Variables and Measurement
Learning Objectives
• Explain and give examples of an operational definition.
• Explain the four properties of measurement and how they are related to the four scales of measurement.
• Explain the difference between a discrete variable and a continuous variable.
An important step when designing a study is to define the variables in your study. A second important step is to determine the level of measurement of the dependent variable, which will ultimately help to determine which statistics are appropriate for analyzing the data collected.
Operationally Defining Variables
Some variables are fairly easy to define, manipulate, and measure. For example, if a researcher were studying the effects of exercise on blood pressure, she could manipulate the amount of exercise by varying the length of time that individuals exercised or by varying the intensity of the exercise (as by monitoring target heart rates). She could also measure blood pressure periodically during the course of the study; a machine already exists that will take this measure in a consistent and accurate manner. Does this mean that the measure will always be accurate? No. There is always the possibility for measurement error. In other words, the machine may not be functioning properly, or there may be human error contributing to the measurement error.
Now let’s suppose that a researcher wants to study a variable that is not as concrete or easily measured as blood pressure. For example, many people study abstract concepts such as aggression, attraction, depression, hunger, or anxiety. How would we either manipulate or measure any of these variables? My definition of what it means to be hungry may be quite different from yours. If I decided to measure hunger by simply asking participants in an experiment if they were hungry, the measure would not be accurate because each individual may define hunger in a different way. What we need is an operational definition of hunger—a definition of the variable in terms of the operations (activities) the researcher uses to measure or manipulate it.
operational definition A definition of a variable in terms of the operations (activities) a researcher uses to measure or manipulate it.
As this is a somewhat circular definition, let’s reword it in a way that may make more sense. An operational definition specifies the activities of the researcher in measuring and/or manipulating a variable (Kerlinger, 1986). In other words, we might define hunger in terms of specific activities, such as not having eaten for 12 hours. Thus, one operational definition of hunger could be that simple: Hunger occurs when 12 hours have passed with no food intake. Notice how much more concrete this definition is than simply saying hunger is that “gnawing feeling” that you get in your stomach. Specifying hunger in terms of the number of hours without food is an operational definition, whereas defining hunger as that “gnawing feeling” is not an operational definition.
In research, it is necessary to operationally define all variables—those measured (dependent variables) and those manipulated (independent variables). One reason for doing so is to ensure that the variables are measured consistently or manipulated in the same way during the course of the study. Another reason is to help us communicate our ideas to others. For example, what if a researcher said that she measured anxiety in her study? I would need to know how she defined anxiety operationally because it can be defined in many different ways. Thus, it can be measured in many different ways. For example, anxiety could be defined as the number of nervous actions displayed in a 1-hour time period, as a person’s score on a GSR (galvanic skin response) machine, as a person’s heart rate, or as a person’s score on the Taylor Manifest Anxiety Scale. Some measures are better than others—better meaning more consistent and valid. Once I understand how a researcher has defined a variable operationally, I can replicate the study if I desire. I can begin to have a better understanding of the study and whether or not it may have problems. I can also better design my study based on how the variables were operationally defined in other research studies.
Properties of Measurement
In addition to operationally defining independent and dependent variables, you must consider the level of measurement of the dependent variable. There are four levels of measurement, each based on the characteristics or properties of the data. These properties include identity, magnitude, equal unit size, and absolute zero. When a measure has the property of identity, objects that are different receive different scores. For example, if participants in a study had different political affiliations, they would receive different scores. Measurements have the property of magnitude (also called ordinality) when the ordering of the numbers reflects the ordering of the variable. In other words, numbers are assigned in order so that some numbers represent more or less of the variable being measured than others.
identity A property of measurement in which objects that are different receive different scores.
magnitude A property of measurement in which the ordering of numbers reflects the ordering of the variable.
Measurements have an equal unit size when a difference of 1 is the same amount throughout the entire scale. For example, the difference between people who are 64 inches tall and 65 inches tall is the same as the difference between people who are 72 inches tall and 73 inches tall. The difference in each situation (1 inch) is identical. Notice how this differs from the property of magnitude. Were we to simply line up and rank a group of individuals based on their height, the scale would have the properties of identity and magnitude, but not equal unit size. Can you think about why this would be so? We would not actually measure people’s height in inches, but simply order them in terms of how tall they appear, from shortest (the person receiving a score of 1) to tallest (the person receiving the highest score). Thus, our scale would not meet the criteria of equal unit size. In other words, the difference in height between the two people receiving scores of 1 and 2 might not be the same as the difference in height between the two people receiving scores of 3 and 4.
equal unit size A property of measurement in which a difference of 1 means the same amount throughout the entire scale.
Lastly, measures have an absolute zero when assigning a score of 0 indicates an absence of the variable being measured. For example, time spent studying would have the property of absolute zero because a score of 0 on this measure would mean an individual spent no time studying. However, a score of 0 is not always equal to the property of absolute zero. As an example, think about the Fahrenheit temperature scale. That measurement scale has a score of 0 (the thermometer can read 0 degrees), but does that score indicate an absence of temperature? No, it indicates a very cold temperature. Hence, it does not have the property of absolute zero.
absolute zero A property of measurement in which assigning a score of 0 indicates an absence of the variable being measured.
Scales (Levels) of Measurement
As noted previously, the level or scale of measurement depends on the properties of the data. There are four scales of measurement (nominal, ordinal, interval, and ratio), and each of these scales has one or more of the properties described in the previous section. We will discuss the scales in order, from the one with the fewest properties to the one with the most properties—that is, from least to most sophisticated. As we will see in later modules, it is important to establish the scale of measurement of your data in order to determine the appropriate statistical test to use when analyzing the data.
Nominal Scale A nominal scale is one in which objects or individuals are broken into categories that have no numerical properties. Nominal scales have the characteristic of identity but lack the other properties. Variables measured on a nominal scale are often referred to as categorical variables because the measuring scale involves dividing the data into categories. However, the categories carry no numerical weight. Some examples of categorical variables, or data measured on a nominal scale, include ethnicity, gender, and political affiliation.
nominal scale A scale in which objects or individuals are broken into categories that have no numerical properties.
We can assign numerical values to the levels of a nominal variable. For example, for ethnicity, we could label Asian Americans as 1, African Americans as 2, Latin Americans as 3, and so on. However, these scores do not carry any numerical weight; they are simply names for the categories. In other words, the scores are used for identity, but not for magnitude, equal unit size, or absolute value. We cannot order the data and claim that 1s are more than or less than 2s. We cannot analyze these data mathematically. It would not be appropriate, for example, to report that the mean ethnicity was 2.56. We cannot say that there is a true zero where someone would have no ethnicity. We can, however, form frequency distributions based on the data, calculate a mode, and use the chi-square test to analyze data measured on a nominal scale. If you are unfamiliar with these statistical concepts, don’t worry. They will be discussed in later modules.
Ordinal Scale An ordinal scale is one in which objects or individuals are categorized and the categories form a rank order along a continuum. Data measured on an ordinal scale have the properties of identity and magnitude but lack equal unit size and absolute zero. Ordinal data are often referred to as ranked data because the data are ordered from highest to lowest, or biggest to smallest. For example, reporting how students did on an exam based simply on their rank (highest score, second highest, and so on) would be an ordinal scale. This variable would carry identity and magnitude because each individual receives a rank (a number) that carries identity, and beyond simple identity it conveys information about order or magnitude (how many students performed better or worse in the class). However, the ranking score does not have equal unit size (the difference in performance on the exam between the students ranked 1 and 2 is not necessarily the same as the difference between the students ranked 2 and 3), or an absolute zero. We can calculate a mode or a median based on ordinal data; it is less meaningful to calculate a mean. We can also use nonparametric tests such as the Wilcoxon rank-sum test or a Spearman rank-order correlation coefficient (again, these statistical concepts will be explained in later modules).
ordinal scale A scale in which objects or individuals are categorized and the categories form a rank order along a continuum.
interval scale A scale in which the units of measurement (intervals) between the numbers on the scale are all equal in size.
Interval Scale An interval scale is one in which the units of measurement (intervals) between the numbers on the scale are all equal in size. When using an interval scale, the properties of identity, magnitude, and equal unit size are met. For example, the Fahrenheit temperature scale is an interval scale of measurement. A given temperature carries identity (days with different temperatures receive different scores on the scale), magnitude (cooler days receive lower scores and hotter days receive higher scores), and equal unit size (the difference between 50 and 51 degrees is the same as that between 90 and 91 degrees.) However, the Fahrenheit scale does not have an absolute zero. Because of this, we are not able to form ratios based on this scale (for example, 100 degrees is not twice as hot as 50 degrees). Because interval data can be added and subtracted, we can calculate the mean, median, or mode for interval data. We can also use t tests, ANOVAs, or Pearson product-moment correlation coefficients to analyze interval data (once again, these statistics will be discussed in later modules).
Ratio Scale A ratio scale is one in which, in addition to order and equal units of measurement, there is an absolute zero that indicates an absence of the variable being measured. Ratio data have all four properties of measurement—identity, magnitude, equal unit size, and absolute zero. Examples of ratio scales of measurement include weight, time, and height. Each of these scales has identity (individuals who weigh different amounts would receive different scores), magnitude (those who weigh less receive lower scores than those who weigh more), and equal unit size (1 pound is the same weight anywhere along the scale and for any person using the scale). These scales also have an absolute zero, which means a score of 0 reflects an absence of that variable. This also means that ratios can be formed. For example, a weight of 100 pounds is twice as much as a weight of 50 pounds. As with interval data, mathematical computations can be performed on ratio data. This means that the mean, median, and mode can be computed. In addition, as with interval data, t tests, ANOVAs, or the Pearson product-moment correlation can be computed.
ratio scale A scale in which, in addition to order and equal units of measurement, there is an absolute zero that indicates an absence of the variable being measured.
Notice that the same statistics are used for both interval and ratio scales. For this reason, many behavioral scientists simply refer to the category as interval-ratio data and typically do not distinguish between these two types of data. You should be familiar with the differences between interval and ratio data but aware that the same statistics are used with both types of data.
