Chapter 22 Variables and Their Measurement 2.1 Levels of Measurement 2.2 Continuous and Discrete Variables Real Limits • Significant Figures • Rounding 2.3 Summation Notation • Computations 2.4 Connections Cumulative Review • Computers • Homework Tips Exercises for Chapter 2 Personal Trainer Lectlet 2A: Variables and Their Measurement LAbS: Lab for Chapter 2 DATAGEN: Statistical Computational Package and Data Generator REViEwMaster 2A Resource 2X: Additional Exercises 15 ch02.indd 15 6/7/17 4:02 PM 16 Chapter 2 Variables and Their Measurement Learning objectives ❶ What is a variable? ❷ What are the characteristics of the three measurement scales (nominal, ordinal, and interval/ratio) used to measure variables? Why do we lump interval and ratio variables together? ❸ What is the difference between a continuous and a discrete variable? ❹ What are the real limits of a measurement? ❺ What are the three parts of the rule about rounding? ❻ How is summation notation used to simplify the communication about sums? “variable” can take on any of several values. There are three kinds of variables (nominal, ordinal, and interval/ratio) based on their level of A measurement. Most statistical computation involves summing values; the shorthand indication for a sum is ∑. variable Recall Pygmalion: Rosenthal and Jacobson (1968) led teachers to believe that the measurement second-graders identified as “bloomers” would show IQ spurts but the other children nominal ordinal would not spurt. There are three variables: the pretest IQ, the posttest IQ, and the intel- interval/ratio lectual growth (IQ gain). All three are interval/ratio variables. ∑ (SIG∙ma) = “sum of” Xi (EX∙sub∙EYE) ∑(2X — 4)2 = “sum of two X minus four tatistics can be thought of as the science of understanding data; data are the that quantity squared” results of a series of measurements on one or more variables. Let us be clear DataGen (DAY∙ta∙jen) Sabout what these terms mean. variable A characteristic A variable is any characteristic of the world that can be measured and that can take that can take on several or on any of several or many different values. For example, height is a variable (defined many different values as the number of inches a person is tall). It takes on the value 70 inches when John is measured and 64 inches when Mary is measured. Values of a variable may sometimes be assigned arbitrarily; for example, biologi- cal sex is a variable to which we may assign the values 1 for female and 2 for male (or 2 for female and 1 for male, or 27 for female and 136 for male, or any other values we choose). measurement The Measurement is the procedure that assigns values to the variable. For our purposes, procedure for assigning a it is enough to realize that a measurement rule must provide a unique and unambiguous value to a variable result for every individual. Thus, in our sex example, although it makes no difference what values are assigned to female and male, the assignment must be the same for all females and for all males. It is not satisfactory measurement to begin by assigning 1 to females and then, halfway through our data collection, change our minds and begin assigning 27. Personal Trainer Click Lectlets and then 2A in the Personal Trainer for an audiovisual discussion of Sec tions 2.1 through 2.4. Lectlets ch02.indd 16 6/7/17 4:02 PM 2.1 Levels of Measurement 17 2.1 Levels of Measurement Statisticians distinguish three kinds of variables and measurement levels: nominal, ordi nal, and interval/ratio. The main characteristics of these kinds of variables are given in Table 2.1. nominal scale Classification The nominal scale of measurement classifies objects into categories based on of unordered variables some characteristic of the object. Examples of nominal measurements of people are male/female, Republican/Democrat/Independent/decline to state, fraternity member/ nonmember, and Californian/Ohioan/Nebraskan/Alaskan/and so on. In each case, the measurement is the placing of an individual into one of the categories in the measure- ment scale. We require that all measurement operations be mutually exclusive; that is, for example, a person cannot be both a Republican and a Democrat or both an Ohioan and a Nebraskan. The order of categories in a nominal variable is not important. For example, it doesn’t matter from a measurement standpoint whether we refer to the sex distinction as male/female or female/male. It may be useful to assign a numerical value to a nomi- nal category; we might let male = 1 and female = 2. Doing that does not change the fact that the order is irrelevant; it would have made just as much measurement sense to let male = 2 and female = 1. ordinal scale Measurement The ordinal scale of measurement classifies objects into mutually exclusive cat- of variables that have an egories based on some characteristic of the object (as does the nominal level of inherent natural order measurement), and furthermore it requires that this classification have some inher- ent, logical order. An example of ordinal measurement of people is class standing (freshman/sophomore/junior/senior) because it is both mutually exclusive (you can’t be both a freshman and a sopho more) and ordered (sophomore is more advanced than freshman). Other examples are class rank (first in class/second/third/etc.), grade in course (A/B/C/D/F), and level of depression (not depressed/slightly depressed/moder- ately depressed/severely depressed). We may assign a numerical value to these categories, and if we do, the order of these categories is important (unlike in nominal variables where order is irrelevant). For example, it does make sense to assign freshman = 1, sophomore = 2, junior = 3, and senior = 4, but it does not make sense to assign sophomore = 1, senior = 2, fresh- man = 3, and junior = 4. Order is inherent in ordinal variables, and the values of the variables must reflect that order. TABLE 2.1 Characteristics of nominal, ordinal, and interval/ratio levels of measurement Level of Measurement Characteristic Nominal Ordinal Interval/Ratio Categories are mutually exclusive. Yes Yes Yes Categories have logical order. No Yes Yes Equal differences in characteristic No No Yes imply equal differences in value. ch02.indd 17 6/7/17 4:02 PM 18 Chapter 2 Variables and Their Measurement interval/ratio scale A The interval/ratio level of measurement classifies objects into mutually exclusive measurement scale for categories based on some characteristic of the object (as do the nominal and ordinal ordered variables that has equal units of measurement levels of mea surement). It requires that this classification have some inherent, logi- cal order (as does the ordinal level of measurement). Furthermore, it requires that the width of all the categories be the same. An example of an interval/ratio variable is temperature as measured in degrees Celsius (°C). When we require that the intervals be equal, we mean, for example, that the temperature difference between 34°C and 35°C is the same as the difference between 77°C and 78°C. It follows that the distance between nonconsecutive measurements must also be equal if the measured differences are equal. For example, the temperature difference between 34°C and 37°C must be the same as that between 75°C and 78°C because both differences are 3 degrees. BOX 2.1 RATIO LEvEL Of MEAsuREMENT c Some statisticians divide the interval/ratio level of measurement into two separate This box may be omitted without loss of continuity. levels, called the interval level and the ratio level. Their “interval” level is the same as what we have labeled “interval/ratio.” ratio scale An interval scale The ratio level of measurement has all the characteristics of the interval level, of measurement that has a and furthermore it requires that the scale have a true zero point. Examples of ratio true zero point measurements are weight (expressed as number of pounds), height (number of inches), and time (number of minutes). A true zero point means that the thing being measured actually vanishes when the scale reads zero. Thus, for example, the vari- able weight measures the heaviness of an individual; when the weight is 0 pounds, there is in fact no weight. Note that the existence of a true zero point on a scale does not imply the exis- tence of an individual whose measurement is zero. There are no 0-pound humans, for example. The existence of a true zero simply means that if there were an indi- vidual with no weight, then the weight scale would read 0. Temperature as measured in degrees Celsius is not a ratio scale because 0°C does not mean the absence of heat; 0°C is instead a relatively arbitrary point, the freezing point of pure water at sea level. The true zero of the Celsius scale (the complete absence of heat) is actually –273°C. By contrast, the Kelvin scale of tem- perature has absolute zero temperature as the 0 K point of the scale (thus making the freezing point of water +273 K). The Kelvin scale is therefore a ratio scale. For most statistical procedures (including all those described in this book), there is no difference between the interval and the ratio levels of measurement. Therefore, we will lump interval and ratio scales together, referring to either as an interval/ratio scale. We must make the distinction among nominal, ordinal, and interval/ratio vari- ables because statistics that are appropriate for variables measured at one level of c measurement may not be appropriate for variables measured at a different level.