Scales of Measurement and Data Display
Spotlight 2.1 Rensis Likert
2.2 Discrete Variables, Continuous Variables, and the Real Limits of Numbers
Discrete Variables
Another important feature of measuring variables concerns how many different values can be assigned. Adiscontinuous(ordiscrete)variablecan take on only a finite number of values. No meaningful values exist between any two adjacent values. For instance, an undergraduate student is a freshman, sophomore, jun- ior, or senior; an adult is single, married, divorced, or widowed; and a roll of a typical die yields a one, two, three, four, five, or six; there are no“in-between” possibilities. One cannot claim to be halfway married or roll a die and hope to
2.2 Discrete Variables, Continuous Variables, and the Real Limits of Numbers 41
get a 4.5. It is permissible, however, to find statistical features of sets of discrete data, even if the number produced is not itself an acceptable value. For instance, it may be true that the average American family size is 2.58 persons, even though a 0.581person is not possible. We just need to keep in mind a correct interpre- tation of this statistic. In this instance, it means that for every 100 American families, there are, on average, 258 people.
Continuous Variables
A continuous variable can theoretically have an infinite number of points between any two numbers. Unlike discrete variables, continuous variables do not have gaps between adjacent numbers. Although 7 and 8 cm may be adjacent options on a ruler, there are an infinite number of values between them. Even if the scale is only marked to the millimeter, there are still an infinite number of values between 7.3 and 7.4 cm. When the underlying dimension of a scale is continuous, any number on the scale is an approximation. Even though we could measure someone’s reaction time down to the tenth of a second, this measurement could still be refined with a more precise instrument. Therefore, one can always theoretically increase the precision of measurements for contin- uous variables. This is not the case when measuring discrete variables. Greater measurement precision will not alter the fact that, for instance, family members exist in whole numbers.
Detecting the continuousness of a variable is not as simple as looking at how it is reported. Age, for instance, is often reported in whole-number years, but the underlying dimension is clearly continuous. The same point can be made with respect to psychological measures. Suppose a psychologist administers an anxiety questionnaire in which scores can range from 16 to 30. Although the measuring tool may only allow the assignment of whole numbers, the underly- ing concept is arguably continuous.
The Midpoint of an Interval and Real Limits of a Number
If a variable is continuous, any assigned number is an approximation. When someone weighs 195 lb, it does not mean that the person is exactly that weight.
A person who weighs 195.1 lb and another who weighs 194.8 lb might both be
1 The use of a“0”in front of a fractional value that is less than one will be the standard practice for this text. This increases reading clarity. However, there will be occasions when this is not the case, in particular in situations where the professional reporting of values is being used (as well as in the tables of Appendix A).
listed as weighing 195 lb. The number 195 lb is located at themidpointof an interval of weights, that is, the balance point of an interval of weights. The upper and lower boundaries of the interval are called thereal limits. The upper real limit of the number is one-half the unit of measurement above the number, and the lower real limit is one-half the unit of measurement below the number. If the unit of measurement is 1, the real limits for the number 13 are 12.5 and 13.5. If the unit of measurement is 0.1, then the real limits for 13 are 12.95 and 13.05.
Figure 2.1 graphically illustrates the concept of upper and lower limits for num- bers with different units of measurement.
■ Question The following data set contains the average temperature and amount of rainfall of several cities for the month of March. For each number, specify the upper and lower real limit.
Place Temperature (°F) Rainfall (in.)
Acapulco 88 0.1
Chicago 43 2.6
Honolulu 77 3.1
Orlando 76 3.4
Unit of Measurement = 1 Lower
real limit
Upper real limit
12.5 13.5
11 12 13 14 15
Unit of Measurement = 0.1 Lower
real limit
Upper real limit
12.95 13.05
12.8 12.9 13.0 13.1 13.2
(a)
(b)
Figure 2.1 The upper and lower limits of a score of 13. (a) The unit of measurement is 1.
The real limits are 13 + 0.5 = 13.5 and 13−0.5 = 12.5. (b) The unit of measurement is 0.1.
The real limits are 13 + 0.05 = 13.05 and 13−0.05 = 12.95.
2.2 Discrete Variables, Continuous Variables, and the Real Limits of Numbers 43
Solution The unit of measurement for temperature appears to be 1 °F. The unit of measurement for rainfall appears to be 0.1 in. Since the boundaries of a number are one-half the unit of measurement, the upper and lower real limits of 1° C are 0.5° C above and below the number used to report temperature.
Therefore, the upper and lower real limits for the temperature in Chicago are 43.5 and 42.5.
Since the unit of measurement for rainfall is 0.1 in., the upper and lower real limits are specified as 0.05 in. When establishing the upper and lower real limits, it helps to think of the decimal place that is one notch greater in precision. If the scale uses whole numbers, the limits will be stated using the tenth decimal place. If the scale of measurement uses one decimal place (e.g. rainfall in this example), the upper and lower limits will be reported using the second decimal place. For example, the upper and lower real lim- its for the March monthly rainfall of Honolulu are 3.15 and 3.05. The number 3.1 is the midpoint between 3.05 and 3.15. Table 2.1 presents the upper and lower real limits for the temperature and rainfall data of this problem. ■
In most research situations, a list (ordistribution) of numbers, calledraw(or original)scores, will be obtained. Unorganized data, though, is hard to inter- pret. However, if the distribution is presented in a tabular or graphical form, summaries and important features of the data set can be communicated to others. The remainder of the chapter presents numerous ways in which data can be presented in tables and on graphs.
Table 2.1 The midpoint, upper, and lower real limits for average temperatures and amount of rainfall for several cities in the month of March.
Temperature Rainfall
Lower limit Midpoint Upper limit Lower limit Midpoint Upper limit Acapulco
87.5 88 88.5 0.05 0.1 0.15
Chicago
42.5 43 43.5 2.55 2.6 2.65
Honolulu
76.5 77 77.5 3.05 3.1 3.15
Orlando
75.5 76 76.5 3.35 3.4 3.45