Social Studies 201

# January 12, 2004

Measurement of variables – Chapter 3

1. Introduction

Chapter 3 of the text discusses different ways of classifying and measuring variables describing a population.  The different characteristics of a population (the variables) can be organized and measured in various ways.  The considerations associated with different forms of measurement are discussed in section 3.2 of the text.

As an example of how forms of measurement differ, consider three variables – sex, age, and attitude.  The variable sex has only two characteristics, male and female, and there are no numerical values necessarily associated with these.  In contrast, age of an individual is ordinarily measured numerically in number of years, so that a student may be 20 years old and a professor 60 years old.  In the case of attitude scales, the measurement may be numerical on a five-point scale from strongly disagree (1) to strongly agree (5) – but the numbers attached are somewhat arbitrary.

In the above examples, sex is an example of a nominal scale of measurement, or a classification or categorization; attitude is ordinal, or an ordering or ranking of responses; and age has an interval and ratio scale of measurement, with meaningful measures of distances and ratios between values.  Each of these will be explained in the following notes.

These forms of measurements are hierarchical in the sense that some forms of measurement permit a more extensive and complete form of quantitative analysis.  The four major forms of measurement have the following hierarchy, with the ratio scale being the highest or strongest level of measurement and nominal the lowest or weakest type of measurement.

 ratio Strongest level of measurement interval h ordinal nominal Weakest level of measurement

In this hierarchy, all forms of measurement are at least nominal, but some are ordinal, in the sense that values of the variable can be ordered relative to each other.  In addition to being nominal and ordinal, some types of measurement permit meaningful measurment of the distance or interval between values.  Finally, variables that are nominal, ordinal, and interval usually have a ratio level of measurement, permitting meaningful calculation of ratios between values of a variable.

As an example of different ways that a researcher can ask survey questions, consider the following questions about political issues:

Survey question

# Variable

Level of measurement

Which political party do you support?

Political party supported

Nominal only – classification of parties.

How strongly do you support the revolutionary party?

Strength of support for party

Ordinal – ordering or ranking of extent of support for a particular party.

What percentage of votes do you think the revolutionary party will obtain?

Anticipated percentage support for party

Interval and ratio – distances and ratios between values can be meaningfully measured.

In the above example, each question provides a way of measuring a variable concerning  political issues.  The questions differ in terms of their intent and are associated with the different levels of measurement.

The reason the type of measurement, or scale, differs for different variables is related (a) to the nature of the phenomenon being investigated and (b) to the manner in which the concept being measured is defined (theoretical and operational issues).  Some physical and social science concepts such as distance, time, and price have well-defined units of measurement such as metre, hour, and dollar, respectively.  Other variables such as university faculty or major, or ethnicity, have no well-defined unit of measure, but are classifications or categorizations of individuals into groups or categories.  In between are variables with an ordinal level of measurement – variables such as degree of alienation, extent of agreement with same-sex marriage, or position on the left-right political spectrum – where values can be ranked or ordered but where there is no well-defined unit of measure.  These ordinal variables are particularly useful for the social sciences, permitting researchers to rank people but, to date, social scientists have not been able to develop or agree on what a unit of measure might be for such variables.

In the following notes, each of the types of scales are defined, with examples of each type of measurement.

2. Nominal scale of measurement – section 3.2.1, pp. 62-63

All variables have a nominal scale of measurement – this is simply a classification of characteristics or values of a variable into different categories.  Since a variable is a way of describing the different ways that members of a population differ on some characteristic, each variable provides a means of categorizing or classifying members of the population.  This is a nominal scale.

Definition of nominal scale (p. 62).  A nominal scale is a scale that classifies the various values of the variable into categories.  These categories are usually given names, or possibly numbers.

In the case where numbers are given to values of a nominal scale, the numbers may be arbitrary.

Examples.  The following examples are variables that have a nominal scale only, that is, they are no more than nominal.

·        Species of animals.  The different species of animals (homo sapiens, common cats, whales) constitute a way of classifying all living animals.  Each animal is a member of one species and not more than one, and members of one species differ in characteristics from those of other species.  Such a variables is not quantitative in that there is no obvious numerical or quantitative value that can be associated with each species.

·        Sex of individuals.  These are two different characteristics of humans, male and female.  While some may consider one sex to have more or some characteristic than another (e.g. males may be considered more aggressive), sex itself has no numerical qualities that make one sex more or less than the other in terms of the characteristic, sex.  As a result, sex is a variable that has no more than a nominal scale of measurement.

·        Religion.  There are many different religions, but again these are just different ways of categorizing the religious preferences of people.  Consequently religion has only a nominal scale of measurement.

·        Ethnicity, nationality, race.  In terms of how classification occurs, this is similar to religion, with each individual having a different characteristic for ethnicity, nationality, or race.  Again, these are only nominal scales.

·        Political party preference.  Again, this is merely a nominal scale – a classification of people into different categories depending on which political party they prefer to support, or not support.

Coding of nominal scales:

Categories of a nominal scale are usually given names.  Numbers can be attached to the categories, but these numbers are arbitrary and used only for convenience.  For example, for the variable sex, each category can be given a number (male = 1 and female = 2) but these numbers are arbitrary.  This is often done in computer analysis of data, where each characteristic of a variable is given a numerical value, so it can be entered into a computer data set.  But in the case of sex, the numbers are arbitrary in that females could be coded as 2 and males as 1.

Example.  In SSAE98, question 6, the variable “Faculty” has no more than a nominal scale of measurement.  Yet each faculty has a numerical value associated with it.  These numbers are arbitrary and appear only as a result of ordering the faculties alphabetically.  If they had been ordered in some other manner, the number associated with each faculty could be different.  The numbers are attached to the faculties only for purposes of keeping track of each category and for entry into a computer data set.

Exhaustive and mutually exclusive:

When constructing nominal scales, it is preferable for the researcher to construct them so the categories are exhaustive of all possible characteristics and the categories are mutually exclusive of each other.

Exhaustive.  If the categories are exhaustive, each individual will be in at least one of the categories of a variable.  For political party preference, this can be done by listing all possible political parties, and also including a category for “none of the parties” and, in case you have forgotten any parties, including an “other” category.  See question 16 of SSAE98.  By including none and other as possibilities, each individual can be classified into some category, even if the category is no more than none or other.   In addition, when entering data into a computer data set, categories such as “no response,” “refused,” or “missed” might be included.  That is, every individual is assigned some category, so the variable is exhaustive of all possibilities.

Mutually exclusive.  If a variable has a mutually exclusive set of categories, these categories will not overlap with each other.  For purposes of statistical analysis, whenever possible, it is advisable to construct variables so the categories are mutually exclusive.  Many of the attitude questions in SSAE98 are constructed this way – for example, the structure of the opinion questions concerning various social and political issues in question 15 is such that each respondent is to circle only one of the numerical responses – each response is to indicate a different attitude than any other category.

An example of a nominal scale that does not have mutually exclusive categories is education when measured by the following list: primary, secondary, post-secondary, university, technical school, business school.  In this case, university and post-secondary overlap and are not mutually exclusive.  In addition, a respondent may have attended several of these types of school.   While education measured by such categories may  be a useful variable, because of the overlapping categories of classificaiton, it is not so readily analyzed

This criterion of classification is often violated in the case of measurement of ethnicity.  That is, many individuals are of mixed ethnic ancestry so an individual might be of Russian, French, and Mexican origin.  While the different categories of ethnicity may be mutually exclusive of others, an individual’s ethnicity may not be easily measured on a single scale of ethnicity.

The following sections of these notes introduce variables that have a nominal scale but are more than nominal – that is, all scales of measurement for variables are nominal, but some are more than this.

3. Ordinal scale of measurement – section 3.2.2, pp. 64-65.

Different values of variables that have an ordinal scale of measurement can be ranked or ordered with respect to each other.  This is a higher or stronger level of measurement than measurement at no more than the nominal level.  Variables having characteristics that can be ranked or ordered are termed ordinal variables.

Definition of ordinal scale (p. 64).  An ordinal scale is a scale that is nominal, and one in which each value of the variable can be ordered, or ranked, as more than, less than, or equal to any other value of the variable.

Associated with an ordinal scale are the words “more than,” “greater than,” “less than,” and “equal to.”  That is, for a characteristic that is measured on an ordinal scale, any member of a population has more than, less than, or equal amounts of this characteristic, as any other member of the population.  Alternatively stated, for an ordinal variable each person can be ordered with respect to any other person.

In mathematical notation, each value of an ordinal scale can be compared to other values of the variable using one of the following mathematical operators:

 < less than = equal to > more or greater than

The idea of ordinality is to order to rank people on some characteristic.  Variables such as sex or ethnicity do not naturally lead to any such ranking; examples of variables that can be ordered or ranked are as follows.

Examples

·        Order of finish in a race.  In a race or competition, contestants are often ranked by whether they finish first, second, third, and so on.  The distance between the rankings may not be important – for example, when gambling on a horse race, in terms of payoff all that matters is whether the horse you selected finished first, second, or third.

·        Attitudes or opinions.  Most attitude and opinion scales used in the social sciences are ordinal scales.  That is, individuals are asked to state their level of agreement or disagreement on some statement about a political or social issue.  SSAE98 contains several examples – questions 15, 17, 24, and 29 are examples.  The format followed for this type of attitude variable is as follows, where attitudes or opinions are ranked as less than, equal to, or more than others in terms of agreement.

 Strongly disagree Strongly agree 1 2 3 4 5

For such an attitude variable, anyone who responds 4 is more in agreement than someone who responds with a 2.  Two individuals that give the same numerical response, say a 2, are equal to each other in terms of agreement or disagreement on this issue, but are less in agreement than someone who responds with a 3.

·        Social class.   If a researcher adopts a stratification approach to the measurement of social class of members of a population, the following ordinal scale might be used.

 upper class 5 upper-middle class 4 middle class 3 lower-middle class 2 lower class 1

Other measures of social class such as socioeconomic status or prestige of occupations, or measures devised for examining Canadian social class structure, such Blishen or Pineo-Porter scores, are ordinal scales.

·        Religiosity.  A researcher might attempt to measure the religiosity of members of a population by asking them how often they attend religious services.  An ordinal scale for religiosity might be constructed as follows.

 Attend daily Attend several times weekly Attend weekly Attend several times monthly Attend once a month Attend several times a year Attend once a year Never attend

In this case, numbers need not be attached.  That is, someone who responds “attend weekly” has a greater religiosity (at least as measured by this scale) than does someone who responds “attend once a month.”  However, numbers from 1 to 8 might be attached to the categories in order to assist with data analysis.

Ordinal scales and numerical values:

Numbers can be attached to different values of an ordinal scale, but they need not be.  Consider question 19 of SSAE98.  The question is “How much of a problem is personal safety on campus?” and the responses are to be “A great problem,” “A minor problem,” or “Not a problem.”  These are ordered or ranked responses even without the numbers that are attached.  Anyone who says indicates that it is a minor problem is lower on the scale for the variable “Personal safety” than is someone who responds that it is “A great problem.”  In the survey questionnaire, numbers are attached for purposes of data entry, but in order to obtain a measure of “Personal safety” they would not have needed to be attached.  Also note the numbers associated with the categories:

A great problem > A minor problem > Not a problem

1            >              2              >            3

In this case, the numbers representing the values are ordered in reverse order to the normal meaning of 1, 2, and 3, that is, a larger number, 3, represents not a problem while smaller numbers indicate a greater extent of problem.  For purposes of analysis, it might have been preferable to reverse the order of the three categories in the questionnaire.

Uses of an ordinal scale.

When analyzing data, it is useful to have variables measured on at least an ordinal scale.  Instead of merely categorizing members of a population into different categories, as is the case with a scale that is no more than nominal, an ordinal scale allows a researcher to say something about who has more or less of the characteristic in question.  This permits a more sophisticated form of analysis, where a researcher can begin to investigate how and why some are ranked higher than are others for the phenomenon being considered.

In terms of statistical measures, percentiles, quartiles, and the median can be calculated for a variable that has an ordinal scale of measurement.  We will examine these measures in Chapter 5.

4. Interval scale of measurement – section 3.2.3, pp. 65-69

Variables with an interval scale of measurement have a well-defined unit of measure, so that distances between values of a variable can be meaningfully measured.  Examples are height or weight where, respectively, the centimetre or the kilogram are well-defined units of measurement that can be used to accurately measure differences in heights or weights of different people.

Definition of interval scale (p. 65).  An interval scale is a scale that is nominal and ordinal, and one in which equal numerical differences represent equal quantities or magnitudes of the characteristic.

While numerical values may be associated with ordinal scales, there is no assurance that the same numerical difference represents the same quantity of the characteristic.  Consider an attitude question measured on a 1-5 scale, from 1 meaning strongly disagree to 5 meaning strongly agree.  Does the difference between an attitude response of 1 and 2 (strongly disagree to mildly disagree) represent the same difference of attitude as the difference between 3 and 4 (neutral to mildly agree).  In both cases, the difference is a value of 1, but it is not clear to that the two 1s (2-1 and 4-3) are equal in an meaningful sense.  For an ordinal attitude scale, the problem is that there is unit of measure – it is not clear what a unit of attitude might be.

In contrast, consider measurement in dollars where, at any point in time, one dollar is equal to any other dollar.  If I have \$4 and you have \$3, this is a difference of \$1.  Similarly, if someone has \$2 and another person has \$1, that is a difference of \$1.  Each of these differences of \$1 represents exactly the same quantity of dollars, in this case one dollar.  While measurements in dollars have a nominal and ordinal level of measurement, they are also interval levels of measurements, since equal dollar differences represent equal differences.  Any variables that can be measured in dollars have an interval scale.  The reason is that the dollar is a well-defined and understood unit, and each dollar really does represent an equal magnitude in monetary terms.

One of the consequences of having an interval scale of measurement is that values of the variable can be added and subtracted, so that these ordinary arithmetic operations on values of the variable are meaningful.  In contrast, for ordinal scales such as attitudes, it may make little sense to add or subtract values of the variable.

Examples

·        Age.  Age in terms of time since birth has an interval scale of measurement.  This is because the measure of age is in time, where time has a well-defined and generally agreed upon unit of measure – the second, or minute, or hour.  These measures of time are constant across geographic boundaries and at different times.  Similarly, other variables measured by time have an interval scale of measurement.

·        Grades.  Class grades in per cent are treated as having an interval scale of measurement.  While there may be some subjective aspect to the grading, most instructors attempt to grade in as objective a manner as possible.  In terms of the grades themselves, the unit is a single percentage point, and each problem or examination has a number of points.  These percentage points are added to compute a total grade for a class.  If someone score 76% for the class, this is 5 percentage points more than someone who scores 71%.  If you obtain a grade of 69%, this means you would have had to obtain 1 percentage point more to get 70%.  Treating grades as interval level measurements also means that the grades for different classes can be added and a student’s grade point average can be meaningfully calculated.

·        Religiosity.  In the examples of ordinal variables, the relative number of times that an individual attended religious services was treated as an ordinal scale of measurement.  By changing the definition of the variable, defining it as the number of times per year that an individual attends religious services, religiosity could be constructed as an interval scale.  Each attendance at a religious service is one unit, and the total number of times attended per year could be counted and treated as an interval scale.  This demonstrates that the scale of measurement depends in part on the structure of the question and the potential responses provided to respondents.  While variables such as sex or ethnicity cannot be constructed as ordinal or interval scales, the definitions for some phenomena can be constructed as ordinal or even interval scales.

·        Income.  This is ordinarily measured in dollars, so income has an interval scale of measurement.

If a well-defined unit of measure is available, meaningful to everyone, and constant across time and space, then the scale of measurement is at least an interval scale.

Scales that may be only ordinal but are sometimes considered interval

In the social sciences there are some forms of measurement that might be treated as being interval level measurements, even though they probably have no more than ordinal scales.  Some examples follow.

Examples

·        IQ.  Intelligence quotient (IQ) is often treated as having an interval scale.  For example, someone with an IQ of 115 may be regarded as having an IQ 10 points less than someone with an IQ of 125.  While statistical analysis of IQ may take this form, it is not clear what a unit of IQ might mean.  As a result, some analysts treat IQ as having only an ordinal scale, so that different individuals can be ranked in terms of intelligence, but how much more or less of IQ any individual has, compared with any other, is not so clear.  Other analysts have less hesitation about treating IQ as an interval scale so that IQs may be added and averaged across a number of individuals.

·        SES.  The same issue as IQ emerges in social stratification research, where socioeconomic status (SES) is sometimes used.  In Canada, the Blishen or Pineo-Porter indexes may be used to rank occupational prestige or status.  Each occupation may be given an SES, for example, a surgeon may have an SES of 87.2 while a custodian may have a status of only 43.1.  Such rankings may be developed by asking people to rank the status or prestige of occupations, or they may be derived from income and education levels of those filling these occupations.  While the ranking may make some sense, as in the case of IQ, it is not clear what the unit of SES is.  Again, some analysts treat such scales as ordinal while other researchers consider them to be interval scales.

·        Attitudes.  Even though attitudes measured on a 1-5 or 1-7 scale from strongly disagree to strongly agree have no more than an ordinal scale of measurement, it is common for researchers to treat them as if they are interval scales.  Average, or mean, attitude may be reported – implying that attitudes of different people can be added and averaged.  In order to do this, attitudes are treated as being interval level scales.

While it is common for researchers to treat ordinal level variables as having an interval scale of measurement, caution should be exercised in doing this.  Strictly speaking, many of the commonly used statistical methods are appropriate only for interval level scales.  In fact, researchers working with attitudes and similar types of variables commonly treat these as if they were ordinal.  Some of the consequences of this assumption will be examined later in the semester.

Uses of interval scales

If a variable can be measured at the interval level, researchers can do much more statistical analysis with the data than in the case of scales that have no more than a nominal or ordinal scale.  In the case of an interval scale, the numerical difference between two values is meaningful and represents a number of units of the variable.   Values of interval scales can be added or subtracted, so the mean value of an interval scale can be meaningfully computed (Chapter 5).  In addition, many other statistical procedures, such as those discussed in Chapters 6-11, can be used to analyze particular variables or the relationships among variables.  When possible, it is advisable for the researcher to define variables so they can be measured at the interval level.

5. Ratio scale of measurement – section 3.2.4, pp. 69-70.

For most variables that are measured at the interval level, ratios between values are also meaningful, so these variables also take on the ratio scale of measurement.  That is, if values of the variable can be added and subtracted (interval scale), then dividing one value by another, to calculate the ratio between these values is also appropriate.

Definition of ratio scale (p. 69).  A ratio scale of measurement is an interval scale with the additional property that equal ratios between two possible values of a variable represent equal magnitudes.

For example, distance as measured in kilometres has a ratio level of measurement – if city A is 500 km. away and city B is 1000 km. away, B is twice as far as A.  Similarly if C is 200 km away and D is 400 km away, D is twice as far away as C.  These ratios of 1000/500 and 400/200 are each equal to 2, and this value of 2 represents the same relative difference between the two pairs of cities.

Interval scales are generally ratio scales as well, so long as the zero point is not arbitrary, but really represents none of the characteristic in question.  In the above example, a distance 0 km, means no distance at all, so the zero point of any distance measurement is not arbitrary.  0 distance represents an absence of any distance.  The same is true with measures in dollars.  An income or amount of money of \$0 represents no money at all.  Since measurements in dollars have an interval level, and since the zero point is not arbitrary, measurements in dollars have a ratio scale of measurement.

A prime example of a scale that is but not ratio is temperature.  In the case of temperature on the Celsius or Fahrenheit scale, the zero point is arbitrary and does not represent the absence of temperature.  As a result, ratios of temperature in degrees Celsius are not necessarily meaningful in a ratio sense.  Does it make sense to say that it is twice as hot a 40° C. as at 20° C.?  It is clearer much hotter, but it is not clear that it is exactly twice as hot.  In the case of comparing positive and negative temperatures, taking ratios is particularly problematic.

In the case of scales such as IQ or attitude scales, while they might be treated as interval scales for purpose of statistical analysis, it would be a more serious error to treat these as ratio scales.  For example, what would it mean to say that someone with an IQ of 140 was twice as intelligent as someone with an IQ of 70.  Similarly with attitude, if there is five point scale from 1 meaning strongly disagree to 5 meaning strongly agree, does it make sense to say that a response of 4 is twice that of 2?  In this case, 2 is on the disagree side and 4 is on the agree side, so a ratio of two such values has little meaning.

When constructing scales, it may be possible to construct the scale so the intervals between values are meaningful (interval level) and the zero point really means none of the characteristic in question (ratio level).  If this is possible, then the scale is a ratio scale, the highest level of measurement.  In this case, it is possible to add, subtract, multiply, and divide values of the variable.  All the statistical procedures are possible with such a scale of measurement, and more detailed statistical manipulations of the data are possible.

6. Conclusion

It is important to be aware of the type of scale associated with each variable – statistical procedures and interpretations differ depending on the type of scale associated with a variable.  The key issues are as follows.

·        The presentation of the data may differ depending on the type of measurement.  In the case of a scale that is no more than nominal, tables may merely list the names associated with the different categories of the variable, and the same may be true of ordinal data.  In the case of interval and ratio level scales, it is likely that the data will be organized into numerical categories – for example, a table of income distribution may be organized into numerical intervals such as \$0 to \$20,000, \$20,000 to \$40,000, and \$40,000 and over.

·        The type of summary statistical measure that can be presented will also differ depending on the type of scale.  In the case of a nominal scale, the summary measure of the mode, the value that occurs most frequently, may be the only meaningful measure.  In the case of an ordinal scale, the median (middle value) can be computed and is meaningful.  For interval and ratio scales, the average value, or mean, can be calculated and other summary measures such as the standard deviation are also useful.  We will define and discuss these in Chapter 5.

·        Different statistical procedures are appropriate for each different type of variable.  In the case of nominal scales, data can be presented as tables, but little more than this.  For ordinal, interval, and ratio scales, it is possible to devise more complex methods of statistical analysis that yield more insight into the connection among variables.  Some of these methods will be examined later in the semester.

The reason why scales differ in the ways examined above is partly the nature of the phenomenon or characteristic being considered and partly the way that the researcher asks questions and constructs variables.  Where possible, it is advisable for the researcher to construct variables at the highest level possible, that is, construct an interval or ratio scale.  But in the case of measures of attitudes or opinions, or some basic human characteristics such as sex and region of origin, this cannot be done.

Next day – continuous and discrete and presenting data.

Last edited January 17, 2004