Variables and Levels of Measurement

by

Adam J. McKee

January, 2004

                Statistics is the study of relationships between things.  Rarely are we interested in the relationship between two things that always stay the same.  Most often, we are interested in the relationship between things that change from person to person, from place to place.  Things that never change are referred to as constants.  Water boils at 212 degrees. This temperature is a constant (given that it is pure water at standard pressure).  Sex can change. Some people are male and some are female.  This means that sex is a variable.  The temperature outside is variable because it can and does change.  Statistics generally requires that we be able to assign a number to a variable.  That is we must be able to quantify it. This does not mean that the number expresses any specific magnitude.  Variables are often divided into two categories: Qualitative (categorical) and Quantitative (continuous).  This can lead to some confusion when we assign numbers to qualitative (categorical) data.  Carefully examine the context of the statement to determine what the author/speaker means when using the terms.  Qualitative or categorical data tells us to what group, kind, or type an element belongs. Sex is a categorical variable.  Usually qualitative and quantitative are used to generally describe a study or method, and categorical and continuous are used to describe specific sets of data.  Four other common descriptions are used to describe the level of information a variable gives us.  Nominal gives us the least amount of information while ratio data gives us the most.  Always use the highest level of data measurement possible.  

Nominal Variables

                Numbers on a football jersey do not express any magnitude or size; they only serve to name the player.  These naming variables are referred to as nominal.  If we divide a sample into groups based on sex, we have group 1 and group 2.  The numbers do not suggest any specific magnitude; they just allow us to tell which group is being considered.  This is less important for us than it is for the computer program that is going to do the hard work for us.  Nominal is the lowest level of measurement.

Ordinal Variables

                Ordinal variables have numbers that allow us to put things in order, but do not refer to a standard difference between the elements.  If you tell me that you graduated high school third in your class, I understand what order you come in, but have no idea how much better you were than the person that came in fourth, and how much worse you are than the person that came in second.  If we arrange a list of items from least to greatest, we have rank-ordered it.  Rank ordered data is ordinal. 

Interval Variables

                Interval level measurement is so called because in addition to telling us the rank of an element, it also tells us the magnitude of that difference.  The distance between 4 feet and 5 feet is the same as the difference between 11 feet and 12 feet: one foot.  The critical idea is that a foot is the same magnitude or size every time.  The intervals along points of the scale are always the same. 

Ratio Variables

                Ratio Level variables provide the same information as Interval variables, as well as having an absolute zero to the scale.  That is, when you reach a value of zero, the variable is no longer present.  Calories in various beverages are an example of ratio level data.  We can determine which beverages have more calories than others do, so we can rank them.  This means that we have met the requirements for the ordinal level of measurement.  We know the size of a calorie does not change along the scale, so we meet the requirements of interval level data.  We can also have a beverage, such as water, that does not have any calories, or an absolute zero.    Sentence lengths would be ratio level data because the judge may impose no sentence.  

                Most statistical procedures used in criminal justice research make no real distinction between interval and ratio levels of measurement. Often they are depicted together with a slash mark between them: interval/ratio. 

Dependent and Independent Variables

                These terms are often very confusing to the student because they imply causation.  These terms were first used by scientists conducting true experiments where the Independent Variable (IV) was thought to cause the Dependent Variable (DV).  Causation is determined strictly by experimental design, not statistical mathematics (more on this later).  Even with this caveat, it is often helpful to think of the DV as the effect and the IV as the cause. The DV in a study is the variable that we want to explain or predict.  Thus if we wanted to explain crime rates, crime rate would be the DV in our study.  If we think poverty causes crime, poverty is our IV.

Types of Statistics based on Number of DVs

                Statistics are generally divided into three general categories. If we want to explain one variable, we say that we are using univariate statistical methods.  Univariate methods are generally descriptive, since no comparison of variables is made.  If we want to describe a relationship between two variables (correlations), we say that we are using bivariate statistical techniques.  If we are interested in two or more dependent variables, we use multivariate statistics.    

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