Most of us are comfortable describing things with descriptive statistics such as means, percentages and ranges. We also often use numbers to help us explore relationships and figure out why things happen as they do. Measurement is the tool that provides such benefits. Measurement is so pervasive in modern life that we usually don't think about it. But if we think carefully about measurement, we'll be better at using it to meet our objectives. This essay aims to provide structure and stimulate thought about measurement. We'll define measurement and discuss how it operates; outline levels of measurement, which tell what mathematics we can use on our observations; and discuss criteria for evaluating and describing qualities of measurement. What Measurement is and how it operates Measurement is typically defined as assigning numerals to observations according to rules. We should discuss the components of this definition because the words in it are chosen carefully to convey a precise meaning.
Critics of the quantitative social sciences often say that "You can't measure people," and they are right. We don't attempt measure people in some global sense to determine the value of their humanity. Rather, we measure attributes of people. Few would argue that we can't measure attributes of people such as their height and weight and most would concede that we can measure less tangible things like their attitudes, and motivations. In this example, people constitute what we call a "unit of observation," or the thing that possesses the attributes that we want to measure. When attributes that we measure take on different values in different observations, we refer to them as variables. Stated more formally: A variable is an attribute of observations that takes on different values in different observations. Sometimes in casual conversation we define variables as "things that vary or are different," but this is a definition that can get us in trouble. Certainly people are different from each other (thank goodness), but that doesn't make them variables. People are units of observation that possess the attributes or characteristics that can constitute variables. People sometimes confuse variables with values or categories of variables. When looking at a hypothesis such as: "Men are taller than women," it's tempting to say that one of the variables is "men," but that not a proper way to think it. "Men" is a value or category or the variable gender, and women is another category. Probably the problem occurs because the variable name, "gender," does not occur in the hypothesis. But the name of the other variable in the hypothesis, "height," does not occur either and probably nobody would say that one of its values, say 70 inches, is a variable. People are certainly the most common units of observation in social science, but it should be stressed that other things can serve that role in research. For example, newspapers could be units of observation varying with regard to such attributes as their number of readers, format (tabloid or broad-sheet), and editorial position (liberal or conservative). Similarly, time periods like weeks or years could be units of observation varying with regard to such things as the number of deaths that occurred during the period, the state of public opinion on any given topic, or the number of news stories published about abortion. Measurement is essential to statistical hypothesis testing, which is one of the main tools for developing understanding of social phenomena. A hypothesis can be defined as a statement of relationship between at least two variables. [The nature of hypotheses and hypothesis testing are the topics for other essays. The definition offered here is adequate for the discussion at hand.] Statistical hypothesis testing requires that both of the variables named in the hypothesis be measured on the same set of units of observation. As an example of the problems that arise when observations are made on differing units of observation, consider the proposition that televised violence causes aggressive behavior. People sometimes argue that this proposition is confirmed because they can document that there is lots of violence on television and there is lots of aggressive behavior in society. Because the unit of observation shifts, with television programs as the unit of observation for violence, and time period as the unit of observation for aggressive behavior, this evidence does not allow for a rigorous test. While measurements on differing units of observation may be relevant to the proposition and lend it plausibility, they can't be used as a proper statistical test. To test the proposition on television violence and aggressive behavior we might test the hypothesis that the more people watch television, the more likely they are to be aggressive. With this hypothesis, people are the units of observation and they are measured on two variables, amount of television watching and likelihood of being aggressive. In fact, there is a lot of research that has tested that hypothesis. Levels of Measurement Levels of measurement are important because they let us make decisions concerning what statistics are appropriate in our research. This is true because different levels of measurement define the kinds of mathematical operations (things like counting, multiplication, and logarithms) When we measure something, we assign numerals to observations so that different numbers reflect differences in the observations. If the numbers do indeed reflect properties of a thing being observed, we can manipulate the numbers to come up with ways to describe and discover things. We could, of course, conduct our research by examining the units of observation themselves, but that would be awkward at best. If we were examining a sample of people, we could have them stand in order from shortest to tallest, which would give sense of a lot of things about them. By doing this, we would see the range of height and what the typical person looked like. But we'd have a very difficult time communicating what we find out to others. Not only is it much easier to manipulate measurements of our sample of people instead of the people themselves, but also such measurements give us access to a variety of mathematical concepts. And computers will do the manipulation for us. For example, we might measure the variable height with regard to person. Typically we would do this by having each person stand next to a stick marked off in inches and recording the number that appears at the top of their heads. We could then mathematically manipulate the measurements to derive descriptive statistics such as the mode, median, and mean. If we have more than one measurement on each person, we can use mathematics to describe relationships among the measurements. Also if we know our sample is representative of some larger population, we can use probability theory and inferential statistics to test hypotheses. If we were measuring people's heights, it would make no sense to arbitrarily change the numbers we assign. While we could decide to assign the number 7 when the scale read 76 inches and the number 180 when the scale read 58 inches, it would be dumb to do that. The order of the numbers assigned to variables such as weight contains information and allows for certain mathematical operations. This is the property of ordinality. In fact most variables that are based in physical properties such as height, weight, distance and time span, have important mathematical properties. For one thing, the differences between units are consistent with regard to height, for example. Units of measure of height (usually things like inches or meters) are consistent. That is, all inches are equal regardless of whether they are assigned to tall people or to short people. This is called intervality. Another important property of measures like height is that ratios of the numbers assigned reflect properties of the underlying attribute. In other words, it makes sense to say that a person who is 36 inches tall (a child, perhaps), is half as tall as someone who is 72 inches tall. This happens because with such measures, we know where to place the zero point. If we didn't know where the zero point is, we couldn't say that 72 is twice as much as 36. Say we set the zero point ten inches too low so that our measures are 46 and 82 inches. We now see that 82 is not twice as much as 46. In other words, our measures of 46 and 82 don't reflect the ratio of the heights of the people being measured. In contrast, we might measure the variable gender with regard to the unit of observation person. We could do this by assigning 1s to persons who have the characteristic of being female and 2s to persons who have the characteristic of being male. It would make just as much sense to reverse this arrangement by assigning 1s to males and 2s to females. For that matter, it would make as much sense to 0s and 1s, or any other pair of different numbers such as 7s and 9s. When we measure gender in this way, we are using numerals simply as names for different categories of people. These numerals have no particular mathematical meaning. If they did, they would be numbers. To summarize, the best of our measurements contain a lot of information and let us use a full range of mathematical operations. Such measures have the following properties:
Ratio Level Measurements: Measurements that have all of these properties are called ratio level measurements. Most measures of the variables that social scientists borrow from the natural sciences are at the ratio level. These include: time, measured in such units such as years or minutes; distance, measured in feet, meters, etc.; and weight measured in such units as pounds or kilograms. It should be emphasized that level of measurement is a property of the measurement, not of the variable. As we'll see below, we can measure such things as time in ways that don't retain all the information of the ratio level. Of course there are measurements that have nothing to do with the natural sciences that also are at the ratio level of measurement. These include such things as income in such units as dollars or pesos, and grade point averages. Interval Level Measurements: Measurements that have all of the properties of the ratio level except for a known zero point are called interval level measurements. They tell us the order of the units of and the differences between the observations with regard to the attribute being measured. Many of the measures used in the social sciences are considered to be at the interval level. These include scores on pencil-and-paper tests measuring such things as achievement and aptitude. Usually we are willing to assume that these measures order people right and that the differences between them are meaningful. Many of the approaches to question wording used by survey researchers yield data considered to be at the interval level. These include:
Of course, when we use questions like these to measure natural science variables, our measures are at the interval level. For example, we might measure the amount of time spent studying on a Likert scale such as: I spend a lot of time studying. [1] Strongly Agree [2] Agree [3] Neutral [4] Disagree Strongly [5] Disagree Obviously we don't know where the zero point is on such a scale but most researchers are willing to assume that the intervals between the numbers represent equal differences in the amount of time students spend studying. Of course, we might be better off to ask an open ended question like: Tell me during the average week, about how many hours do you spend studying? While this might be harder to code, it would yield a ratio level measure. Ordinal Level Measurements: Measurements that indicate the order of units along a continuum but don't tell how far apart they are or have a known zero point are called ordinal. Examples include such things as ranking of athletic teams in polls and position on top-ten lists. Such things definitely tell us order, but we don't know if the top ranked item is as much better than the second ranked item, as the second is better than the third, and so forth. Some would researchers say that such measurements approaches as Likert scales should be treated as ordinal, but most are willing to assume that they are interval. Because the mathematics associated with the ordinal level of measurement are complex and not easy to program into computers, researchers usually avoid them. This is done by treating measures with five or more categories that have a definite order as interval measures and those with four or fewer categories as nominal. In general this approach is acceptable. Nominal Measurements: The weakest form of measurement simply uses numerals as names for categories. Such measures don't tell us the order of observations, the difference between them, or their magnitude in proportion to each other. Many demographic variables such as gender, race, and national origin can be measured only at the nominal level. (Other demographics such as age and income, of course, can be measured at higher levels.) Nominal measures have less information than the other kinds, but that doesn't mean that they have an unimportant role in research. Such things as counting and calculating percentages are about the only mathematical operations we can use with such measurements, but researchers have developed powerful ways of using these operations to describe things and test hypotheses. Of course, we sometimes like to arrange things in alphabetical order, but that doesn't mean that assigning numbers to observations in that order constitutes ordinal measurement. If you don't understand something in this Web note, please e-mail Dr. Sitton.
Revised 092811 — http://www.uamont.edu/FacultyWeb/sitton/crz/mrea/measure.html |
