Scales of measurement

Table 1: Results of a vocabulary test among 10 students

Name	number of correct items	percentage of correct items	rank	mark	pass
John	20	1	1	1	yes
Peter	19	0.95	2	2 +	yes
Ellis	15	0.75	3	3	yes
Sara	14	0.7	4	3	yes
Martin	13	0.65	5	3 -	yes
Matty	12	0.6	6	4	yes
Eve	12	0.6	6	4	yes
Nancy	10	0.5	7	5	no
Adam	9	0.45	8	5	no
Mike	2	0.1	9	6	no

Source: Albert & Koster (2002: 74)

Interval scale

• The number of correct items is displayed on an interval scale.
• An interval scale gives you the following information:
• who’s the best and who’s the worst student in the test (rank order),
• the interval between the individual students.
• Typical interval scale data are: number of years learning a foreign language, words per minute in a reading exercise, number of occurrences of a particular word in a corpus, temperature of water in degrees celsius, ...
• Changing the absolute scores from “raw points” (tokens) to relative percentages doesn’t change the scale of measurement.

Ordinal scale

• Rank is displayed on an ordinal scale: John is better than Peter, Peter is better than Ellis, etc.
• We cannot say by how much Peter is better than Ellis.
• We cannot calculate a mean rank.
• A typical ordinal scale item are grades at school: The interval between “gut (2)” and “befriedigend (3)” needn’t be the same as the one between “sehr gut (1)” and “gut (2)”.

Nominal scale

• Pass or fail is shown on a nominal scale.
• Each result is assigned one category, in this case “yes” and “no”.
• Typical nominal scale data are: gender, native language, type of phrase, ...
• Even though you could encode “pass” as 1 and “fail” as 0, it makes no sense to calculate a mean score.

Valid operations

Interval scale: =, ≠, >, <, +, –, mode, median, arithmetic mean
Ordinal scale: =, ≠, >, <, mode, median
Nominal scale: =, ≠, mode

Transformations between scales

• You can transform a “higher” scale into a lower one, e.g. the interval scaled variable age into four individual age groups: up to 20, 21–40, 41–60, older than 60
• Whenever you do a downward transformation like this you’ll lose information!
• You cannot convert a “lower” scale upwards into a “higher” one!

Exercise 1: Which scale of measurement would you assume for the following variables?

• informant’s gender
• informant’s dialect
• number of occurrences of different types of relative clauses in a corpus
• grading of the same essays by different teachers
• length of headlines in The Sun and The Guardian

Exercise 2: Which scales can you use to describe the following dataset?


Figure 1: Percentage of EU citizens who can have a conversation in a foreign language (Source: Albert & Koster 2002: 76)