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 would 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

TimeV

Nominal

A 10 23

B 11 12

C 13 20

D 20 19

E 13 20

S 0 26

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

TimeV

NominalW

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

TimeV

NominalW

OrdinalX

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

TimeV

NominalW

OrdinalX

OrdinalY

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

TimeV

NominalW

OrdinalX

OrdinalY

IntervalZ

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

, 0Y aX b a

Permissible Transforms (Ratio)

For ratio data, any multiplicative transform of the form

, 0Y aX a