In talking about variables, sometimes you hear variables being described as categorical(or sometimes nominal), or ordinal, or interval. Below we will define theseterms and explain why they are important.
Categorical or nominal
A categorical variable (sometimes called a nominal variable) is one that has two ormore categories, but there is no intrinsic ordering to the categories. For example,a binary variable (such as yes/no question) is a categorical variable having two categories (yes or no) and there is nointrinsic ordering to the categories. Hair color is also a categorical variablehaving a number of categories (blonde, brown, brunette, red, etc.) and again, there is noagreed way to order these from highest to lowest. A purely nominal variable isone that simply allows you to assign categories but you cannot clearly order thecategories. If the variable has a clear ordering, then that variable would be anordinal variable, as described below.
Ordinal
An ordinal variable is similar to a categorical variable. The difference betweenthe two is that there is a clear ordering of the categories. For example, suppose youhave a variable, economic status, with three categories (low, medium and high). Inaddition to being able to classify people into these three categories, you can order thecategories as low, medium and high. Now consider a variable like educational experience(with values such as elementary school graduate, high school graduate, some college andcollege graduate). These also can be ordered as elementary school, high school, some college,and college graduate. Even though we can order these from lowest to highest, thespacing between the values may not be the same across the levels of the variables.Say we assign scores 1, 2, 3 and 4 to these four levels of educational experience and wecompare the difference in education between categories one and two with the difference ineducational experience between categories two and three, or the difference betweencategories three and four. The difference between categories one and two (elementary andhigh school) is probably much bigger than the difference between categories two and three(high school and some college). In this example, we can order the people in level ofeducational experience but the size of the difference between categories is inconsistent(because the spacing between categories one and two is bigger than categories two andthree). If these categories were equally spaced, then the variable would be aninterval variable.
Interval (also called numerical)
An interval variable is similar to an ordinal variable, except that the intervalsbetween the values of the numerical variable are equally spaced. For example, supposeyou have a variable such as annual income that is measured in dollars, and we have threepeople who make \$10,000, \$15,000 and \$20,000. The second person makes \$5,000 more than thefirst person and \$5,000 less than the third person, and the size of these intervalsis the same. If there were two other people who make \$90,000 and \$95,000, the sizeof that interval between these two people is also the same (\$5,000).
Why does it matter whether a variable is categorical, ordinal or interval?
Statistical computations and analyses assume that the variables have a specific levelsof measurement. For example, it would not make sense to compute an average haircolor. An average of a nominal variable does not make much sense because thereis no intrinsic ordering of the levels of the categories. Moreover, if you tried tocompute the average of educational experience as defined in the ordinal section above, youwould also obtain a nonsensical result. Because the spacing between the four levelsof educational experience is very uneven, the meaning of this average would be veryquestionable. In short, an average requires a variable to be numerical.Sometimes you have variables that are “in between” ordinal and numerical, forexample, a five-point Likert scale with values “strongly agree”,“agree”, “neutral”, “disagree” and “stronglydisagree”. If we cannot be sure that the intervals between each of these fivevalues are the same, then we would not be able to say that this is an interval variable,but we would say that it is an ordinal variable. However, in order to be able to usestatistics that assume the variable is numerical, we will assume that the intervals areequally spaced.
Does it matter if my dependent variable is normally distributed?
When you are doing a t-test or ANOVA, the assumption is that the distribution of thesample means are normally distributed. One way to guarantee this is for thedistribution of the individual observations from the sample to be normal. However,even if the distribution of the individual observations is not normal, the distribution ofthe sample means will be normally distributed if your sample size is about 30 orlarger. This is due to the “central limit theorem” that shows that evenwhen a population is non-normally distributed, the distribution of the “samplemeans” will be normally distributed when the sample size is 30 or more, for examplesee Central limit theorem demonstration .
If you are doing a regression analysis, then the assumption is that your residuals arenormally distributed. One way to make it very likely to have normal residuals is tohave a dependent variable that is normally distributed and predictors that are allnormally distributed; however, this is not necessary for your residuals to be normallydistributed. You can see the following resources for more information:
- Regression with Stata: Chapter 2 – Regression Diagnostics
- Regression with SAS: Chapter 2 -Regression Diagnostics
- Introduction to Regression with SPSS: Lesson 2 – Regression Diagnostics
