Campbell, one of the first to deal with the issues of measurement as the assigment of numerals to represent properties of material systems other than numbers, in virtue of the laws governing these properties. Stevens, a noted theorist in th area of measurement in the social sciences, referred to measurement as the assignment of numerals to objects or events according to rules. Campbell make a distinction between systems and the properties of the systems. That systems in Campbell’s definition are what Stevens call ‘object or events’. These could include houses, tables, people, assets, or distance travelled. Properties are the specific aspect or characteristics of the systems, such as weight, length, widht or colour. We always measure properties and not the systems themselves. In the respect, Campbell’s definition is more precise than Steven’s. Campbell’s definition requires numerals to be assigned to properties according to the laws governing the properties, whereas Steven’s definition requires only that the assigment be done ‘according the rules’. Sterling objects to the broadness of Steven’s definition, arguing, ‘one need restrictions upon the kind of rule that can be used. Otherwise, any assigment of numbers can be called measurement. In the usual understanding of measurement, semantic rules (operational definitions) are devised and used to link the formal number the system with the property (of objects or events) to be measured. When the semantic rules assaign numbers to objects or events in such a way that the relationships among the objects or events (in respect of a given property) correspond the mathematical relationships, a scale has been estabilished and the property is said to be measured. Steven’s states: when this correspondence between the formal model and its empirical counterpart is close and tight, we find ourselves able to discover truth about matters of fact by examining the model itself. Under this view, the measurement procc is similiar to the approach to theory formulation and testing mentioned earlier. A statement, expessed mathematical, is advanced. Semantic rules (operations) are devised to connect the symbols of the statement to particular objects or events. When it is demonstrased that the relationships in the mathematical statement correlate with the relationships of the objects or events, then measurement of the given aspect of the objects or events has been made. In accounting we measure profit by first assigning a value to capital and then calculating profit as the change in capital over the period after accounting for economic events that affect the wealth of the firm. Scale every measurement is made on a scale. A scale is created when a semantic rule is used to relate the mathematical statement to objects or events. The scale shows what information the numbers represent, thus giving meaning to the numbers. The type of scale created depends on the semantic rules used. According to Steven’s, scale can be described in general term as nominal, ordinal, interval or ratio. These classifications were married at by examining the mathematical group structure of scales. The mathematical structure is determind by considering the kind of transformation that leaves the structure of the scale invariant,i.e. unchanged. Nominal scale in the nominal scale, numbers are used only as labels. The numbering of football players is an example given by Steven’s. Many theorist objects to the nominal scale as representing measurement. torgerson states: in measurement, as we use the term, the number assigned refers to the relative amount or degree of a property possessed by the objects, and not to the objects itself, whereas in the nominal scale the number often denote the objects themselves, such as numbering or naming players in sporting teams. The major property the numbers have is to identify players or objects. In the accounting system, the closest we have to the nominal scale is the classifications of assets and liabilities into different classess. Ordinal scale ordinal scale is created when an operation ranks the objects in question with respect to a given property. For example, suppose a certain investor has three feasible investment opportunities for a given amount of money to invest. They are ranked 1,2,3 according to their net present values, with the highest ranked as 1 and the lowest as 3. The operation (calculation of net present values) gives rise to ordinal scale, which is the set of numbers referring to the investment alternatives. The numbers indicate the order of the size of the net present value of the options and, therefore, their profitability. A weakness of the ordinal scale is that the intervals between the numbers (1 to 2, 2 to 3,and 1 to 3) do not tell us anything about the differences in the quantity of the property they represent. In our example, in terms of the aspect measured (net present value), option 2 may be very close to option 1, and option 3 may be considerably less than option 2. Another weakness is that the numbers do not signify ‘how much’ of the attribute the objects possess. Torgerson argues that some ordinal scales have a ‘natural origin’, that is a natural zero point. Applied to our example of ranking investment alternatives, the natural zero point could be a neutral point where in one direction are all the expected profitable alternatives and in the other direction are the expected unprofitable ones. The numbers assigned to the options no one side of the zero point would have positive signs and, on the other, negative signs. Interval scale the interval scale imparts more information than the ordinal scale. Not only is the ranking of the objects known with respect to the given property, but the distance between the intervals on the scale is equal and known. A select zero point also exists on the scale. An example is the Celcius scale of temperature. Equal intervals of temperature are note by equal volume of expansion with an arbitrary zero point agreed on for the scale. The temperature differential is divided between freezing and boiling into 100 degree, with the freezing point arbitrality set at zero degrees. If the temperature of two different room is measured with a Celcius thermometer and gives reading of 22 degrees and 30 degrees, we can say not only that the second room is hotter, but also that it is 8 degrees higher in temperature. The differences in the characteristic of the objects. The weakness of the interva scale is that the zero point is arbitrarily established. For example, suppose we were to measure the height with respect to thr average of the group. The average representing the zero point on the scale. If A is 3 cm above the average, then we would assign him the number +3; and if B is 5 cm below the everage, then we would assign him the number -5. On this scale, we do not know how tall A or B is in an absolute (actual) sense. B may be the shortest man in the group, but the group may consist of tall basketball players. Mattessich mentioned standard cost accounting as one example where the interval scale is used in accounting. The standard may based on theoritical, average, practica, or normal performance. Because the choice is more or less arbitrary, the calculation of standards and variances geberates an interval scale. If the variance is zero, this signifies neutrality, but this point is arbitrarily selected. Ratio scale a ratio scale is one where: 1. The rank order of the objects or events with respect to a given property is known 2. The intervals between the objects are equal and are known 3. A unique origin, a natural zero point, exists where the distance from it for at least one object is known. There ratio scale conveys the most information. The measurement of length is a good example of a ratio scale. When is 10m long and B is 20m, we can say not only that B is 10m longer but also that it is twice as long as A. The ratis of the numbers are also directly interpretable as ratio of the quantities of the property measured. Thus, it make sense to say that A is half as long as B or B is twice as long as A, whereas we cannot say that 40 degrees Celcius is twice as hot as 20 degrees Celcius. An example of the ratio scale in accounting is the use of dollars to represent cost and value. If assets A cost $10000 and assets B $2000, we can state that B cost twice as much as A. A natural zero point exists, because 0 denotes absence of cost or value, just as 0 for length means no length at all. PERMISSIBLE OPERATIONS OF SCALES One reason for discussing scales is that certain mathematical applications are permissible only for different types of scales. The ratio scale allows for all of fundamental arithmetical of addition, substraction, multiplication an dvision, and also algebra, analytic geometry, calculus an statistical methods. A ratio scale remains invariant (fixed) over all transformations when multiplied by a constant. For example, consider the following: X’= cX If X represents all the points on a given scale, and each point is multiplied by a constant c, the resulting scale X’ will also be a ratio scale. The reason is that the stucture of the scale is left invariant, that is: – The rank order of the points is unchanged – The ratios of the points are unchanged – The zero point is unchanged This means that if we measureed the lenght of a room and found it to be 400 cm and then converted than 400 cm to 4 m by multiplying by the constant 1/100., we can be assured that the lenght of the room is unchanged, even though the number representing lenght has changed. This is the same point we wake in chapter 6 regarding the the conversion of historical cost of, say, $100.000 of equipment under the nominal dollar scale to the purchasing power of the dollar scale by applying a constant, say, 120/100, to derive $ 120.000. The $120.000 is still hisorical cost. The invariance of a scale permits us to know the extent to which a theory or rule remains basically the same, even though the scale is expressed in different units, such as from centimetres to metres or from nominal dollars to constant dollars. An variant transformation of a ratio scale will leave intact the same general form the relationsip of the variables. Whithout invariance, it is possible to find that X is twice as long as Y when measured on centimetres, but three times as long when measured in metres. In accounting, the scale for current cost is variant from that of historical cost, because the atributes to be measured are different. When machine A is measured under historical cost it may be $90.000, but when measured under current cost it may be $110.000. The unit of measure, the dollar, is used in the both cases but the scales are different: they are variant. But changing from the nominal dollar scale to purchasing power the dollar scale from the same atribute (historical cost or current cost) leaves the structure invariant. With an internal scale, not all arithmetical operations of permissible. Addition and substruction can be used with respect to the particular numbers on the scale as well as the intervals. However, multiplication and division cannot be used with reference to the particular numbers, only to the intervals. The reason are due to the conditions of variance. An interval scale is invariant under any linear transformation of the form: X’= cX + b The transformation of one interval scale for maesuring a spescific property to another interval scale for maesuring the same property is made by multiplying each point of the first scale X by a constant c and adding to it a constant b. The reason for b is that there is no absolute zero point on an interval scale. For example, to transform from Celcius temperature to Fahrenheit temperature, we multiply each degree by 9/5 and add 32. The 9/5 is used because the Celcius scale has 100 degrees as opposed to 180 degrees for Fahrenheit , and 32 is added because that is the freezing for the latter scale. The conditions of invariance show that we can multipley and divide with respect to the intervals, but these aritmatical operations cannot be used the particular numbers on the scale. To illustrate, consider the following transformation: X’=x+10 Consider the objects on points 3 and 6 on scale X. Transforming to scale X, we now have 13 and 16. The ratio of 13 to 16 is not the same as the ratio of 3 to 6 because of the addition of the constant. Multiplication and division (i.e.ratios) are therefore not permissible for the particular numbers. Thus, if Robyn receives 90 points on her accounting exam and Maria receives 45 points, we cannot say that Robyn knows twice as much as Maria regarding the subject matter of the exam. The reason is that there is no natural zero point for the exam for “no knowledge”. Even if a student receives 0 on the exam, we cannot say that he has no knowledge of the subject matter. In this example, what we can say is that Robyn passed the exam and Maria failed the exam, but we cannot infer comparatively the ammount of knowladge to the numbers. Likewise, if the quantity variance is $ 5000 favourable, as opposed to the previous month’s variance of $ 10000 favourable, we can say that the use of materials this month is only half as efficient as in the preceding month. With ordinal scales, none of the arithmetical operations can be used. We cannot add, subtract, multiply or divide the numbers or the intervals on the scale. Ordinal scales, therefore, convey limited information.