Newsom
Psy 521/621 Univariate Quantitative Methods, Fall 2020 1
Levels of Measurement and Choosing the Correct Statistical Test
Most textbooks distinguish among nominal, ordinal, interval, and ratio scales based on a classification
system developed by Stevens (1946). Choice of the statistical analyses in the social sciences typically
rests on a more general or cruder classification of measures into what I will call “continuous” and
“discrete.” Continuous refers to a variable with many possible values. By "discrete" I mean few
categories. I, as well as others, often use the terms “dichotomous,” “binary,” “categorical,” or “qualitative”
synonymously with “discrete.” 1 This general characterization of a dependent (response) variable as
discrete or continuous relates to two general classes of commonly employed statistical tests—those
based on the normal distribution and those based on the binomial distribution (or its relatives, the
multinomial and Poisson distributions). Normal theory plays an important role in statistical tests with
continuous dependent variables, such as t-tests, ANOVA, correlation, and regression, and binomial
theory plays an important role in statistical tests with discrete dependent variables, such as chi-square
and logistic regression.2
Ordinal scales with few categories (2,3, or possibly 4) and nominal measures are often classified as
discrete and are analyzed using binomial class of statistical tests, whereas ordinal scales with many
categories (5 or more), interval, and ratio, are usually analyzed with the normal theory class of statistical
tests. Although the distinction is a somewhat fuzzy one, it is often a very useful distinction for choosing
the preferred statistical test, especially when you are starting out.
Type of Dependent
Variable (or Scale)
Level of
Measurement
General Class of
Statistic
Examples of Statistical
Procedures
Discrete
(binary and categorical)
nominal, ordinal
with 2, 3, or 4
levels
Binomial
(as well as multinomial and
Poisson)
chi-square, logistic regression
Continuous
ordinal with more
than 4 categories,
interval, ratio
normal
ANOVA, regression, correlation,
t-tests
Classifying the independent and the dependent variable as continuous or discrete will determine the type
of analyses that are likely to be appropriate in a given situation.
Dependent Variable
Discrete
Continuous
Independent
variable
Discrete
(binary and categorical)
Chi-square
Logistic Regression
Phi
Cramer's V
t-test
ANOVA
Regression
Point-biserial Correlation
Continuous
Logistic Regression
Point-biserial Correlation
Regression
Correlation
Controversies and Common Practice
There is a longstanding debate about how to classify measurements and whether levels of
measurement can be a successful guide to the choice of data analysis type (e.g., Borgatta & Bohrnstedt,
1980; Michell, 1986; Townsend & Ashby, 1984; see Hayes & Embretson, 2012; Scholten & Borsboom,
2009 for recent discussions of the controversy). 3 In my view, there are two relevant issues that should
1 Mathematicians will define discrete variables more generally in a way that will include many if not most of the variables that social scientists
view as “continuous” in common practice. For example, Hays (1994) gives “If a random variable can assume only a particular finite or a
countably infinite set of values, it is said to be a discrete random variable.” (p. 98)
2 As we will discover later, the Pearson chi-square test really uses a normal distribution as an approximation, but the binomial (or multinomial)
distribution is central to most statistics used with categorical dependent variables. I have placed chi-square with the binomial theory class of
statistics, therefore, because the normal distribution is really just used as an efficient substitute for the binomial distribution.
3 My intention is not to try to resolve the debate, but to offer a general simple heuristic as a starting place for deciding which type of analysis is
used in common practice in the social sciences for general types of dependent variables. In reality, there are a number of other factors that
