What is Measurement?
 Mapping of (a property of) objects to numbers. If we call this
mapping F, then the measured value of object x is f(x)
 e.g., the height of person A can be written f(A), and the height of
someone twice as tall as A might be written 2*f(A)
 The mapping is like an isomorphism in the sense that certain
relationships among the objects are also mapped to mathematical
relationships among the numbers.
 Since not all relationships among the objects will correspond to
relationships among the measured values, measurement is a kind of model
 there is an implicit theory about what is important and what is not
 Exactly which relationships among objects are preserved as
arithmetic relationships among measured values is what defines different
kinds of measurements, such as the familiar measurement scales of
Stevens (nominal, ordinal, interval, ratio)
Examples of Mapping Relationships
 When we measure weight, two objects that balance each other at each
distance on a balance beam are assigned the same measured value (i.e.,
f(x) = f(y), and if two objects have the same measured value (f(x)=f(y),
they balance each other  mirror image
 If objects X and Y are placed together on the left side of a balance
scale and they perfectly balance an object Z on the right side, then it
will be the case that f(x)+f(y) = f(z)  the combining operation is
mirrored by arithmetic addition
 Suppose an object weighs 5 pounds. Then five of these objects will
weight 5*5 = 25. So multiplication of measured scores by a constant maps
to the physical operation of putting that many items on the scale
 But not all mathematical relationships among measured values have a
counterpart in physical operations. For example, the breakdown of an
object's weight into its prime multipliers doesn't say anything about
the objects themselves. Similarly, if the weight of one object happens
to be the log of the weight of another, this does not imply any special
relationship among the objects
Meaningfulness
 Scales of measurement are distinguished from each other by which
arithmetic operations are meaningful (because they map to physical
properties) and which are not.
 Scales of measurement are independent of the properties they
measure. I can choose to measure mass using a ratio scale (which is what
we usually do) or an interval scale, or an ordinal scale ... etc.
 Meaningfulness also dictates the uniqueness of measurement  what
other sets of measurement could be seen as equally valid
Family of Measurement Scales
 For pedagogical reasons, the following discussion is ordered a
little differently
Nominal Scale Measurement
 Also known as classification and categorical measurement. Some
controversy over whether classification is measurement or something else
entirely
 Only equality preserved. A system of measurement in which only
equality of numbers has meaning. If we measure weight with a nominal
scale, then if f(A) = f(B) we can be sure that A and B weigh the same,
but if f(A) = 12 and f(B) = 13, we can't tell whether B weighs more than
A  there is no "more than" in nominal measurement. There is just same
or not.
 Lack of uniqueness: many alternative measurements equally valid.
Suppose A weighs same as B and C, D weighs twice as much as A, and E
weighs 3 times as much as D. Then any of the following measurements are
equally valid:
Object 
M1 
M2 
M3 
A 
22 
0 
99 
B 
22 
0 
99 
C 
22 
0 
99 
D 
23 
1 
98 
E 
24 
67 
12 
 There is no simple normalization technique with nominal scales that
lets you compare variables measured on different scales. What you have
to do is 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 call into just one category of the
column variable, you can establish a 11 mapping of one measurement to
the other.
Ordinal Scale
 preserves 2 properties: equality and ordinality
 Equality property (like Nominal scaling): If we measure weight with
an ordinal scale, then if f(A) = f(B) we can be sure that A and B weigh
the same, and viceversa
 Ordinality property: If we measure weight with an ordinal
scale, then if f(A) > f(B), we can be sure that A weighs more than B.
But we don't know how much more.
 Uniqueness. Ordinal scales are unique up to a monotone
transformation. A monotone transformation T is one that assigns new
values such that if f(x) > f(y) in the original scale, then T(f(x)) >
T(f(y)) in the newly transformed scale. The following measurements are
equally valid:
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
Interval Scale
 preserves 3 properties: equality, ordinality, and interval ratios
 Equality property (like Nominal scaling): If we measure weight with
an interval scale, then if f(A) = f(B) we can be sure that A and B weigh
the same, and viceversa
 Ordinality property: If we measure weight with an interval
scale, then if f(A) > f(B), we can be sure that A weighs more than B.
 Difference property. If f(A)f(B) = 10, and the difference between B
and C is 20, then we know the difference in mass between B and C is
greater than the difference in mass between A and B
(A......B..........C), and in fact we know that the difference between B
and C is twice as much as between A and C.
 the intervals between measured values of objects have ratio scale
properties
 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 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 
 Ratios of values are not meaningful in interval scales. Consider
asking whether the temperature in one city is twice as hot as in
another. Measure temperature in Fahrenheit. City A is 80 degrees and
City B is 40 degrees. So it looks like A is twice as hot as B. But
suppose instead we
measure temperature in centigrade. City A is 28 deg, and City B is 4
degrees. So now it looks like A is 7 times as hot as B. This contradicts
our previous result of twice as hot. Yet centigrade and fahrenheit are
both valid measurements. In fact, one is just a linear transformation of
the other: F = 9/5C + 32. The contradiction indicates that ratios
of interval measurements are not meaningful.
 It is often said that interval scales lack of zero point. That's
kind of sloppy language. What it means is that the value of zero has no
special meaning in an interval scale  it is just a value one unit
above 1 and two units below +2. The constant "b" in the linear
transformation allows you to slide the scale up and down what is zero in
one scale is another value in a different, equally valid, interval
scaling of the same object property.
 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
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
 Preserves 4 properties: equality, ordinality, interval ratios, and
value ratios
 Equality property (like Nominal scaling): If we measure mass with a
ratio scale, then if f(A) = f(B) we can be sure that A and B weigh the
same, and viceversa
 Ordinality property: If we measure mass with a ratio scale,
then if f(A) > f(B), we can be sure that A weighs more than B.
 Interval ratios property. If f(A)f(B) = 10, and the difference
between B and C is 20, then we know the difference in mass between B and
C is greater than the difference in mass between A and B
(A......B..........C), and in fact we know that the difference between B
and C is twice as much as between A and C.
 Value Ratios property. If we measure mass with a ratio scale, then
if A weighs twice as much as B, then f(A) = 2*f(B) and vice versa. I.e.,
ratios of the measured values correspond to ratios of the actual
properties being measured.
 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 
 Ratio scales are said to have a defined zero point. This is because
the admissible transformations (of the form Y = mX) do not include
adding a constant, so no sliding of the scale up and down is permitted
without changing the meaning of the values.
 To normalize a ratio scale, you perform a particular congruence
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). 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, from say exactly the same thing. The
last column is different however, indicating that it measures something
else.
 Note all 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.
Additive Scale
 Preserves 4 properties: equality, ordinality, interval ratios, and
interval equalities
 Equality property (like Nominal scaling): If we measure mass with a
ratio scale, then if f(A) = f(B) we can be sure that A and B weigh the
same, and viceversa
 Ordinality property: If we measure mass with a ratio scale,
then if f(A) > f(B), we can be sure that A weighs more than B.
 Interval ratios property. If f(A)f(B) = 10, and the difference
between B and C is 20, then we know the difference in mass between B and
C is twice as much as between A and C.
 Interval equalities property. If we measure mass with an additive scale, then
if the difference between A and B is 10 units using one scale, then the
difference is 10 units using any valid scale.
 Uniqueness. Additive scales are unique up to a scale 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 
 Additive scales are said to lack a defined zero point. This is because
the admissible transformations (of the form Y = X +b) effectively allow
sliding the scale up and down without changing the meaning of the values
 it is only the gaps between the values that matter.
 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. 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 three columns are just rescalings (in a ratio
sense) of each other and therefore, from say exactly the same thing. The
last column is different however, indicating that it measures something
else.
 Note all 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.
Absolute Scale
 preserves all properties discussed above.
 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 absolute scales
exists.
