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This tutorial discusses a classification system that is often used to describe the measurement of concepts or variables that are used in social sciences and behavioral research. This classification system categorizes the variables as being measured on either a nominal, ordinal, interval, or ratio scale. After introducing the classification system and providing examples of variables which are typically measured on each type of scale, we note the implications of these measurement scales for the analysis of data. Specifically, we discuss the statistical tests which are most appropriate for data measured on each type of scale. Finally, we will briefly consider some of the limits and criticisms of this classification system.

In the social and behavioral sciences, as in many other areas of science, we typically assign numbers to various attributes of people, objects, or concepts. This process is known as measurement. For example, we can measure the height of a person by assigning the person a number based on the number of inches tall that person is. Or, we can measure the size of a city by assigning the city a number which is equal to the number of residents in that city.

Sometimes the assignment of numbers to concepts we are studying is rather crude, such as when we assign a number to reflect a person's gender (i.e., Male = 0 and Female = 1). This type of measurement is known as a Nominal measurement scale. A Nominal measurement scale is used for variables in which each participant or observation in the study must be placed into one mutually exclusive and exhaustive category. For example, categorizing study participants into "male" and "female" categories demonstrates that 'sex' is measured on a nominal scale. Every observation in the study falls into one, and only one, Nominal category.

With a nominal measurement scale, there is no relative ordering of the categories -- the assignment of numeric scores to each category (Male, Female) is purely arbitrary. The next level of measurement, Ordinal measurement scales, **do** indicate something about the rank-ordering of study participants. For example, if you think of some type of competition or race (swimming, running), it is possible to rank order the finishers from first place to last place. If someone tells you they finished 2nd, you know that one person finished ahead of them, and all other participants finished behind them.

Although ordinal variables provide information concerning the relative position of participants or observations in our research study, ordinal variables do not tell us anything about the absolute magnitude of the difference between 1st and 2nd or between 2nd and 3rd. That is, we know 1st was before 2nd, and 2nd was before 3rd, but we do not know how close 3rd was to 2nd or how close 2nd was to 1st. The 1st place finisher could have been a great deal ahead of the 2nd place finisher, who finished a great deal ahead of the 3rd place finisher; or, the 1st, 2nd, and 3rd place finishers may have all finished very close together. The image below illustrates the ordinal ranking of individuals in a competition. The tick mark to the far right illustrates the person who finished in first place, while the tick mark to the far left represents the person who finished sixth out of six.

The limits of ordinal data are most apparent when one looks at the distance between the third and the fourth place finishers. Although the absolute distance between third and fourth was not that large, the measurement of ordinal data does not indicate this detail.

The next level of measurement, Interval scales, provide us with still more quantitative information. When a variable is measured on an interval scale, the distance between numbers or units on the scale is equal over all levels of the scale. An example of an Interval scale is the Farenheit scale of temperature. In the Farenheit temperature scale, the distance between 20 degrees and 40 degrees is the same as the distance between 75 degrees and 95 degrees.

With Interval scales, there is no absolute zero point. For this reason, it is inappropriate to express Interval level measurements as ratios; it would not be appropriate to say that 60 degrees is twice as hot as 30 degrees. Our final type of measurement scales, Ratio scales, **do** have a fixed zero point. Not only are numbers or units on the scale equal over all levels of the scale, but there is also a meaningful zero point which allows for the interpretation of ratio comparisons. Time is an example of a ratio measurement scale. Not only can we say that difference between three hours and five hours is the same as the difference between eight hours and ten hours (equal intervals), but we can also say that ten hours is twice as long as five hours (a ratio comparison).

Test your understanding of measurement scales in the question and answer panel which follows:

For a helpful hint to remember the four measurement scales, click here.

One of the primary purposes of classifying variables according to their level or scale of measurement is to facilitate the choice of a statistical test used to analyze the data. There are certain statistical analyses which are only meaningful for data which are measured at certain measurement scales. For example, it is generally inappropriate to compute the mean for Nominal variables. Suppose you had 20 subjects, 12 of which were male, and 8 of which were female. If you assigned males a value of '1' and females a value of '2', could you compute the mean sex of subjects in your sample? It is possible to compute a mean value, but how meaningful would that be? How would you interpret a mean sex of 1.4? When you are examining a Nominal variable such as sex, it is more appropriate to compute a statistic such as a percentage (60% of the sample was male).

When a research wishes to examine the relationship or association between two variables, there are also guidelines concerning which statistical tests are appropriate. For example, let's say a University administrator was interested in the relationship between student gender (a Nominal variable) and major field of study (another Nominal variable). In this case, the most appropriate measure of association between gender and major would be a Chi-Square test. Let's say our University administrator was interested in the relationship between undergraduate major and starting salary of students' first job after graduation. In this case, salary is not a Nominal variable; it is a ratio level variable. The appropriate test of association between undergraduate major and salary would be a one-way Analysis of Variance (ANOVA), to see if the mean starting salary is related to undergraduate major.

Finally, suppose we were interested in the relationship between undergraduate grade point average and starting salary. In this case, both grade point average and starting salary are ratio level variables. Now, neither Chi-square nor ANOVA would be appropriate; instead, we would look at the relationship between these two variables using the Pearson correlation coefficient.

As a final comment, we alert you to what is perhaps the most common criticism of the measurement scales discussed in this tutorial. Even though this comment might seem at odds with much of what has been covered in this tutorial, it is an important issue that we must deal with.

In the social and behavioral sciences, much of what we study is measured on what would be classified as an ordinal level. We often ask if people "Strongly Disagree", "Slightly Disagree", or are "Neutral" to a series of statements. We then assign a value of '1' if they Strongly Disagree with a statement, up to a '5' if they Strongly Agree with a statement. To be sure, this type of measurement is ordinal, in the sense that "Strongly Agree" reflects more agreement than "Slightly Agree". This type of measurement is **not** an interval or a ratio level of measurement, because we can not state for certain that the interval between "Strongly Disagree" and "Slightly Disagree" is equivalent to the interval between "Slightly Disagree" and "Neutral". Nor can we say that there is an absolute zero point for level of agreement. However, if we were to rigidly follow the rules of "permissible" analyses for ordinal variables, many of the analyses we conduct in social sciences research would be deemed impermissible. On the other hand, some scientists have conducted computer simulations to try and find out what would happen if we violated certain "rules" of data analysis. They have found that for the most part, it is alright to treat ordinal data (such as variables which have been measured using Strongly Disagree to Strongly Agree response alternatives) as though it were interval level data, and conduct statistical tests that are appropriate for interval level data.

The point of this concluding note is to inform you that while the classification of variables according to their measurement scales is useful to assist you in choosing an analytic procedure, it is not meant to be a substitute for using sound judgment when choosing a statistical analysis. In summary, when choosing a statistical analysis procedure, you should consider the level of measurement of your variables, but you also need to consider the assumptions of the statistical analytic procedure you are considering, and you also need to consider the substantive meaning and the interpretability of the statistics you are computing. There is no substitute for informed, sound judgment when choosing a statistical test for analyzing your data.