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In Section 1.1 we said that a variable is quantitativeA variable having values that are numbers representing quantities. if its possible values are numbers that represent quantities (that is, “how much” or “how many”). In general, a quantitative variable is measured on a scale having a fixed unit of measurement between its possible values. For example, if we measure employees’ salaries to the nearest dollar, then one dollar is the fixed unit of measurement between different employees’ salaries. There are two types of quantitative variables: ratioA quantitative variable such that ratios of its values are meaningful and for which there is an inherently defined zero value. and interval.A quantitative variable such that ratios of its values are not meaningful and for which there is not an inherently defined zero value. A ratio variableA quantitative variable such that ratios of its values are meaningful and for which there is an inherently defined zero value. is a quantitative variable measured on a scale such that ratios of its values are meaningful and there is an inherently defined zero value. Variables such as salary, height, weight, time, and distance are ratio variables. For example, a distance of zero miles is “no distance at all,” and a town that is 30 miles away is “twice as far” as a town that is 15 miles away. An interval variableA quantitative variable such that ratios of its values are not meaningful and for which there is not an inherently defined zero value. is a quantitative variable where ratios of its values are not meaningful and there is not an inherently defined zero value. Temperature (on the Fahrenheit scale) is an interval variable. For example, zero degrees Fahrenheit does not represent “no heat at all,” just that it is very cold. Thus, there is no inherently defined zero value. Furthermore, ratios of temperatures are not meaningful. For example, it makes no sense to say that 60° is twice as warm as 30°. In practice, there are very few interval variables other than temperature. Almost all quantitative variables are ratio variables. In Section 1.1 we also said that if we simply record into which of several categories a population (or sample) unit falls, then the variable is qualitative (or categorical)A variable having values that indicate into which of several categories a population unit belongs.. There are two types of qualitative variables: ordinalA qualitative variable for which there is a meaningful ordering or ranking of the categories. and nominative.Aqualitative variable for which there is no meaningful ordering, or ranking, of the categories. An ordinal variableA qualitative variable for which there is a meaningful ordering or ranking of the categories. is a qualitative variable for which there is a meaningful ordering, or ranking, of the categories. The measurements of an ordinal variable may be nonnumerical or numerical. For example, a student may be asked to rate the teaching effectiveness of a college professor as excellent, good, average, poor, or unsatisfactory. Here, one category is higher than the next one; that is, “excellent” is a higher rating than “good,” “good” is a higher rating than “average,” and so on. Therefore, teaching effectiveness is an ordinal variable having nonnumerical measurements. On the other hand, if (as is often done) we substitute the numbers 4, 3, 2, 1, and 0 for the ratings excellent through unsatisfactory, then teaching effectiveness is an ordinal variable having numerical measurements. In practice, both numbers and associated words are often presented to respondents asked to rate a person or item. When numbers are used, statisticians debate whether the ordinal variable is “somewhat quantitative.” For example, statisticians who claim that teaching effectiveness rated as 4, 3, 2, 1, or 0 is not somewhat quantitative argue that the difference between 4 (excellent) and 3 (good) may not be the same as the difference between 3 (good) and 2 (average). Other statisticians argue that as soon as respondents (students) see equally spaced numbers (even though the numbers are described by words), their responses are affected enough to make the variable (teaching effectiveness) somewhat quantitative. Generally speaking, the specific words associated with the numbers probably substantially affect whether an ordinal variable may be considered somewhat quantitative. It is important to note, however, that in practice numerical ordinal ratings are often analyzed as though they are quantitative. Specifically, various arithmetic operations (as discussed in Chapters 2 through 14) are often performed on numerical ordinal ratings. For example, a professor’s teaching effectiveness average and a student’s grade point average are calculated. In Chapter 15 we will learn how to use nonparametric statistics to analyze an ordinal variable without considering the variable to be somewhat quantitative and performing such arithmetic operations. To conclude this section, we consider the second type of qualitative variable. A nominative variableA qualitative variable for which there is no meaningful ordering, or ranking, of the categories. is a qualitative variable for which there is no meaningful ordering, or ranking, of the categories. A person’s gender, the color of a car, and an employee’s state of residence are nominative variables.
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