1. Chapter 2: Levels of Measurement

2. Researchers classify variables according to the extent to which the values of the variable measure the intended characteristics. In the table below, which particular option do you think precisely measure the intended characteristic of age? Clearly option D is the most precise measure of age, but we do see how the variable age can be measured using different levels of precision.

3. The term levels of measurement refers to the precision with which a variable measures the intended empirical characteristic. Each level corresponds to how the measurement can be treated mathematically. Knowing the levels of measurement used in a study helps to determine the kinds of statistical methods that can be applied to the analysis of the data collected, and this also affects the conclusion that can be drawn from the data.

4. The Four Levels of Measurement The four levels of measurement are nominal, ordinal, interval, and ratio. Nominal is at the lowest of the four levels, followed by ordinal, interval, and ratio.

5. Nominal-level Variables A variable is classified as nominal when values or responses of the variable only differ in names, labels, or categories and there is no intrinsic or obvious order between the values or categories of the variable. Examples: gender (male, female), favorite color (purple, blue, red, etc…), religious affiliation (Catholic, Protestant, Judaism, etc…) If numbers are assigned on the nominal scale, they are simply as identifiers or for names only. For example, the numbers on the back of baseball jerseys. This is for convenience and does not measure any quantity possessed by the object.

6. Ordinal-level Variables A variable is ordinal when the values or categories of the variable can be meaningfully or logically listed in order and there is no intention to determine the numerical value of the characteristic being measured. An example is a student’s class standing in high school. They can be a freshman, sophomore, junior, or senior. This would be ordinal because the category names are rank ordered. Other examples include letter grades (A, A-, B+,…, F), political philosophy (very liberal, liberal, moderate, conservative, very conservative), social class, etc…

7. Interval-level Variables The values of an interval-level variables are numeric and more precise than nominal and ordinal. The values are actual numerical amounts of the characteristics being measured. Think of interval-level variables in some sense as a number line. There are equal intervals between values. Each tick mark is an equal distance apart and they increase by the same amount. There is an arbitrary starting point (usually 0).

8. The best example of an interval-level variable is temperature in degrees Celsius. The numbers correspond to levels of mercury in the thermometer and they are equal distance apart (one degree). 0 is an arbitrary value (it does not mean the absence of heat). Another example is time measured in calendar years. The interval between categories is one year. 0 is an arbitrary point (point between BC and AD). Time still existed at time 0.

9. Ratio-level Variables Objects measured on the ratio scale possess all the properties of the interval scale AND, in addition, an absolute zero point. This is the scale generally used in physical sciences, such as distance, weights, heights, age, etc… We can use division to compare the qualities being measured and make a statement about “how many times” one object is greater than or less than another. Example: A student who has completed 30 credits has twice as many credits as a student who has completed 15 credits.

10. Numerical vs Categorical Variables Numerical variables are just as they sound; they’re number. They consist of numbers and can be interpreted as numbers. Interval and ratio variables are numerical variables because they provide a quantity or a dimension. Note: Don’t get confused. Just because something is a number doesn’t automatically make it a numerical variable. For example, if I let 1=male and 2=female, it doesn’t mean that gender is now numerical. It is not. It is still nominal, which is categorical.

11. Categorical variables are variables that record the quality of something. The predetermined non-overlapping categories have been set up ahead of time by the researcher. Examples include gender, blood type, ethnicity, outcome of a plate appearance in baseball, etc… The variables are non-numerical in nature (but can be coded numerically). Each observation must be able to fit into only one class or category. (ex: male or female) Nominal and ordinal variables are categorical.

12. Discrete vs Continuous Variables Numerical variables can be classified as discrete or continuous. Discrete variables take on a finite number of values and are usually countable objects. For example, you can count chairs, people, oranges. Continuous variables deal with characteristics that cannot be counted directly, like age, weight, height, volume, heat, speed, and area. A continuous variable is one whose values are obtained as a result of measuring some characteristic of an object. A continuous variable can take on an infinite number of values. Think about reporting your age up to the second, or your weight with the decimal part.

13. Example: Determine whether each variable is categorical or numerical. If it is categorical, decide whether it is nominal or ordinal. If it is numerical, decide whether it is discrete or continuous. a) Marital status (single, divorced, married, separated, widowed) Categorical, nominal b) Number of courses you are taking this semester (0, 1, 2, 3, 4, 5, or 6) Numerical, discrete (also ratio for fun) c) Age (below 18, 18-19, 20-21, over 21) Categorical (being reported as categories), ordinal