score of from a distribution with a mean of 0 and a variance, and standard deviation, of
1—denoted N(0,1). Then anything that is true of is also true of , and zand Xare com-
parable variables. (Statisticians sometimes call za pivotal statisticbecause its distribution
does not depend on the values of mand s^2 .)
From Exercise 2.34 we know that subtracting a constant from each score in a set of
scores reduces the mean of the set by that constant. Thus, if we subtract 50 (the mean) from
all the values for X, the new mean will be 50 – 50 5 0. [More generally, the distribution of
(X– m) has a mean of 0 and the (X– m) scores are called deviation scoresbecause they
measure deviations from the mean.] The effect of this transformation is shown in the second
set of values for the abscissa in Figure 3.6. We are halfway there, since we now have the
mean down to 0, although the standard deviation (s) is still 10. We also know from Exercise
2.35 that if we multiply or divide all values of a variable by a constant (e.g., 10), we multi-
ply or divide the standard deviation by that constant. Thus, if we divide all deviation scores
by 10, the standard deviation will now be 10/10 5 1, which is just what we wanted. We will
call this transformed distribution zand define it, on the basis of what we have done, asFor our particular case, where m550 and s 5 10,The third set of values (labeled z) for the abscissa in Figure 3.6 shows the effect of this
transformation. Note that aside from a linear transformation of the numerical values, the
data have not been changed in any way. The distribution has the same shape and the obser-
vations continue to stand in the same relation to each other as they did before the transfor-
mation. It should not come as a great surprise that changing the unit of measurement does
not change the shape of the distribution or the relative standing of observations. Whether
we measure the quantity of alcohol that people consume per week in ounces or in milli-
liters really makes no difference in the relative standing of people. It just changes the nu-
merical values on the abscissa. (The town drunk is still the town drunk, even if now his
liquor is measured in milliliters.) It is important to realize exactly what converting Xto z
has accomplished. A score that used to be 60 is now 1. That is, a score that used to be one
standard deviation (10 points) above the mean remains one standard deviation above the
mean, but now is given a new value of 1. A score of 45, which was 0.5 standard deviation
belowthe mean, now is given the value of 2 0.5, and so on. In other words, a zscore repre-
sents the number of standard deviations that is above or below the mean—a positive z
score being above the mean and a negative zscore being below the mean.
The equation for zis completely general. We can transform any distribution to a distri-
bution of zscoressimply by applying this equation. Keep in mind, however, the point that
was just made. The shapeof the distribution is unaffected by a linear transformation. That
means that if the distribution was not normal before it was transformed, it will not be nor-
mal afterward. Some people believe that they can “normalize” (in the sense of producing a
normal distribution) their data by transforming them to z. It just won’t work.
You can see what happens when you draw random samples from a population that is
normal by going to http://surfstat.anu.edu.au/surfstat-home/surfstat-main.html and clicking
on “Hotlist for Java Applets.” Just click on the histogram, and it will present another his-
togram that you can modify in various ways. By repeatedly clicking “start” without clear-
ing, you can add cases to the sample. It is useful to see how the distribution approaches a
normal distribution as the number of observations increases. (And how nonnormal a distri-
bution with a small sample size can look.)Xiz=X2m
s=
X 250
10
.
z=X2m
s.
zi Xizi72 Chapter 3 The Normal Distribution
pivotal statistic
deviation scores
zscores