Measurement in Accounting: What is the Scale? In the last few years the field of accounting has been the subject of a critical review,the intensity of which is increasing as time goes by. Somehow, both within and without the accounting profession, there is a feeling of dissatisfaction with the information generated by the accounting process. Such dissatisfaction is leading to questions about the fundamental bases of Accounting. The practitioner claims to identify,collect,measure,record,analyse and report financial data useful for a variety of purposes. Unfortunately, recent financial turmoil emanating from some of the uses applied to accounting reports is causing accountants and other stakeholders to question the foundations of accounting. Some ask questions like is Accounting a Science or Art? If it is a science, what is its foundation in Science? Are their theories guiding the focus of accounting scholars? Are these theories the result of empirical findings and observations? Are the findings the outcome of diligent and rigorous measurement of phenomena? Does accounting measure any phenomena? Is the accounting phenomena amenable to measurement? Inshort is Accounting a measurement Discipline? Scholars are therefore re-examining the foundations of accounting to determine its suitability as a Science. A Science Discipline is founded on theories which are prediction of phenomena. These predictions are possible because rigorous and precise measurements of phenomena is achieved using instruments or tools. Therefore measurement is critical to any science discipline.

STATEMENT OF THE PROBLEM
Accounting is regarded as a Social Science. The Social Science Disciplines are also regarded as Measurement Disciplines. These disciplines have theories based on quantitative and qualitatively observed phenomena. The phenomena observed in accounting is performance of economic activities. Thus accounting claims to measure performance achieved by attaching number values to economic activities. The problem is whether the accounting criteria for measuring economic performance is relevant and valid for a measurement discipline. All measurement disciplines must have a measurement theory. Therefore, we ask which theory of measurement is accounting measurement borrowing from?

OBJECTIVE OF THE STUDY
The objective of this paper is to determine whether the belief that accounting is a measurement discipline is justified;
To examine whether accounting measurement has any scale of measurement and;
To examine the relationship between accounting measurement and measurement theory.

SIGNIFICANCE OF THE STUDY
This work is important to both academics, practicing accountants and other stakeholders because it will determine whether accounting reports can be considered objects of scientific endeavours.Scientific endeavours are based on objectivity. Objectivity is crucial because accounting reports affect economic activities.
SCOPE AND LIMITATION
The scope of this work is a review of literature which the scholar accessed on measurement theory and accounting measurement. Unfortunately, the span of literature review is limited by; a) Limited library resource on the relevant field b) Materials available online on limited access for preview only. Full access available on paid subscription which is unfortunately affordable.
LITERATURE REVIEW
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the meter, second or degree Celsius. The science of measurement is called metrology. With the exception of a few seemingly fundamental quantum constants, units of measurement are essentially arbitrary; in other words, people make them up and then agree to use them. Nothing inherent in nature dictates that an inch has to be a certain length, or that a mile is a better measure of distance than a kilometre. Over the course of human history, however, first for convenience and then for necessity, standards of measurement evolved so that communities would have certain common benchmarks. Laws regulating measurement were originally developed to prevent fraud in commerce. Today, units of measurement are generally defined on a scientific basis, overseen by governmental or supra-governmental agencies, and established in international treaties, pre-eminent of which is the General Conference on Weights and Measures (CGPM), established in 1875 by the Treaty of the metre and which oversees the International System of Units (SI) and which has custody of the International Prototype Kilogram. The metre, for example, was redefined in 1983 by the CGPM as the distance travelled by light in free space in 1⁄299,792,458 of a second while in 1960 the international yard was defined by the governments of the United States, United Kingdom, Australia and South Africa as being exactly 0.9144 metres.

In the classical definition, which is standard throughout the Physical Sciences, measurement is the determination or estimation of ratios of quantities. Quantity and measurement are mutually defined: quantitative attributes are those possible to measure, at least in principle. This definition is not suitable for the Social Sciences which measure attributes of human behaviour.Social Sciences therefore base there measurements on a different theory.

For measurement to be valid scientifically, there must be a scale. The scale concept is integral to measurement. Ryan,Scapens & Theobald (2002) show that every measurement process begins with scale specification. The specification of a scale of measurement is proof that the process is indeed one of measurement. IASB(2009) state that accounting is considered to be a measurement discipline. Thus, one would expect to find specified scales of measurement in accounting. A scale distinguishes the amount of a specified property in a specified class of phenomena. But it is said that such variation of specified property into specified classes does not exist in accounting.The works of Staubus(2004) and Ryan et al. (2002) all lend credence to this. Studies by Chambers (1997) and Willet (1987) reached similar conclusions. This contradicts the view that accounting is a measurement discipline. Narens (2002) also makes the point that a theory of measurement consists of a precise specification of how a scale is formed. The absence of specified scales of measurement in accounting therefore points to the absence of a theory of measurement in accounting. According to Luce, Krantz, Suppes and Tversk (1971), every process of measurement must have a theory of measurement. This means that every measurement discipline must have an underlying theory of measurement. The presence of a theory of measurement evidently guarantees the empirical validity of information produced by a process of measurement. Authors such as Staubus (1985) and Ijiri (1975) indicate a consensus that the accounting discipline has not succeeded in inferring a comprehensive and coherent theory of measurement from the observation of accounting measurement practices. Consequently, this suggests that researchers have not succeeded in specifying either how scales are formed in accounting measurement or whether the empirical validity of accounting information can be guaranteed. The above discussion points out that accounting is considered to be a measurement discipline in the absence of evidence (scales) confirming that it is such a discipline. As a result, it is necessary to investigate whether this accounting perspective is a delusion. Accounting is classified as a social science (Ryan et al., 2002). This means that measurement in accounting is viewed from the perspective of social science. The theory that establishes measurement in social sciences is the representational theory of measurement (Luce et al., 1971, 1989, 1990). The concept of scale in accounting is thus discussed from the perspective of representational measurement theory.

THEORETICAL FRAMEWORK
This paper views the representational theory of measurement as the framework under which accounting may or may not fit into the realm of measurement science. Unlike the real theory of measurement, the representational theory is more amenable to social sciences. Recall that we earlier stated that accounting is regarded as a social science..

The Representational Theory of Measurement
The classical or real theory of measurement is relevant to the physical sciences because it only recognises measurement of quantitative phenomena. To facilitate measurement in the Social Sciences, the representational theory of measurement was developed in 1958. Scott and Suppes(1958) developed this theory of ordering for the social sciences influenced by application of ordering to psychology(Luce,1956).This group of measurement theories requires in common a scale that can be defined by a set of structure preserving mappings from a qualitative or empirically-based structure onto a structure from mathematics. The representational measurement theory offers an abstract theory of the kinds of well-behaved scales encountered in science. It places great emphasis on the uniqueness of representations. Empirical examples of all scale types can be found in the representational theory of measurement. These scale types are also commonly found in science. They were conceptualised by Stevens(1946,1951) as ordinal, interval, ratio and nominal scales. Other works of Naren(1981a) and Naren(1981b) reveal that there can be scales between the ratio and interval scales which have not yet been applied to scientific measurement. Luce and Suppes(2001) agree to the existence of these scales and their non usage in scientific measurement.

The Concept of a Representational Scale
The concept of a scale is of fundamental importance in representational measurement. A scale is a rule used for the assignment of numerals to properties of objects or events(Stevens, 1951). This equates a scale to a specific method of measurement. It is also noteworthy that measurement always occurs in a specific way, which means that every measurement process must have a rule of measurement. Consequently, it is clear that in a process of measurement the random assignment of numbers to objects is excluded. Luce et al. (1971) argue that measurement can take place only if the rule that maps an empirical relational structure onto the numerical relational structure is specified. This indicates that the process of measurement takes place only in the presence of a standardized rule of measurement. From the foregoing, we can assume that the word rule as used above means the presence of a scale in every measurement process. Therefore it suffices to conclude that every process of measurement must have a scale of measurement. We can define a representational scale as that which exists when there is a qualitative structure X and a mathematical representing structure N for X such that the scale is a subset of one to one homomorphisms from X to N.

This definition points out that a scale of measurement can exist only when the quality in a structure can be represented by a numerical relational structure. It is also clear that a scale is part of the homomorphisms that map a qualitative structure onto a numerical relational structure. A homomorphism is a function that maps an algebraic structure onto another in a way that preserves the properties of the algebraic structure being mapped(Bhattacharya, Jain & Nagpaul, 1986). It is evident from this that a scale indicates the relationship that enables a qualitative structure to be mapped onto a numerical relational structure. It follows, therefore, that a scale explains how the properties of a qualitative structure are represented by an algebraic structure or a rule that explains the representation of an empirical relational structure by a numerical relational structure. According to Luce et al. (1971) the scales of measurement are subject to arbitrary conventions. It can be argued from this that scales of measurement are socially constructed. It also suggests that each process of measurement has its own rules of measurement. As a result, there is the possibility of a proliferation of rules of measurement for a single process of measurement. Thus we see that we can have different scales and different kinds of measurement because we can have different rules for assigning numerals to qualities of structures(Stevens,1951:1). This may result in the proliferation of measurement rules.If we agree with this then there are several measurement methods. Thus, each type of scale distinguishes one form of measurement from another. These numerous methods of measurement lead to different kinds of scales. It can also be inferred that each frame of reference can have its own rules of measurement. In addition, it should be noted that the rules of measurement are not part of the phenomenon being measured. The empirical relational structure and its empirical properties are not a matter of convention. Luce et al. (1971) argue that the empirical relational structure and its empirical properties should be treated as a set of qualitative empirical laws. This means that the phenomenon being measured should be invariant under any set of measurement procedures. A set of measurement procedures does not change the underlying property it is measuring (e.g., a man’s height is not changed in being measured by a metre rule). Stevens (1951) characterized scales into four types, namely, nominal, ordinal interval and ratio scales. The type of scale achieved in measurement depends upon the character of the basic empirical operations performed on the property being measured. Ordinarily, these operations are limited by the peculiarities of the thing being scaled and by our choice of the concrete procedures. However, once selected, these procedures determine the type of scale that will eventuate (Stevens, 1951). The type of a scale indicates the level of measurement.Associated with each level of measurement is a set of mathematical operations that may be performed on a measure. Each level of measurement involves different properties(relations and operations) of the numbers or symbols that constitute the measurements. The mathematical operations that may be performed on a measure without changing its meaning are termed permissible transformations on its scale. Permissible transformations on a scale preserve the relevant relationships of the measurement process (Luce et al., 1971). For example, changing the unit of measurement of distance(say, from inches to centimetres) multiplies the measurements by a constant factor. This multiplication does not alter the correspondence of the relationships “greater than” or the correspondence of addition and concatenation. Hence, it follows that the change of units is a permissible transformation with respect to these relationships. The concept of a representational scale is inextricably linked to the uniqueness and the existence of theorems of representational measurement. A number assigned to measure a property is unique once a unit of measurement has been chosen (Luce et al., 1990). This means that a scale of measurement makes a measure unique. It follows that the proof of the uniqueness theorem is equivalent to identifying all possible scales for the measurement of the elements of a given empirical relational system. It can therefore be concluded that the type of measurement can be known if, and only if, the scale of measurement is known.

CHAPTER THREE
DISCUSSIONS
The discussions in this section seeks to highlight the significance of absence of a scale in accounting measurement and the wider implication of non conformity with the theory of representational measurement. Critical literature review will serve as our aid on these discussions.
3.1 Importance of a Measurement Scale
From review of different studies it is apparent that a scientific measurement cannot hold without a scale. The scale serves as rule to guide every measurement process. It serves to direct the process of relating empirical properties to the assignment of numerical properties. Scales are classified into different types according to information implied in their usage(Stevens,1951).We know that accounting reports economic performance using numerical assignments. Are there any scales used to map out the relations between the properties measured and the numbers attached?Mattessichi(1964) seems to believe that accounting communicates its information using the media of the four known scale types. He cites nominal, ordinal, interval and ratio scales as those used in accounting. He however fails to specify how each property measured is related to the numbers assigned based on these scale types.

The theory of measurement suggests that every measurement scheme should specify the type of scale used in order to indicate the amount of information contained by the measures it produces. It can thus be argued that a scale of measurement is an embodiment of the properties of the phenomenon being measured. It follows that, without the specification of a scale of measurement, it would be not possible to know what a particular numerical assignment represents. Therefore, if accounting were a measurement discipline, it should have been capable of specifying the rules of measurement employed in its measurement processes to give meaning to its numerical assignments. Failure to do so calls into question the status of accounting as a measurement discipline. Measurement is about stating the relationship between the numerals and the objects. A rule of measurement states the relationship between the numerals and objects(Luce,1961) This means that, if accounting has no specified rules of assigning numerals to objects, it implies that the relations between the numerals and objects are not known. If these relations are not known it would be difficult to determine the meaning of a measurement. It follows that the concept of a scale influences the meaningfulness of a measure. This indicates that a scale used in a process of measurement must be specified. The scale gives empirical significance to measurements. Thus, a scale specifies the conditions under which a measurement has been made. It is clear from the above that the essence of meaningfulness is embodied in the description of the scale type and permissible statistics. This evidently highlights the fact that the statistics that can be performed on a measure depends on the scale type used to get the measure.The scale also affects the meaningfulness or lack of it in a measure.

The meaning of a measure is therefore embodied in the description of the meaning of the standard unit. For example, the meaning of a length will be embodied in the description of a metre. The meaning of 0 C(0 degree Celsius) is embedded in the description of what a degree Celsius is. 0 C is the temperature at which water freezes to ice. This is a standard measure which is acceptable all over the world. Such measure facilitates comparism with say 100 C which is the temperature at which water boils. This measure(100 C) is also regarded as standard and acceptable all around the world. Therefore any mention of temperature in degrees Celsius any where in the world will inspire the understanding of the level of coldness or hotness of the location.. Where-as scientific measurements are amenable to comparism, this cannot be said of a valuation of assets in currency. For starters most currencies are heavily influenced by politics. There values are mostly fixed by fiat and may have nothing to do with actual worth. How then do we measure values of assets and liabilities based on these currencies.

Lets take a case of a hypothetical regime which may fix the value of its currency against two others by fiat. It may as it wishes fix it lower against one and higher against the other. But it may be the case that the exchange rate against both currencies are the same or reverse of the fixed rates. Comparing values of assets in the fixed currency. against the values in the other two currencies might be problematic. We may not know the basis for comparism. Therefore without specifying the scales, measurements in accounting lack meaning and are not amenable to comparism. In the light of this, it is surprising that accounting information is considered comparable. One objective of establishing international standards in accounting reports is to facilitate comparism(IASB, 2009).This comparism can be done within one firm through time using time series. Comparism can also be done between different firms using a cross sectional analysis. This suggests that it is possible for users of accounting information to compare information from different entities in the absence of specified scales of measurement. The IASB (2009) points out that the measurement and display of the financial effect of like transactions and other events must be carried out in a consistent way throughout an entity and over time for that entity, and in a consistent way for different entities. This highlights the existence of a belief in accounting that measurement is possible in the absence of a specified scale of measurement.However, a scientific discipline cannot be based on mere belief. There has to be an objective base for carrying out a scientific endeavour. It can thus be concluded that the concept of a scale of measurement is not recognized in the accounting discipline. The nature of accounting measurements demands that the scales of measurement should be specified before they are compared. This is because accounting measurements are dependent on the intuition of the accountant.

The value attached to accounting measurement is even more so questionable in the light of different valuation methods that may yield different values for same asset or liability. This issue is corroborated by Mattessich (1964:79).He says accounting measurement is in the category of measurement by fiat where values are arbitrarily fixed subject to the whims and caprice of the market or of the regime that determines the value of the currency in which the item of interest is valued. He insists accounting measurement cannot be inferred through natural laws or a combination of fundamental measures which result in a derived measure. The conclusion to this lack of natural or fundamental laws guiding measurement in accounting is a certain definitional arbitrariness to accounting discipline. It means that the meaning of a measure in accounting is subject to several interpretations.

This emphasizes that accounting measurements are dependent on the intuition of the accountant, as accounting is not a natural, but a social science. The use of the phrase “definitional arbitrariness of our discipline” implies that accounting definitions are not based on consistent rules or plans, but are instead dependent on the context in which they are used. It is clear that accounting measurements are socially constructed. Consequently, this suggests a need to specify clearly the nature of the social context of accounting measurements before they are evaluated. There could be a difference between the kinds of assigning of numbers arising from different procedures of measurement. Therefore, if a scale of measurement was not specified in a measurement discipline, it would not possible to tell whether there were any other numbers that could be assigned as measures of the same property. Such knowledge of other numbers that might be assigned is important in determining the uniqueness of a measure. Luce et al.(1971) state that the number assigned to measure a property of an object is unique once a unit has been assigned to it. The concept of a scale is thus also important for the quality of uniqueness of measures. A lack of specified scales of measurement implies that the uniqueness of numbers assigned to represent the properties of accounting objects cannot be determined. The lack of specified scales of measurement has negative implications for the mathematical operations that could be carried out on accounting measurements. All known measurement scales from ordinal to ratio scales have permissible mathematical operations.

It is usual to add up income and subtract expenses in order to arrive at net income in income statements. What is not known is whether such mathematical operations are permissible on these items which may have been valued using different methods. The scale used to measure incomes and expenses is unspecified. Such unspecification makes any mathematical operation arbitrary.Every scale of measurements or rather measurements taken in them have different mathematical characteristics(Chambers,1997). As a result, it is necessary to consider the conditions under which addition and other forms of relation is mathematically permissible. That is, the values of assets and liabilities are added in the balance sheet and in the income statement without first verifying whether these measurements have been made under the same scale of measurement. The lack of specified scales causes inconsistencies in the classification of measures in accounting.
AAA(1971) provides some examples of accounting measures. They include primary measures which are counts of physical quantities and prices of non-monetary goods. These prices maybe historical, current or expected prices. It however does not tell us if these apply to physical counts also. In any case if these are applicable to physical counts, how do we physically count a measure in the past or the future unless we can travel through time. If we assume they are not applicable then it means they do not belong in same class of measurement. Hence we cannot add or subtract physical measures to/from non-monetary measures. It appears that accounting measures that differ are grouped in the same class, and that physical counts and prices are regarded as measures of the same property. It is also clear that there is no specified property represented by physical counts or by prices. Furthermore, there is no specification of the scale of measurement that could be used to distinguish the extent to which physical counts and prices possess a particular property. Consequently, it is not clear whether physical counts and prices are measures of the same property. This leads to incorrect classification of measures. In this way, the lack of specified scales in accounting casts doubt on the current belief in the literature that accounting is a measurement discipline.

Recall that Luce et al(1971) insists that the empirical property being measured must not change with the measurement method. It does not matter whether we measure distance in metres or kilometres.The distance being measured remains the same. All we need do is to convert distance measured in metres to kilometres using a constant factor. This constant factor will not change whether it is summer or winter and whether it is boom or depression.Here in lies the dilemma of accounting valuation methods. Converting from historical costs to current or market prices may not necessarily be hinged on a constant factor. Infact the factor may depend on time and place.Thus the conversion factor at a particular time may be different from a previous or future period. Also the conversion factor at a particular locale may be different from another locale both intra and international.
For illustrations, the value paid for rainboots bought during dry season will be less than it will cost during the rainy season. Again this value will keep changing depending on stage of the season at hand. Assuming an Accountant seeks to measure the value of such an asset, he will have no constant factor to guide his conversion of historical cost of purchase to any of either current or future prices. What he does is to wait for an event to happen before he gets the factors to use for his conversions. If we take the fact that value is attached to quality of assets,then valuation of assets is an indication of quality.Hence,measuring the value of an asset is akin to measuring the quality of an asset.In measurement, the empirical property being measured does not alter with the method of measurement.Since we are measuring quality by valuing assets in accounting and these measures sometime depend on time and location,then the property being measured may be said to vary with the method of measurement.

CHAPTER FOUR
RECOMMENDATION AND CONCLUSION
The concept of measurement is often misunderstood as merely the assignment of a value, but it is possible to assign a value in a way that is not a measurement in terms of the requirements of additive conjoint measurement. One may assign a value to a person's height, but unless it can be established that there is a correlation between measurements of height and empirical relations, it is not a measurement according to additive conjoint measurement theory. Likewise, computing and assigning arbitrary values, like the "book value" of an asset in accounting, is not a measurement because it does not satisfy the necessary criteria. The criteria for measurement is the presence of a scale which serves as a rule for measurement. The absence of a scale in accounting makes it difficult to qualify as a measurement discipline.
In this work we have reviewed previous studies that sought to situate accounting measurement. The result of such studies have been negative or problematic. Problematic because the theory underlying accounting measurement is unknown. The theory is unknown because the scale of measurement is unspecified. This study is also reaching the same conclusion that accounting measurement is at best still at infancy and in the process of developing a measurement theory or in the worst case scenario not fit for a measurement discipline.

4.2 Recommendations
However, choosing the best case scenario, this paper recommends that further studies be undertaken to unravel the theory and specify scales of accounting measurement. Such discovery will elevate the status of our discipline to that noble height of scientific disciplines and shut the mouth of our detractors once and for all.

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