Handout
Normalizing Variables
This is a discussion of how to normalize (aka standardize)
variables. The point of normalization is to make variables comparable to each
other. The reason this is a problem is that measurements made using such scales
of measurement as nominal, ordinal, interval and ratio are not unique. For
example, you can measure temperature in both Fahrenheit and Centigrade. Both are
valid, but they produce different numbers. If you want to know whether it is
warmer in Seattle or Paris on a given day, and one is 68 degrees Fahrenheit and
the other is 25 degrees Centigrade, you can't just say 68 is bigger than 25 so
Seattle is warmer. Instead, you need to reduce the measurements to the same
scale, and then compare. Normalization is the process of reducing measurements
to a "neutral" or "standard" scale.
Normalizing is done differently depending on the level of
measurement of the variables, and is intimately related to the uniqueness
properties of the measurement level. See the handout on
measurement theory for more information.
Nominal Scale Variables
 To compare two nominal variables that may be measured
using different scales, you would like to "normalize" the values so you
can see how well they correspond to each other. There is no simple normalization technique
to do this, but it can be done.
 One approach: Construct a contingency table to crossclassify observations of
one variable against the other. Then, if all observations falling into a
given category or the row variable fall into just one category of the
column variable, you can establish a 11 mapping of one measurement to
the other. i.e., in variable A, a "2" corresponds to a "92" means in
variable B
 More sophisticated approach: use a technique called
correspondence analysis (particularly a variant called optimal scaling)
to work out a set of scores that maximize correspondence
Ordinal Scale
Object 
M1 
M2 
M3 
A 
22 
0 
99 
B 
22 
0 
99 
C 
22 
0 
99 
D 
23 
1 
150 
E 
24 
67 
152 
 To normalize an ordinal scale, you convert the values to rank order
values, for example, normalizing each of the scales above would yield:
Object 
M1* 
M2* 
M3* 
A 
1 
1 
1 
B 
1 
1 
1 
C 
1 
1 
1 
D 
2 
2 
2 
E 
3 
3 
3 
 By normalizing variables, you can see whether a set of measured
variables are really measuring the same thing. i.e., you take away
numerical differences that are arbitrary (due to different measurement
properties) and leave only the differences that reflect differences in
the underlying property being measured.
 Note: we tend to use an asterisk after a variable
name to indicate the normalized version of the variable
Interval Scale
 Uniqueness. Interval scales are unique up to a
linear transformation (Y = mX+b). In
other words, if you measure a set of objects on an interval scale, and
then multiply and/or add a constant to each value, the resulting values
are equally as valid as the original values. This is because the ratios of
the intervals between the numbers are not affected by linear
transformations. The following measurements are equally valid:
Object 
M1 
M2 
M3 
M3 
A 
22 
32 
220 
230 
B 
22 
32 
220 
230 
C 
22 
32 
220 
230 
D 
23 
33 
230 
240 
E 
24 
34 
240 
250 
 To normalize an interval scale, you perform a linear transformation
that creates a normalized version of the variable with the property that
the mean is zero and the standard deviation is one. This linear
transformation is called standardizing or reducing to zscores.
Normalizing each of the variables above would yield:
Object 
M1 
M2 
M3 
M4 
A 
.75 
.75 
.75 
.75 
B 
.75 
.75 
.75 
.75 
C 
.75 
.75 
.75 
.75 
D 
0.50 
0.50 
0.50 
0.50 
E 
1.75 
1.75 
1.75 
1.75 
 Note that all the values are the same  this indicates that all
four columns are just linear transformations of each other and
therefore, from an interval scaling point of view, say exactly the same
thing.
 Note all also that the standardized values can be interpreted as
(standard) deviations from the mean. D is just slightly above the mean of all
objects on this variable, while E is quite a bit higher than the mean.
Ratio Scale
 Uniqueness. Ratio scales are unique up to a congruence or
proportionality transformation (Y = mX). In other words, if you measure
a set of objects on a ratio scale, and then multiply each value by a
constant, the resulting values are equally as valid as the original
values. This is because the ratios of the intervals between the numbers
are not affected by congruence transformations. The measurements M1, M2
and M3 are equally valid measures of given object property, but M4 is
not measuring the same thing:
Object 
M1 
M2 
M3 
M4 
A 
22 
220 
11 
12 
B 
22 
220 
11 
12 
C 
22 
220 
11 
12 
D 
23 
230 
11.5 
13 
E 
24 
240 
12 
14 
 To normalize a ratio scale, you perform a particular
"congruence" or
"similarity"
transformation that creates a normalized version of the variable with
the property that the length of the vector is 1 (i.e., the Euclidean or
L2 norm equals 1.0). In other words, to normalize a ratioscaled
variable, we divide each value of the variable by the square root
of the sum of squares of all the original values. Normalizing each of the variables above would
yield:
Object 
M1 
M2 
M3 
M4 
A 
0.44 
0.44 
0.44 
0.43 
B 
0.44 
0.44 
0.44 
0.43 
C 
0.44 
0.44 
0.44 
0.43 
D 
0.45 
0.45 
0.45 
0.46 
E 
0.47 
0.47 
0.47 
0.50 
 Note that all the values except the last column are the same  this
indicates that the first three columns are just rescalings (in a ratio
sense) of each other and therefore, measure exactly the same thing. The
last column is different however, indicating that it measures something
else.
 Note also that other ways of normalizing accomplish the same
goal of making different measurements comparable. So we could just
divide each column by the column sum, creating a new variable whose
values add to 1. This allows interpretation of the rescaled values as
proportions or shares of the whole. This is not the usual way but it
works fine.
Difference Scale (aka Additive Scale)
 Uniqueness. Additive scales are unique up to a "translation"
transformation (Y = X + b). In other words, if you measure a set
of objects on an additive scale, and then add a
constant to each value, the resulting values are equally as valid as the original
values. This is because the intervals between values are not affected by
translation transformations. The measurements M1 and M2 are equally valid measures of given object property, but M3 is
not measuring the same thing:
Object 
M1 
M2 
M3 
A 
22 
12 
11 
B 
22 
12 
11 
C 
22 
12 
11 
D 
23 
13 
11.5 
E 
25 
15 
12 
 To normalize an additive scale, you perform a particular translation transformation that creates a normalized version of the variable with
the property that the mean of the transformed vector is 0. To do this,
we just subtract the mean of the original values. Normalizing each of the variables above would
yield:
Object 
M1 
M2 
M3 
A 
0.8 
0.8 
0.3 
B 
0.8 
0.8 
0.3 
C 
0.8 
0.8 
0.3 
D 
0.2 
0.2 
0.2 
E 
2.2 
2.2 
0.7 
 Note that all the values except the last column are the same  this
indicates that the first two columns are just rescalings of each other and therefore, say exactly the same thing. The
last column is different however, indicating that it measures something
else.
 Note also that other ways of normalizing accomplish the same
goal of making different measurements comparable. So we could just
subtract the column sum from each value
Absolute Scale
 Uniqueness. Absolute scales are unique up to an identity
transformation (Y = X). In other words, they are completely unique and
no (nontrivial) transformation of the numbers is permissible.
 As a result of their uniqueness, no normalization of absolutescaled
variables is needed (nor exists).
