What is Measurement

Mapping empirical to numerical relations

=> The mapping function delivers a distinct (homomorph) mapping of the empirical to numerical relations

Thus, each empirical relation corresponds to a numerical relation

4 types of scales

nominal scales

ordinal scales

interval scales

ratio scales

Nominal scale

Simplest kind of ”measuring“

Not really scaling, rather labelling – objects with the same attribute value get the same measurement value (equivalence relation)

each transformation which assigns different numbers to objects with different attributes is allowed

Examples – Category systems(ICD,DSM)

person with anxiety disorder is not better or worse than person with depression

=> disorders are qualitatively different / distinct

Ordinal scale

Scale orders objects along some continuum (order relation)

Possible statements:

two objects are equal or different

larger or smaller relations

Monotone transformations of the measurement values is allowed

order remains intact

1,2,3,4 or 8,9,10,11 => order relation invariant (unverändert)

Example: soccer table

Interval scale

Differences can be interpreted

30–10 equals 80–60

But: Does the difference in ability between grades 'very good' and 'good' equal the difference between 'fair' and 'failure‘?

=> The interval scale property of school grades can be doubted.

You may argue that the numerical intervals are equal. However, it's important to remember that these intervals refer to the numerical relational system, not to the empirical relational system.

In this example, the mapping function is incorrect because the relations of the empirical relative are not mapped properly

There is no predefined measurement unit and no absolute but an arbitrary (!) yet meaningful zero point

Example: Temperature scales

Celsius scale: centered on freezing point of water as zero point

Fahrenheit: Centered on the temperature produced by the mixing of equal weights of snow and common salt

Ratio scale

In addition: true zero point

unit not fixed

ratios can be interpreted

object a is twice as large as object b

Example: physical scales (length, volume, Kelvin scale)

Allows us to compare ratios between numbers

Example,if you measured the time it takes 3 people to run a race, their times may be

10 seconds (Racer A),

15 seconds (Racer B) and

20 seconds (Racer C)

=> You can say that it took Racer C twice as long as Racer A. Unlike the interval scale, the ratio scale has a true/natural zero value

Interval vs. Ratio scale: Often

confused

Temperature example

Interval scale: Are ratios of scale values meaningful?

No, because the zero point is arbitrary (but not meaningless)

Illustration: Is the following statement justified when using a Celsius or Fahrenheit scale?: ”It is twice as hot as yesterday“

Celcius °

First day: 10 degrees Celsius

Second day: 20 degrees Celsius

The second day was twice as hot (20°C / 10°C = 2) as the first day, right?

Fahrenheit

We could also use the corresponding values of the Fahrenheit scale

First day: 50 degrees Fahrenheit

Second day: 68.

It looks like the second day was 68°F / 50°F = 1.36 as hot as the first one, not twice as hot

Which of the following statements is meaningful and which is not? Explain why.

a) ”Studying abroad gave me three times more confidence“

b) ”After studying abroad I had three times more friends than before“

True of false?

Teachers use a percentage scale runing from 1 to 100 to rate the importance of students' class-room behavior.

A colleague argues that, since a percentage scale was used, scale values can be interpreted as units of a ratio scale.

Thus, for example, if teacher A assigned a weight of 70% to ”don't disturb the lesson”, while teacher B assigned only 35%, teacher A perceives this behavior as twice as important as teacher B does (i.e., 0.70/0.35 = 2)

Hint #1: Measurement as assignments of numbers to represent relations between objects.

Hint #2: Does 0% (i.e., number) correspond to zero importance (i.e., empirical property)? What‘s the natural zero point of “perceived importance“?

Yet another confusion: Continuous scale = interval scale

What is completely obscured here

Using real numbers (as seen in previous examples) pertains to the numerical relational system

However, nothing is necessarily implied regarding the empirical relational system, which concerns relationships between properties of sets of objects (such as perceived pain, attitude statements, intelligence, etc.).

Examples

response time

Factor scores (i.e., z-scores) from factor analysis (real number line)

Extraversion scale, expressed as z-scores; participants A, B, and C

A: z-score = 0

B: z-score = 1

C: z-score = 2

=> Difference between A and C is twice the difference between B and C

=> However, does it follow that the difference in extraversion (i.e., empirical relational system) between A and C is twice the difference between B and C?

By merely assigning these numbers to objects, we have not at all established a mapping of the empirical to numerical relations

We just assume relations between these numbers to correspond to relations between degrees of extraversion

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