# Level of Measurement

I am a Math Phd student and a Teaching Assistant for Statistics in Psychology Deparment.

The problem I have is to determine the level of measurement. Let me first present the question:

Assume that a researcher measured self-esteem and obtained the scores 49, 53, 67, 20, 27, 36, 49, 27, 61, and 28. Determine the level of measurement...

I believe it should be an ordinal measurement, since there is no guarantee on how people perceive the scale and hence it is questionable to talk about differences. However, we consider it as an interval scale.

My apologies for a simple question. I really appreciate if you can provide some references for me to read and learn more so that I can guide students better.

Thank you very much.

• You must read Stevens' original paper, On the Theory of Scales of Measurement. Then read F. Lord, On the statistical treatment of football numbers. Finally, study Velleman and Wilkinson, Nominal, Ordinal, Interval, and Ratio Typologies are Misleading. (All can be found freely on the Web--start with Google Scholar.) As a math student you will make short work of this; you will appreciate George Birkhoff's contribution, which apparently few psychologists do :-); and you will get a clearer picture of how misunderstood levels of measurement are.
– whuber
Jan 31, 2018 at 0:10
• I can't resist quoting Velleman and Wilkinson's conclusions: "A careful data analyst should not assume that the scale type of a variable is what it appears to be even when clear assurances are made about the data. ... [P]rograms based on Stevens’s typology suggest that doing statistics is simply a matter of declaring the scale type of data and picking a model. Worse, they assert that the scale type is evident from the data independent of the questions asked of the data. They thus restrict the questions that may be asked of the data. Such restrictions lead to bad data analysis and bad science."
– whuber
Jan 31, 2018 at 0:13
• @whuber Thank you very much, I will start to read references now. Jan 31, 2018 at 1:09

Although there is no guarantee, but most people interpret 80 two times more than 40 if he/she asked to evaluate anything on a 0-100 scale. (Moreover, I assume most people in the US believe 20 °F is two times as warm as 10 °F and most people believe the same with °C, both obviously false.) And they interpret 0 as the total lack of the given feature. So people tends to interpret it automatically as a ratio scale. It does not matter they can't evaluate that feature consequently, they use it as a ratio scale.

But still, it's a good question, it would be even the subject of a nice little psychology study.

• Your argument, if correct, is that this is a ratio scale. So what would twice a self-esteem of 80 be measured as?
– whuber
Jan 30, 2018 at 23:26
• I am just saying people will interpret it as a ratio scale. I'm not saying it's a rational thing to do. And although height is a ratio scale there is no twice as high person as a two meter high person. Jan 30, 2018 at 23:34

I would like to answer this question as both a statistician AND a psychologist. I am assuming this question came from a introductory statistics textbook FOR psychology majors (or some other social science focus).

In this context, the question is not just asking if the variable has been measured on some measurement level, but also the nature of that which is being measured.

Self-esteem is not a concrete quantity that can be measured. And, it is not the case that you have no self-esteem (at least that is not how most psychologists would state it). A person may have low or weak self-esteem, but they never have NO self-esteem. So, from this context, we have to rule out a ratio level measurement.

Clearly, this data would not be nominal. So, it remains to decide if it would be ordinal or interval. Because the data here has been quantified to a rather precise degree, the textbook authors are directing the students toward the interval option for the answer. The differences should be comparable, 10 to 20 should be interpreted as as much of a gain as from 70 to 80...but there is no "fixed" zero indicating nothing of the quantity.

While there always will be some questions about the validity of the measurement instrument being used, the idea of psychometrics is that we can measure along a unidimensional scale, and the observed scores from such instruments as these are reliable proxy scores for those latent (directly unobservable) scores.

My hope is that my answer sheds light into the different epistemological approaches to questions such as this across the two disciplines.