Chapter 4

Measurement

Measurement consists of rules for assigning numbers to attributes of objects based upon rules.

The language of algebra has no meaning in and of itself. The theoretical mathematician deals entirely within the realm of the formal language and is concerned with the structure and relationships within the language. The applied mathematician or statistician, on the other hand, is concerned not only with the language, but the relationship of the symbols in the language to real world objects and events. The concern about the meaning of mathematical symbols (numbers) is a concern about measurement.

By definition any set of rules for assigning numbers to attributes of objects is measurement. Not all measurement systems are equally useful in dealing with the world, however, and it is the function of the scientist to select those that are more useful. The physical and biological scientists generally have well-established, standardized, systems of measurement. A scientist knows, for example, what is meant when a "ghundefelder fish" is described as 10.23 centimeters long and weighing 34.23 grams. The social scientist does not, as a general rule, have such established and recognized systems. A description of an individual as having 23 "units" of need for achievement does not evoke a great deal of recognition from most scientists. For this reason the social scientist, more than the physical or biological scientist, has been concerned about the nature and meaning of measurement systems.

S. S. Stevens (1951) described properties of measurement systems that allowed decisions about the quality or goodness of a measurement technique. A property of a measurement system deals with the extent that the relationships that exist between the attributes of objects in the "real world" are preserved in the numbers that are assigned these objects. For an example of relationships existing in the "real world", if the attribute in question is height, then objects (people) in the "real world" have more or less of the attribute (height) than other objects (people). In a similar manner, numbers have relationships to other numbers. For example 59 is less than 62, 48 equals 48, and 73 is greater than 68. One property of a measurement system that measures height, then, is whether the relationships between heights in the "real world" are preserved in the numbers which are assigned to heights; that is, whether taller individuals are given bigger numbers.

Before describing in detail the properties of measurement systems, a means of symbolizing the preceding situation will be presented. The student need not comprehend the following formalism to understand the issues involved in measurement, but mathematical formalism has a certain "beauty" which some students may appreciate.

Objects in the real world may be represented by O_{i} where "O" is a shorthand notation for "object" and "i" is a subscript referring to which object is being described. The value of "i" may take on any integer number. For example O_{1} is the first object, O_{2} the second, O_{3} the third and so on. The symbol M(O_{i}) will be used to symbolize the number, or measure (M), of any particular object that is assigned to that object by the system of rules; M(O_{1}) being the number assigned to the first object, M(O_{2}) the second, and so on. The expression O_{1} > O_{2} means that the first object has more of something in the "real world" than does the second. The expression M(O_{1}) > M(O_{2}) means that the number assigned to the first object is greater than that assigned to the second.

In mathematical terms measurement is a functional mapping from the set of objects {O_{i}} into the set of real numbers {M(O_{i})}.

The goal of measurement systems is to structure the rules for assigning numbers to objects in such a way that the relationship between the objects is preserved in the numbers assigned to the objects. The different kinds of relationships preserved are called properties of the measurement system.

The property of magnitude exists when an object that has more of the attribute than another object, is given a bigger number by the rule system. This relationship must hold for all objects in the "real world". Mathematically speaking, this property may be described as follows:

The property of *Magnitude* exists

when for all i, j if O_{i} > O_{j}, then M(O_{i}) > M(O_{j}).

The property of intervals is concerned with the relationship of differences between objects. If a measurement system possesses the property of intervals it means that the unit of measurement means the same thing throughout the scale of numbers. That is, an inch is an inch is an inch, no matter were it falls - immediately ahead or a mile down the road.

More precisely, an equal difference between two numbers reflects an equal difference in the "real world" between the objects that were assigned the numbers. In order to define the property of intervals in the mathematical notation, four objects are required: O_{i}, O_{j}, O_{k}, and O_{l} . The difference between objects is represented by the "-" sign; O_{i} - O_{j} refers to the actual "real world" difference between object i and object j, while M(O_{i}) - M(O_{j}) refers to differences between numbers.

The property of *Intervals* exists, for all i, j, k, l

if O_{i}-O_{j}
³
O_{k}- O_{l} then M(O_{i})-M(O_{j}) ³
M(O_{k})-M( O_{l} ).

A corollary to the preceding definition is that if the number assigned to two pairs of objects are equally different, then the pairs of objects must be equally different in the real world. Mathematically it may be stated

If the property of *Intervals *exists if for all I, j, k, l

if M(O_{i})-M(O_{j}) = M(O_{k})-M( O_{l} ) then O_{i}-O_{j} = O_{k}- O_{l} .

This provides the means to test whether a measurement system possesses the interval property, for if two pairs of objects are assigned numbers equally distant on the number scale, then it must be assumed that the objects are equally different in the real world. For example, in order for the first test in a statistics class to possess the interval property, it must be true that two students making scores of 23 and 28 respectively must reflect the same change in knowledge of statistics as two students making scores of 30 and 35.

The property of intervals is critical in terms of the ability to meaningfully use the mathematical operations "+" and "-". To the extent to which the property of intervals is not satisfied, any statistic that is produced by adding or subtracting numbers will be in error.

A measurement system possesses a rational zero if an object that has none of the attribute in question is assigned the number zero by the system of rules. The object does not need to really exist in the "real world", as it is somewhat difficult to visualize a "man with no height". The requirement for a rational zero is this: if objects with none of the attribute did exist would they be given the value zero. Defining O_{0} as the object with none of the attribute in question, the definition of a rational zero becomes:

The property of *Rational Zero* exists if M(O_{0}) = 0.

The property of rational zero is necessary for ratios between numbers to be meaningful. Only in a measurement system with a rational zero would it make sense to argue that a person with a score of 30 has twice as much of the attribute as a person with a score of 15. In many application of statistics this property is not necessary to make meaningful inferences.

In the same article in which he proposed the properties of measurement systems, S. S. Stevens (1951) proposed four scale types. These scale types were *Nominal, Ordinal, Interval,* and *Ratio, *and each possessed different properties of measurement systems. Scale types were originally proposed as a way to classify measurement systems with respect to whether the properties would be preserved when various mathematical operations were used with the numbers that the system produced. For example, if a measurement system possessed the property of magnitude, would it still possess the property if all the numbers it produced were multiplied by three? Would is still posses the property if all the numbers it produced were scrambled? Others used scale types to classify measurement systems with respect to appropriateness for various kinds of statistical analysis. For example, they argued that unless a measurement system possessed the property of intervals, it was not appropriate to do many of the statistical analyses that will be discussed in later chapters. Assigning people numbers by placing a book on their head and observing where it crossed a tape measure placed on the wall could be considered an interval scale if the tape measure started out at a value other than zero. Assume that the dog chewed off the bottom two inches of the ruler, such that everyone was assigned a number two inches bigger than his or her actual height. In this case a person with no height would be assigned the number "2" and the measurement system would not possess the property of a rational zero.

Even though, in the opinion of the author, scale types have limited value as a conceptual framework for understanding measurement systems, they have an important historical value. The reader may encounter these concepts and terms in the research literature

Nominal scales are measurement systems that possess none of the three properties discussed earlier. Nominal scales may be further subdivided into two groups: Renaming and Categorical.

**Nominal-Renaming** occurs when each object in the set is assigned a different number, that is, renamed with a number. Examples of nominal-renaming are social security numbers or numbers on the back of a baseball player. The former is necessary because different individuals have the same name, i.e. Mary Smith, and because computers have an easier time dealing with numbers rather than alpha-numeric characters.

**Nominal-categorical
**occurs when objects are grouped into subgroups and each object within a subgroup is given the same number. The subgroups must be mutually exclusive, that is, an object may not belong to more than one category or subgroup. An example of nominal-categorical measurement is grouping people into categories based upon stated political party preference (Republican, Democrat, or Other,) or upon sex (Male or Female.) In the political party preference system Republicans might be assigned the number "1", Democrats "2", and Others "3", while in the latter females might be assigned the number "1" and males "2".

In general it is meaningless to find means, standard deviations, correlation coefficients, etc., when the data is nominal-categorical. If the mean for a sample based on the above system of political party preferences was 1.89, one would not know whether most respondents were Democrats or whether Republicans and Others were about equally split. This does not mean, however, that such systems of measurement are useless, for in combination with other measures they can provide a great deal of information.

An exception to the rule of not finding statistics based on nominal-categorical scales types is when the data is dichotomous, or has two levels, such as Females = 1 and Males = 2. In this case it is appropriate to both compute and interpret statistics that assume the interval property is met, because the single interval involved satisfies the requirement of the interval property.

Ordinal Scales are measurement systems that possess the property of magnitude, but not the property of intervals. The property of rational zero is not important if the property of intervals is not satisfied. Any time ordering, ranking, or rank ordering is involved, the possibility of an ordinal scale should be examined. As with a nominal scale, computation of most of the statistics described in the rest of the book is not appropriate when the scale type is ordinal. Rank ordering people in a classroom according to height and assigning the shortest person the number "1", the next shortest person the number "2", etc. is an example of an ordinal scale.

Interval scales are measurement systems that possess the properties of magnitude and intervals, but not the property of rational zero. It is appropriate to compute the statistics described in the rest of the book when the scale type is interval.

Ratio scales are measurement systems that possess all three properties: magnitude, intervals, and rational zero. The added power of a rational zero allows ratios of numbers to be meaningfully interpreted; i.e. the ratio of John's height to Mary's height is 1.32, whereas this is not possible with interval scales. A system of measurement that assigned people a number for the attribute of height based on where a book crossed a tape measure placed on the wall could be considered a ratio scale if the tape measure started at zero. In this case, a person with no height would be assigned the number zero and the rational zero property would be satisfied.

It is at this point that most introductory statistics textbooks end the discussion of measurement systems, and in most cases never discuss the topic again. Taking an opposite tack, some books organize the entire text around what is believed to be the appropriate statistical analysis for a particular scale type. The organization of measurement systems into a rigorous scale type classification leads to some considerable difficulties. The remaining portion of this chapter will be used to point out those difficulties and a possible reconceptualization of measurement systems.

The following present a list of different attributes and rules for assigning numbers to objects. Try to classify the different measurement systems into one of the four types of scales before reading any further.

- Your checking account number as a name for your account.
- Your checking account balance as a measure of the amount of money you have in that account.
- Your checking account balance as a measure of your wealth.
- The number you get from a machine (32, 33, ...) as a measure of the time you arrived in line.
- The order in which you were eliminated in a spelling bee as a measure of your spelling ability.
- Your score on the first statistics test as a measure of your knowledge of statistics.
- Your score on an individual intelligence test as a measure of your intelligence.
- The distance around your forehead measured with a tape measure as a measure of your intelligence.
- A response to the statement "Abortion is a woman's right" where "Strongly Disagree" = 1, "Disagree" = 2, "No Opinion" = 3, "Agree" = 4, and "Strongly Agree" = 5, as a measure of attitude toward abortion.

If you encountered difficulty in answering some of the above descriptions, it does not mean that you lack understanding of the scale types. The problem resides in the method used to describe measurement systems; it simply does not work in many applied systems. Michell (1986) presents a recent discussion of the controversy still present in Psychology involving scale types and statistics.

Usually the difficulty begins in deciding the scale type of wealth as measured by bank account balance. Is it not possible that John has less money in the bank than Mary, but John has more wealth? Perhaps John has a pot of gold buried in his back yard, or perhaps he just bought a new car. Therefore the measurement system must be nominal-renaming. But if Mary has $1,000,000 in her account and John has -$10, isn't it more likely that Mary has more wealth? Doesn't knowing a person's bank account balance tell you *something *about their wealth? It just doesn't fit within the system.

Similar types of arguments may be presented with respect to the testing situations. Is it not possible that someone might score higher on a test yet know less or be less intelligent? Of course, maybe they didn't get enough sleep the night before or simply studied the wrong thing. On the other hand, maybe they were lucky; whenever they guessed they got the item correct. Should test scores not be used because they do not meet the requirements of an interval scale?

What about measuring intelligence with a tape measure? Many psychologists would argue that it is interval or ratio measurement, because it involves distance measured with a standard instrument. Not so. If your child were going to be placed in a special education classroom, would you prefer that the decision be made based on the results of a standardized intelligence test, or the distance around his or her forehead? The latter measure is nonsense, or almost entirely error.

Suppose a ruler was constructed in a non-industrialized country in the following manner: peasants were given a stick of wood and sat painting interval markings and numbers on the wood. Would anything measured with this ruler be an interval scale? No, because no matter how good the peasant was at this task, the intervals would simply not be equal.

Suppose one purchases a ruler at a local department store. This ruler has been mass-produced at a factory in the United States. Would anything measured with this ruler be measured on an interval scale? No again, although it may be argued that it would certainly be closer than the painted ruler.

Finally, suppose one purchases a very precise Swiss caliper, measuring to the nearest 1/10,000 of an inch. Would it be possible to measure anything on precisely an interval scale using this instrument. Again the answer is "no", although it is certainly the closest system presented thus far.

Suppose a molecular biochemist wished to measure the size of a molecule. Would the Swiss caliper work? Is it not possible to think of situations where the painted ruler might work? Certainly the ruler made in the United States would be accurate enough to measure a room to decide how much carpet to order. The point is that in reality, unless a system is based on simple counting, an interval scale does not exist. The requirement that all measures be an interval or ratio scales before performing statistical operations makes no sense at all.

Measurement, as a process in which the symbols of the language of algebra are given meaning, is one aspect of the modeling process described in the chapter on models. Remembering the definition of a model as a representation of the "essential structure" of the real world, and not the complete structure, one would not expect that any system of measurement would be perfect. The argument that an interval scale does not exist in reality is not surprising viewed from this perspective.

The critical question is not whether a scale is nominal, ordinal, interval, or ratio, but rather whether it is useful for the purposes at hand. A measurement system may be "better" than another system because it more closely approximates the properties necessary for algebraic manipulation or costs less in terms of money or time, but no measurement system is perfect. In the view of the author, S. S. Stevens has greatly added to the understanding of measurement with the discussion of properties of measurement, but an unquestioning acceptance of scale types has blurred important concepts related to measurement for decades.

A discussion of error in measurement systems is perhaps a more fruitful manner of viewing measurement than scale types.

Different measurement systems exhibit greater or lesser amounts of different types of error. A complete discussion of measurement error remains for future study by the student.

The bottom line with respect to the theory of measurement that is of concern to the introductory statistics student is that certain assumptions are made, but never completely satisfied, with respect to the meaning of numbers. An understanding of these assumptions will allow the student to better evaluate whether a particular system will be useful for the purposes intended.