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Classical measurement theory authors have used both versions. The advantage of the second version is that partial orders in the empirical system can be mapped to integers or reals that are both totally ordered.
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EXAMPLE 5.1 HEIGHT OF PEOPLE
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The classic example of mapping an empirical system to a numerical system is the height of people. In the empirical system, there is a well-understood height relationship among people. Given two people who are standing next to each other, everyone would agree about who is taller. This is the empirical system: people are the entities and the well-understood relation is shorter or the same height. The numerical system is the real number system (either metric or imperial units) with the standard relation of less than or equal. The mapping is just the standard measured height of people. This is usually measured barefoot, standing straight against a wall. The representation condition (either version) is satisfied, since if Fred is shorter than or equal to Bill, then Fred s measured height is less than or equal to Bill s measured height.
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Develop a measure, BIG, for people that combines both weight and height. Empirically, if two people are the same height, the heavier is bigger, and if two people are the same weight, the taller is bigger. If we use this notion, we can have a partial order that most people would agree with. The only pair of persons that we would not order by this would be if one was heavier and the other was taller. Numerically, we can use a tuple, < height, weight >. Each part of the tuple would be a real number. Two tuples would be related if both parts were related in the same direction. That is, if x ; y are tuples, than x is less than or equal to y in bigness if x height < y height and x weight < y weight. This is also a partial order, and both versions of the representation condition are satisfied.
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An important characteristic of a measure is monotonicity. It means that the value of the measure of an attribute does not change direction as the attribute increases in the object. For example, the count of lines of code will not decrease as more code is added.
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EXAMPLE 5.3
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A linear function is monotonic, since it always goes in the same direction. A quadratic function is usually not monotonic. For example, y 5x x 2 is not monotonic in the range x 0 to x 10. From x 0 to x 5, y increases. From x 5 to x 10, y decreases.
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MEASUREMENT SCALES
There are ve di erent scale types: nominal, ordinal, interval, ratio, and absolute. The least restrictive measurement is using the nominal scale type. This type basically assigns numbers or symbols without regard to any quantity. The classic
CHAPTER 5 Software Metrics
example for a nominal scale measure is the numbers on sports uniforms. We do not think that one player is better than another just because the number on one uniform is bigger or smaller than the number on the other uniform. There is no formula for converting from one nominal scale measure to another nominal scale measure. In an ordinal scale measure, there is an implied ordering of the entities by the numbers assigned to the entity. The classic example is class rank. If a student is ranked rst, her performance has been better than a student who is ranked second, or third, or any other number greater than 1. However, we never assume that the numerical di erence in rank is signi cant. That is, we don t assume that the di erence between the rst and second student is the same as the di erence between the 100th and 101st student. Any formula that converts from one ordinal scale measure to another ordinal scale measure for the same entity must preserve the ordering. In an interval scale measure, the amount of the di erence is constant. An example is temperature. There are two instances of temperature measures that are commonly used, Fahrenheit and Celsius. The formula for converting from a Celsius scale measure to a Fahrenheit scale measure is 9=5 x 32. With any two interval scale measures for the same attribute, the formula for conversion must be of the form ax b. In a ratio scale measure, the amount of the di erence is constant and there is a well-understood zero that any scale measure would use. For example, money, length, and height are measurements using ratio scales. These measurements have well-understood notions of zero: zero money, zero height, and zero length. Any formula for converting from one set of units to another from centimeters to inches, for example would just use a multiplicative constant. The absolute is a counting scale measure. The units are obvious and well understood. Counting marbles is an example of an absolute scale measure.
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