Levels of Measurement
James H. Steiger
Measurement
Measurement is the process of assigning
numbers to quantities. The process is so
familiar that perhaps we often overlook its
fundamental characteristics.
Properties of a Quantity
Quantities that we can measure have a
number of properties. For example, a
quantitity can be discrete or continuous.
Discrete Quantities
A discrete quantity can be placed in 1-1
correspondence with integers. For example,
number of children given birth to, number
of atoms in a bar of soap, number of cars in
your driveway.
Continuous Quantities
Quantities that are continuous can take on
(effectively) infinitely many values over
their range. An example is height or weight.
Height is frequently reported only to the
nearest whole inch. So when a person is
reported as being 71 inches tall, that person
could, for example, have a height of
71.114512312…. inches.
Dangers to Avoid
Attaching unwarranted significance to aspects of
the numbers that do not convey meaningful
information
Failing to simply data when we could easily do so
Manipulating our data in ways that destroy
information
Performing meaningless statistical operations on
the data
Levels of Measurement
Attributes have properties that are similar to
numbers.
When we assign numbers to attributes, we
can do so poorly, in which case the
properties of the numbers to not correspond
to the properties of the attributes.
In such a case, we achieve only a “low level
of measurement
Levels of Measurement
On the other hand, if the properties of our
assigned numbers correspond properly to
those of the assigned attributes, we achieve
a high level of measurement.
A simple example should help clarify the
above.
Properties of Numbers and
Attributes
Nominal (Same-Different). My income is the same
as yours or different.
Ordinal (Ordering). If our incomes are different,
mine is greater or less than yours.
Interval (Relative Differences). The difference
between my income and yours might be, say,
twice as great as the different between my income
and the governor’s.
Ratio (Ratios and Zero Point). My brother’s
income is about 10 times what mine is.
A Simple Example
Six athletes try out for a sprinter’s position
on a local track team.
They all run a 100 meter dash, and are
timed by several coaches each using a
different stopwatch.
A Simple Example (Nominal)
Athlete
True
Time
Nominal
A
10
23
B
11
12
C
13
20
D
20
19
E
13
20
S
0
26
V
A Simple Example (Nominal)
Watch V is virtually useless, but it has
captured a basic property of the running
times. Namely, two values given by the
watch are the same if and only if two actual
times are the same.
Watch V has achieved only a nominal level
of measurement.
A Simple Example (Ordinal)
Athlete
True
Time
V
W
Nominal
Ordinal
A
10
23
11
B
11
12
14
C
13
20
15
D
20
19
18
E
13
20
15
S
0
26
9
A Simple Example (Ordinal)
Besides capturing the same-difference
property, Stopwatch W has the correct
ordering.
We say that Stopwatch W has achieved an
ordinal level of measurement.
A Simple Example (Ordinal)
Athlete
True
Time
V
W
X
Nominal
Ordinal
Ordinal
A
10
23
11
2
B
11
12
14
3
C
13
20
15
4
D
20
19
18
5
E
13
20
15
4
S
0
26
9
1
A Simple Example (Ordinal)
Stopwatch X is also at the ordinal level of
measurement!
What does this tell you?
A Simple Example (Interval)
Athlete
True
Time
V
W
X
Y
Nominal
Ordinal
Ordinal
Interval
A
10
23
11
2
21
B
11
12
14
3
23
C
13
20
15
4
27
D
20
19
18
5
41
E
13
20
15
4
27
S
0
26
9
1
1
A Simple Example (Interval)
The relative spacing (not the absolute spacing) of
the values given by stopwatch Y matches the relative
spacing of the actual times. So the intervals are in
correct proportion.
When the numbers capture same-difference, have
the correct order, and have the correct relative
interval spacing, we say they have achieved an
interval level of measurement.
A Simple Example (Ratio)
Athlete
True
Time
V
W
X
Y
Z
Nominal
Ordinal
Ordinal
Interval
Ratio
A
10
23
11
2
21
20
B
11
12
14
3
23
22
C
13
20
15
4
27
26
D
20
19
18
5
41
40
E
13
20
15
4
27
26
S
0
26
9
1
1
0
A Simple Example (Ratio)
The data produced by stopwatch Y do not capture
ratios correctly. Person D took twice as long as
person A, but the stopwatch did not assign a value
that was twice as large.
The zero point for stopwatch Y is also incorrect. S
took no time at all, but is assigned a time of 1.
Both deficiencies are corrected by stopwatch Z. It
has nominal, ordinal and interval properties, but
also has correct ratios and a correct zero point.
Permissible Transforms
Some of the information in numbers at a
particular level of measurement is valuable,
but, as we have seen, some is arbitrary or
superfluous.
What can we do to a list of numbers without
destroying valuable information?
Permissible Transforms
Each level of measurement has a
permissible transform.
These transforms are hierarchical. If you
perform a transform that is only permissible
at a lower level, you will automatically drop
the level of measurement to that lower
level.
Permissible Transforms (Ordinal)
For ordinal data, any monotonic functional
transform
Y  f (X )
Permissible Transforms (Interval)
For interval data, any linear transform of
the form
Y  aX  b, a  0
Permissible Transforms (Ratio)
For ratio data, any multiplicative transform
of the form
Y  aX , a  0