Broken Pencils: Properties of Data in Statistical Analysis: Three Levels of Measurement (Nominal, Ordinal and Interval/Ratio)

S.B. asked me to comment on what are the “levels of measurement” I have been referring to in class, and importantly, what they have to do with our statistical forums conducted using SPSS. Good question, grasshopper. Let me try to put it in perspective.

So you plan to conduct a statistical analysis on a dependent variable (Y) and several independent variables (Xs) in your statistical forum due shortly. Where do you start? To be able to determine what statistical test is most appropriate for your task at hand, first you must assign a level of measurement to your study’s dependent variable: nominal, ordinal or interval/ratio. Your choice will depend on the type of variable that’s involved in your analysis. You should keep in mind that not only are variables measured differently but many variables can be measured at more than one level (see “Note” at the end of Pros and Cons to a Univariate Analysis). Although levels of measurement differ in many ways, they have certain similarities as well and can be classified using a few basic principles.

What do I mean by “measurement”? Here are a few orienting points:

Measurement can be defined as the assignment of numbers to a variable according to sets of predetermined rules
The things we observe (gender or race, e.g.) are variables; any particular observation of that variable is an assigned number (e.g., "1" = male)
We view the property of numbers that define the values of a variable

Here are the three fundamentally different ways in which numbers are used in statistical research:

Nominal Level Data: To name or identify
Ordinal Level Data: To represent position in a series or scale
Interval/Ratio: To represent quantity

Data Measured at the Nominal Level

When measuring a variable at the nominal level, the properties of the variables you’re working are categories. A number is then assigned to each category (e.g., for the variable “sex,” 1=male and 2=female). Race, region, and religion are additional examples of the numerous variables measured at the nominal level (sometimes referred to as the nominal scale). The main principle underlying nominal data is that they do not imply any ordering among the responses. Using “Party Affiliation" as an example (see inset chart below), the value of “1” for Republican is no more or less of the property of party affiliation than the value of "2" for Independent or “3” for Democrat. These numeric values are simply categories of the variable “Party Affiliation.” Data measured at the nominal level represent the lowest level of measurement.

Principles of Nominal Data: If the measurement tells only what class a case (e.g., a person) falls into with respect to the variable. Categories of nominal data are mutually exclusive and exhaustive.

Examples of Nominal Data:

1. Are you: __ (1) Male __ (2) Female
2. Are you: __ (1) Protestant __ (2) Catholic __ (3) Jewish __ (4) Muslim __ (5) Other

Data Measured at the Ordinal Level

A researcher who measures job satisfaction might ask employees to specify whether they are 1= "very dissatisfied", 2="somewhat dissatisfied", 3="somewhat satisfied" or 4="very satisfied" with one or more aspects of their work environment. The item responses in this job satisfaction scale are numbered in order, ranging from least to most satisfied with the value of the number indicating that one case (e.g., a person) has more or less of the variable’s property than does another case. This is what distinguishes ordinal from nominal data. There is no such ordering with nominal level data, since the only use of nominal data is to identify a category of the variable (such 1= Male and 2= Female) for analytical purposes.

While more powerful than nominal data, ordinal data does not capture important information that will be present in the next and most powerful level of measurement we’ll examine – interval/ratio. Specifically, the difference between two numbers in an ordinal scale are not assumed to be the same as the difference between any two other numbers in the same scale. In our job satisfaction scale, for example, the difference between the responses 1 (“very dissatisfied") and 2 ("somewhat dissatisfied") cannot be compared to the difference between 2 ("somewhat dissatisfied") and 3 ("somewhat satisfied"). Nothing in our measurement procedure allows us to determine whether the two differences reflect the same difference in job satisfaction. Researchers point out that the differences between ordinal level values do not represent equal appearing intervals.

Principles of Ordinal Data: In addition to the nominal data principles, it also tells us when one case has more or less of the variable’s property than does the other. Ordinal data, therefore, provides a ranked or ordered appearance.

Examples of Ordinal Data:

1. In total, how much income did your family earn before taxes last year?

__ (1) Less than $10,000 __ (2) $10,001 – $20,000 __ (3) $20,001 – $30,000
__ (4) $30,001 – $50,000 __ (5) $50,001 – $80,000 __ (6) $80,001 – $100,000
__ (7) $100,001 or Greater

2. Your highest level of education attained is:

__ (1) H.S. Incomplete __ (2) H.S. Diploma __ (3) Some College
__ (4) College Degree __ (5) Some Graduate School __ (6) M.A./M.S. or Greater

Important Point: We know higher values are more of the property of “income” or “educational attainment” than lower values, but we cannot say that a “4” is twice as much as a “2” or a “6” is twice as much as a “3” in either variable. Only when we use interval/ratio level data can we make such claims.

Data Measured at the Interval/Ratio Level (Two Levels are Combined)

Interval level data are numerical scales (or indexes) in which the intervals have the same quantitative value throughout. As an example, consider the variable “age.” The difference between a 30 year old and a 40 year old represents the same age difference as the difference between an 80 year old and a 90 year old (i.e., 10 years). This is because all 10 year intervals have the same quantitative value or meaning. Likewise, the difference between a 30 year old and a 31 year old represents the same age difference as the difference between an 80 year old and a 81 year old (i.e., one year). This is because all one year intervals have the same quantitative value or meaning.

Interval level data, such as scales and indexes used in the social and behavioral sciences, are far from perfect. They do not have a true zero point even if one of the scaled values happens to carry the name "zero". The Fahrenheit scale illustrates the issue. Zero degrees Fahrenheit do not represent the complete absence of temperature (the absence of any molecular kinetic energy). In reality, the label "zero" is applied to temperature for reasons connected to the history of temperature measurement. Since an interval scale has no true zero point, it does not make sense to compute ratios. It doesn’t make sense to say that 80 degrees is "twice as hot" as 40 degrees does it? The same can be said about the use of attitudinal scale scores in the social and behavioral sciences. Their use also precludes calculating ratios.

Ratio level data does have a true zero point (i.e., an “absolute zero”). That is, true ratio level data is interval level data with the added property that its zero position indicates the absence of the quantity being measured. You can think of ratio level data as the three prior levels of measurement rolled into one. Like nominal data, it provides a name or category for each object where numbers serve as labels. Like ordinal data, the objects are ranked in terms of the order of the numbers. Like an interval scale, the same difference at two places on the scale has the same meaning (equal appearing intervals) and, in addition, the same ratio at two places on the scale also carries the same meaning due to the presence of an absolute zero.

Principles of Interval Data: In addition to the principles of nominal and ordinal data, it also tells us that one case differs by a certain amount of the variable’s property from the others. Interval level data also provides equal appearing intervals

Principles of Ratio Data: ... and it also tells us that one case has so many times as much of the variable’s property as does the other (i.e., it has an absolute zero).

Examples of Interval/Ratio Data:

1. As of your last birthday, how old are you? __ (fill in age)
2. In total, how much income did your family earn before taxes last year? __ (fill in $)
3. In total, how many years of education have you attained? __ (fill in # of years)

In sum, knowledge of the three levels of measurement (nominal, ordinal and interval/ratio) assigned to independent and dependent variables in a study will help you to determine the most appropriate tests of statistical significance to use in your analysis. Of the three, the strongest level of measurement is interval/ratio because of its versatility with quantitative information (see Table inset below). The weakest (but still highly useful) level of measurement is nominal data which provides categorical information and limited arithmetic operations.