown below it. They are almost exactly
the same.

The Point: Transforming raw scores to ZScores
doesn't change their distribution. But we
can now tell that most of you (all but 4) are above
the population mean.
Looking at the summary statistics for the ZScores
we see that: the mean of your scores is above
the large majority of the general population of
scores (since your mean is 1.29 and z=1 is at
the 84% and z=2 is at the 97%  we will see
that z=1.29 is at the 90%).
 Here is another frequency histogram: This
shows exactly the same Math ZScores as are in the
preceeding histogram, but the two histograms look
different.
The Point: Histogram are not a very reliable
(stable) visualization.

SAT Verbal Example
Consider the SATVerbal variable that we observed
in the survey done on the first day of class.
Sample Statistics, Population Parameters
and Sample Frequency Distribution for SAT Verbal

Statistics & Parameters 
Sample Frequency Distribution 
Sample Statistics
Samp. Mean = 551.34
Samp. Stand. Dev. = 64.74 

Population Parameters
Pop. Mean = 430
Pop. Stand. Dev. = 100 
Again, red is for males, blue for females. Note
that there are no males in the lower portion of the
distribution!
 TScores
 Definition:
 TScores are a transformation of raw scores into
a standard form, where the transformation is made
when there is no knowledge of the population's mean
and standard deviation.
The scores are computed by using the sample's
mean and standard deviation, which is our best
estimate of the population's mean and standard
deviation.
 TScore Formula:

 Example:
 We must compute TScores for GPA, rather than
ZScores, because we don't know what the population
mean and standard deviation are for GPA.
This means we must estimate the population parameters
by using the sample's mean and standard deviation.
These are, respectively, 3.06 and .44, as shown
in the summary
report given earlier.
Thus, you can calculate your standardized GPA
TScore as
You can view the histogram for the entire distribution
of GPA TScores. It should look like:
Note once again: There are no males in the lower
range of GPA scores!
 Compare GPA and SAT Verbal scores.
 Finally, we have now converted both GPA and SAT
scores to standard scores, and we can compare them
to each other easily. For example, we can form a
scatterplot of one variable versus the other.
This gives:
We see that there is some relationship between these
two sets of scores: People who have higher Verbal
SAT scores also have higher GPA's, and conversely.
Also, we see that males are absent in the lower
ranges of these two variables.
