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The specialist, writing the protocol of his observations in a form of a
data table «object-property», is free in the selection of protocol-keeping
language. More often the tables contain numerical data, but one can also
meet the letters, figures, texts and so on. If the protocol is proposed
for the «manual» use by the author himself only, then it is
only important for him to remember what properties he was measuring and
how he encoded them by numbers or signs. If the protocol is proposed to
be used by other people or by computer software then it is necessary to
provide strict understanding of the protocol by any user. Under the effect
of this evident requirement some discipline of protocol-keeping was elaborated.
It is studied by theory of measurements [1]. We will present here the basic
knowledge of this theory, which is necessary to describe data analysis
methods.
Two objects participate in the process of measurement: measuring device and measured object. In the result of their interaction the device comes into some state, which is fixed by one or another methods, depending upon the type of device and measurement procedure: by position of a pointer at the physical device scale, by the colour of litmus paper, by numbers at the numerical board, positive or negative answer on the question of a sociologist etc. Then this state is reflected in the protocol in a form of one or another symbols – numbers, letters, words an so on. The theory of measurements operates with the meaning «empirical system with relations» (E), including the multitude of measured objects (A) and the set of relations between these objects (R), interesting for the researcher: E={A,R}. For example, the multitude A is the multitude of physical bodies and the set R is the relations between them in respect to weight, density, sizes etc. To write down the results of observations the «symbolic system with relations» (N) is used, consisting of multitude of symbols (M), e.g., multitude of all real numbers, and finite set of relations (P) at these symbols N = {M,P}. The relations P are selected in the way, providing convenient revealing of observed empirical relations R. If the body t is heavier than body q, i.e. the relation R(t>q) exists, then the numerical representation of the bodies weight t=5 and q=3 demonstrates visually this empirical event, written as P(5>3). The agreement to use exactly such revealing of the system E into the system N means the selection of some definite revealing rule g. The trio of elements <E,N,g> is called the «scale» (don’t mix it up with the physical device scale). But we can agree as well about some other method of revealing w and then we will deal with another scale <E,N,w>, For example, g recommends to write the weight of the body in kilograms, while w – in grams or tonnes. The numerical record in the protocols will be different in this case, but the empirical content of the protocols will be equal. It means, that we have selected not any way of revealing (g, w an others) but only such one, connected with each other by mutually strict transformations. In other words, there is such transformation f, by means of which the records in language w may be accurately defined according to the record in language g (and vice versa): g=f(w) and w=f’(g). Transformation f unites the differently looking scales into the definite group, which is called the «scale type». Fixing the admissible transformation f we therefore are fixing the concrete scale type. In practical research studies the scales of quite a few types are distributed. Let us describe the scales of basic types. 1. Absolute scale. The admissible transformation for the scales of this type is an identity, i.e. if the protocol contains the record «y» in one language and record «x» in another one, then between them the simple relation must be fulfilled: y =x. This scale type is convenient for writing the number elements in some finite multitude. If, having counted the number of apples, one will write «6» and another one will write «VI», then it will be sufficient for us to know, that «6» and «VI» mean the same, i.e. that there exists some identity relation between these records: 6=VI. 2. Scale of relations. Between different protocols, fixing the same empirical fact in different languages, the following relation must be fulfilled for this type of scale: «16 kg», «16000 g», «0.016 t» and so on. One can move from one record to any another, having found the corresponding multiplier «a». This type of the scale is convenient for measurement of weights, lengths etc. If we don’t know exactly the units, which were used for recording of bodies weights in different protocols, then we can rely upon the relation of the two bodies weight: for example, the body with the weight of 10 units is twice more heavy, than the body with weight of 5 units, independently from the initially taken measurement unit – tonne or gram. Invariance of relations is revealed in the name of the scale of this type. If the measurement unit is written in the protocol, then such protocol reflects the properties of bodies in the absolute scale. 3. Interval scale. Here the linear transformations between protocols y and x are admissible: y=a*x+b, where a is any positive number and b may be both positive and negative. It means that different scale of units (a) and different zero points (b) may be used in different protocols. The example of such scales are the scales for measurement of temperature. If the protocol contains degrees, which are not specified (Celsius, Kelvin, Fahrenheit etc.), then to avoid the misunderstanding in description of regularities it is possible to use the interval relations only, because for any value of a and b the equity (y1-y2):(y3-y4) = {(a*x1+b)-(a*x2+b)}:{(a*x3+b)-(a*x4+b)} is satisfied. If the protocol records are supplied by an information of what degrees are meant (for example, «18 degrees C»), then we are dealing with the protocol in the absolute scale. 4. Scale of order. The admissible transformations for this type of scales are all monotonous transformations, i.e such transformations, which do not disturb the order of the sequence of the measured values. Such protocols may arise, for example, after the comparison of material hardness. The records «1*2*3» and «5.3*12.5*109.2» contain similar information, that the first material is the hardest one, the second one being less hard and third one – the softest one. And there is no information on numerical difference of these properties (in how many times or how much units of hardness difference) in these records and it is impossible to rely upon both concrete values and their relations and differences. The variety of the order scale is the rank scale, where the sequentially increasing numbers starting from 1 are used. If there is no equal objects among m measured ones, then the rank place of each object in the protocol will be given by one of numbers from 1 to m. For the equal values of measured property for k objects, taking current numbers from t-th to (t+k)-th, their ranks will be indicated by equal number, equivalent to their average rank x k where x = (1/k)* S (i+t-1). i=1 Such type of the order scale is called "normalised" rank scale. The widely used scale of marks also relates to the orders scale type. Here the integer numbers are used in limited range of their values: from 1 to 5 in education, from 1 to 6 or 10 in sports etc. In any of these cases the protocol contains the information only about three empirical relations: The scales of three first types contain more reach information, their data may undergo the definite mathematical transformations and thus they are often called «strong», «quantitative» and «arithmetic» ones. The scales of order and names type are less informative and reveal the qualitative properties, and they are named «weak» or «qualitative». However, it is impossible to recommend to use the «strong» scales only. Devices for measurement of strong properties are usually more expensive and sometimes devices for measurements of properties in strong scales do not exist at all (especially, in humanitarian areas). 1. Supes P., Zines G. Psychological measurements. «Mir», Moscow, 1967. |