Measurement scales
In practical activities, it is necessary to carry out measurements of various quantities that characterize the properties of bodies, substances, phenomena and processes. As was shown in the previous sections, some properties appear only qualitatively, others - quantitatively. Various manifestations (quantitative or qualitative) of any property form sets, the mappings of whose elements onto an ordered set of numbers or, in a more general case, conventional signs form measurement scales these properties. The quantitative property measurement scale is the PV scale. Physical quantity scale is an ordered sequence of PV values, adopted by agreement based on the results of accurate measurements. The terms and definitions of the theory of measurement scales are set out in document MI 2365-96.
In accordance with the logical structure of the manifestation of properties, five main types of measurement scales are distinguished.
1. Naming scale (classification scale). Such scales are used to classify empirical objects whose properties appear only in relation to equivalence. These properties cannot be considered physical quantities, therefore scales of this type are not PV scales. This is the simplest type of scale, based on assigning numbers to the qualitative properties of objects, playing the role of names.
In naming scales in which the assignment of a reflected property to a particular equivalence class is carried out using human senses, the most adequate result is the one chosen by the majority of experts. In this case, the correct choice of classes of the equivalent scale is of great importance - they must be reliably distinguished by observers and experts assessing this property. The numbering of objects on a scale of names is carried out according to the principle: “do not assign the same number to different objects.” Numbers assigned to objects can be used to determine the probability or frequency of occurrence of a given object, but they cannot be used for summation or other mathematical operations.
Since these scales are characterized only by equivalence relations, they do not contain the concepts of zero, “more” or “less” and units of measurement. An example of naming scales are widely used color atlases designed for color identification.
2. Order scale (rank scale). If the property of a given empirical object manifests itself in relation to equivalence and order in increasing or decreasing quantitative manifestation of the property, then an order scale can be constructed for it. It is monotonically increasing or decreasing and allows you to establish a greater/lesser ratio between quantities characterizing the specified property. In order scales, zero exists or does not exist, but in principle it is impossible to introduce units of measurement, since a proportionality relation has not been established for them and, accordingly, there is no way to judge how many times more or less specific manifestations of a property are.
In cases where the level of knowledge of a phenomenon does not allow one to accurately establish the relationships that exist between the values of a given characteristic, or the use of a scale is convenient and sufficient for practice, conditional (empirical) order scales are used. Conditional scale is a PV scale, the initial values of which are expressed in conventional units. For example, the Engler viscosity scale, the 12-point Beaufort scale for sea wind strength.
Order scales with reference points marked on them have become widespread. Such scales, for example, include the Mohs scale for determining the hardness of minerals, which contains 10 reference (reference) minerals with different hardness numbers: talc - 1; gypsum - 2; calcium - 3; fluorite - 4; apatite - 5; orthoclase - 6; quartz - 7; topaz - 8; corundum - 9; diamond - 10. The assignment of a mineral to a particular gradation of hardness is carried out on the basis of an experiment, which consists of scratching the test material with a supporting one. If after scratching the tested mineral with quartz (7) a trace remains on it, but after orthoclase (6) there is no trace, then the hardness of the tested material is more than 6, but less than 7. It is impossible to give a more accurate answer in this case.
In conventional scales, the same intervals between the sizes of a given quantity do not correspond to the same dimensions of the numbers displaying the sizes. Using these numbers you can find probabilities, modes, medians, quantiles, but they cannot be used for summation, multiplication and other mathematical operations.
Determining the value of quantities using order scales cannot be considered a measurement, since units of measurement cannot be entered on these scales. The operation of assigning a number to a required value should be considered assessment. Assessment on order scales is ambiguous and very conditional, as evidenced by the example considered.
3. Interval scale (difference scale). These scales are further development order scales and are used for objects whose properties satisfy the relations of equivalence, order and additivity. The interval scale consists of identical intervals, has a unit of measurement and an arbitrarily chosen beginning - the zero point. Such scales include chronology according to various calendars, in which either the creation of the world, or the Nativity of Christ, etc. is taken as the starting point. The Celsius, Fahrenheit and Reaumur temperature scales are also interval scales.
The interval scale defines the actions of adding and subtracting intervals. Indeed, on a time scale, intervals can be summed or subtracted and compared by how many times one interval is greater than another, but adding up the dates of any events is simply pointless.
The Q interval scale is described by the equation
where q is the numerical value of the quantity; - start of the scale; - unit of the quantity under consideration. Such a scale is completely determined by specifying the origin of the scale and the unit of the given value.
There are practically two ways to set the scale. In the first of them, two values and quantities are selected that are relatively simply implemented physically. These values are called reference points, or main rappers and interval() - main interval. The point is taken as the origin, and the value 
per unit Q. In this case, n is chosen such that it is an integer value.
Translation of one interval scale 
, to another 
carried out according to the formula
(2.2)
The numerical value of the interval between the starting points on the scales under consideration, measured in degrees Fahrenheit ( 
, equals 32. The transition from temperature on the Fahrenheit scale to temperature on the Celsius scale is carried out according to the formula 
.
In the second way of specifying the scale, the unit is reproduced directly as an interval, a certain fraction of it, or a certain number of intervals of the size of a given value, and the starting point is chosen differently each time depending on the specific conditions of the phenomenon being studied. An example of this approach is a time scale in which 1 s = 9,192,631,770 periods of radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom. The beginning of the phenomenon being studied is taken as the reference point.
4. Relationship scale . These scales describe the properties of empirical objects that satisfy the relations of equivalence, order and additivity (scales of the second kind are additive), and in some cases proportionality (scales of the first kind are proportional). Their examples are the scale of mass (second kind), thermodynamic temperature (first kind).
In ratio scales, there is an unambiguous natural criterion for the zero quantitative manifestation of a property and a unit of measurement established by agreement. From a formal point of view, the ratio scale is an interval scale with a natural origin. All arithmetic operations are applicable to the values obtained on this scale, which is important when measuring EF.
Relationship scales are the most advanced. They are described by the equation Q = q[Q], where Q is the PV for which the scale is constructed, [Q] is its unit of measurement, q is the numerical value of the PV. The transition from one scale of relations to another occurs in accordance with the equation 
.
5. Absolute scales. Some authors use the concept of absolute scales, by which they mean scales that have all the characteristics of ratio scales, but additionally have a natural unambiguous definition of the unit of measurement and do not depend on the adopted system of units of measurement. Such scales correspond to relative values: gain, attenuation, etc. To form many derived units in the SI system, dimensionless and counting units of absolute scales are used.
Note that the scales of names and order are called non-metric (conceptual), and interval and ratio scales - metric (material). Absolute and metric scales belong to the category of linear. The practical implementation of measurement scales is carried out by standardizing both the scales and units of measurement themselves, and, if necessary, the methods and conditions for their unambiguous
Types and methods of measurements
Types and methods of measurements.
Measurements as experimental procedures for determining the values of measured quantities are very diverse, which is explained by the multitude of measured quantities, the different nature of their changes over time, different requirements and accuracy of measurements, etc.
Measurements, depending on the method of processing experimental data to find the result, are classified as direct, indirect, joint and cumulative.
Direct measurement – a measurement in which the desired value of a quantity is found directly from experimental data as a result of the measurement.
(Example: measuring source voltage with a voltmeter).
Indirect measurement – measurement in which the desired value of a quantity is found based on known dependence between this quantity and the quantities subjected to direct measurements.
(For example: the resistance of a resistor R is found from the equation R=U/I, into which the measured values of the voltage drop U across the resistor and the current I through it are substituted).
Joint measurements – simultaneous changes in several quantities of different names to find the relationship between them. In this case, a system of equations is solved.
(For example: determine the dependence of the resistor resistance on temperature R t = R 0 (1 + At + Bt 2); by measuring the resistance of the resistor at three different temperatures, they create a system of three equations, from which the parameters R 0 , A and B dependencies are found).
Aggregate Measurements – simultaneous measurements of several quantities of the same name, in which the desired values of the quantities are found by solving a system of equations composed of the results of direct measurements of various combinations of these quantities. (For example: measuring the resistance of resistors connected in a triangle by measuring the resistance between different vertices of the triangle; the results of three measurements determine the resistance of the resistors).
The interaction of measuring instruments with an object is based on physical phenomena, the totality of which constitutes the measurement principle, and the set of techniques for using the principle and measuring instruments is called measurement method .
The numerical value of the measured quantity is obtained by comparing it with a known quantity reproduced by a certain type of measuring instrument - measure.
Depending on the method of applying a measure of a known quantity, a distinction is made between the method of direct assessment and methods of comparison with the measure.
At direct assessment method the value of the measured quantity is determined directly from the reading device of a direct conversion measuring device, the scale of which was previously calibrated using a multi-valued measure that reproduces the known values of the measured quantity.
(Example: measuring current using an ammeter).
Comparison methods with a measure - methods in which a comparison is made of the measured value and the value of the reproducible measure.
Comparison can be direct or indirect through other quantities that are uniquely related to the first.
A distinctive feature of comparison methods is the direct participation in the measurement process of a measure of a known quantity that is homogeneous with the one being measured.
The group of methods for comparison with a measure includes the following methods: null , differential , substitution And coincidences .
At zero method measurement, the difference between the measured quantity and the known quantity or the difference in the effects produced by the measured and known quantities is reduced to zero during the measurement process, which is recorded by a highly sensitive device - a null indicator.
With high accuracy of measures reproducing a known value and high sensitivity of the null indicator, high measurement accuracy can be achieved.
(Example: measuring the resistance of a resistor using a four-arm bridge, in which the voltage drop across a resistor of unknown resistance is balanced by the voltage drop across a resistor of known resistance.)
At differential method the difference between the measured value and the value of a known, reproducible measure is measured using a measuring device.
The unknown quantity is determined from the known quantity and the measured difference. In this case, the balancing of the measured value with a known value is not carried out completely, and this is the difference between the differential method and the zero method. Differential method can also provide high measurement accuracy if the known quantity is reproduced with high accuracy and the difference between it and the unknown quantity is small.
Example: measurement of DC voltage U x using a discrete voltage divider R U and a voltmeter V
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Fig.1.1. Voltage measurement circuit using the differential method.
Unknown voltage U x =U 0 + U x , where U 0 is the known voltage, U x is the measured voltage difference.
At substitution method The measured quantity and the known quantity are alternately connected to the input of the device, and the value of the unknown quantity is estimated from the two readings of the device. The highest measurement accuracy is obtained when, as a result of selecting a known value, the device produces the same output signal as with an unknown value.
Example: measuring a small voltage using a highly sensitive galvanometer, to which a source of unknown voltage is first connected and the pointer deflection is determined, and then the same pointer deflection is obtained using an adjustable source of known voltage. In this case, the known voltage is equal to the known one.
At matching method measure the difference between the measured value and the value reproduced by the measure, using the coincidence of scale marks or periodic signals.
Example: measuring the rotation speed of a part using a flashing strobe lamp: observing the position of the mark on the rotating part when the lamp flashes, but the frequency of the flashes and the displacement of the mark determine the rotation speed of the part.
Measurement error. Basic concepts and types of errors
. Basic concepts and types of errors.
The measurement procedure consists of the following main steps:
- accepted models of the measurement object;
- choice of measurement method;
- selection of measuring instruments;
- conducting an experiment to obtain a numerical value of a measurement result.
Various shortcomings inherent in these stages lead to the fact that the measurement result differs from the true value of the measured value.
The reasons for the error may vary.
Measurement conversions are carried out using various physical phenomena, on the basis of which it is possible to establish the relationship between the measured quantity of the object of study and the output signal of the measuring instrument, by which the measurement result is evaluated.
It is never possible to accurately establish this relationship due to insufficient knowledge of the object of study and the inadequacy of its adopted model, the impossibility of accurately taking into account the influence of external factors, insufficient development of the theory of physical phenomena underlying the measurement, the use of simple but approximate analytical dependencies instead of more accurate but complex and etc.
The concept of “error” is one of the central ones in metrology, where the concepts of “error of the measurement result” and “error of the measuring instrument” are used. Measurement result error is the difference between the measurement result X and the true (or actual) value Q of the measured quantity:
It indicates the limits of uncertainty in the value of the measured quantity. Measuring instrument error- the difference between the SI reading and the true (actual) value of the measured PV. It characterizes the accuracy of the measurement results carried out by this tool.
These two concepts are in many ways close to each other and are classified according to the same criteria.
By nature of manifestation errors are divided into random, systematic, progressive and gross (misses).
Note that from the above definition of error it does not in any way follow that it must consist of any components. The division of the error into components was introduced for the convenience of processing measurement results based on the nature of their manifestation. In the process of forming metrology, it was discovered that the error is not a constant value. Through elementary analysis, it was established that one part of it appears as a constant value, while the other changes unpredictably. These parts were called systematic and random errors.
As will be shown in Sect. 4.3, the change in error over time is a non-stationary random process. Dividing the error into systematic, progressive and random components is an attempt to describe different parts of the frequency spectrum of this broadband process: infra-low frequency, low frequency and high frequency.
Random error- component of the measurement error that changes randomly (in sign and value) in a series of repeated measurements of the same EF size, carried out with the same care under the same conditions. There is no pattern observed in the appearance of such errors (Fig. 4.1); they are detected during repeated measurements of the same quantity in the form of some scatter in the results obtained. Random errors are inevitable, irremovable and always present as a result of measurement. Description of random errors is possible only on the basis of the theory of random processes and mathematical statistics.
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Unlike systematic, random errors cannot be eliminated from measurement results by introducing a correction, but they can be significantly reduced by increasing the number of observations. Therefore, to obtain a result that differs minimally from the true value of the measured value, multiple measurements of the required value are carried out, followed by mathematical processing of the experimental data.
Of great importance is the study of random error as a function of observation number i or the corresponding time point 1 of measurements, i.e. D; = A(t.). Individual error values are values of the function A(t), therefore, the measurement error is a random function of time. When carrying out multiple measurements, one realization of such a function is obtained. This is exactly the implementation shown in Fig. 4.1. Repeating a series of measurements will give us another implementation of this function, different from the first, etc. The error corresponding to each i-th measurement is the cross section of the random function A(t). In each section of this function one can find the average value around which the errors in various implementations are grouped. If a smooth curve is drawn through the average values obtained in this way, it will characterize the general trend of changes in the error over time.
Systematic error- component of the measurement error that remains constant or changes naturally with repeated measurements of the same PV. Constant and variable systematic errors are shown in Fig. 4.2. Their hallmark is that they can be predicted, detected and thanks to this almost completely eliminated by introducing an appropriate correction.
It should be noted that in lately The above definition of systematic error is subject to justifiable criticism, especially in connection with technical measurements. It is quite reasonably proposed to consider systematic error as a specific, “degenerate” random variable (see Section 5.1), which has some, but not all, properties of a random variable studied in probability theory and mathematical statistics. Its properties, which must be taken into account when combining error components, are reflected by the same characteristics as the properties of “real” random variables: dispersion (standard deviation) and cross-correlation coefficient.
Progressive (drift) error is an unpredictable error that changes slowly over time. This concept was first introduced in the monograph by M.F. Malikov “Fundamentals of Metrology”, published in 1949. Distinctive features of progressive errors:
They can be corrected by amendments only at a given point in time, and then change again unpredictably;
Changes in progressive errors over time are a non-stationary random process, and therefore, within the framework of a well-developed theory of stationary random processes, they can
be described only with certain reservations.
The concept of progressive error is widely used in studying the dynamics of SI errors and the metrological reliability of the latter.
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Gross error (miss)- this is a random error in the result of an individual observation included in a series of measurements, which, for given conditions, differs sharply from the other results of this series. They usually arise due to errors or incorrect actions of the operator (his psychophysiological state, incorrect readings, errors in recordings or calculations, incorrect switching on of devices or malfunctions in their operation, etc.). Short-term sudden changes in measurement conditions can also be a possible cause of errors. If errors are detected during the measurement process, the results containing them are discarded. However, most often errors are identified only during the final processing of measurement results using special criteria, which are discussed in Chapter. 7.
By way of expression , distinguish between absolute, relative and reduced errors.
Absolute error is described by formula (4.1) and is expressed in units of the measured quantity.
However, it cannot fully serve as an indicator of measurement accuracy, since the same value, for example, D = 0.05 mm at X = 100 mm corresponds to a fairly high measurement accuracy, and at X = 1 mm - low. Therefore, the concept of relative error is introduced. Relative error is the ratio of the absolute measurement error to true meaning measured quantity:
This visual characteristic of the accuracy of the measurement result is not suitable for normalizing the SI error, since when the values change, Q takes on different values up to infinity at Q = 0. In this regard, to indicate and normalize the SI error, another type of error is used - reduced.
Given error - this is a relative error in which the absolute error of the SI is related to the conventionally accepted one, constant over the entire measurement range or part of it:
The conventionally accepted value of Q N is called normalizing. Most often, it is taken to be the upper limit of measurements of a given SI, in relation to which the concept of “reduced error” is mainly used.
Depending on place of origin distinguish between instrumental, methodological and subjective errors.
Instrumental error due to the error of the used SI. Sometimes this error is called hardware
Methodological error measurements are determined by:
The difference between the accepted model of the measured object and the model that adequately describes its property, which is determined by measurement;
The influence of methods of using SI. This occurs, for example, when measuring voltage with a voltmeter with a finite value of internal resistance. In this case, the voltmeter shunts the section of the circuit on which the voltage is measured, and it turns out to be less than it was before connecting the voltmeter;
The influence of algorithms (formulas) by which measurement results are calculated;
The influence of other factors not related to the properties of the measuring instruments used.
A distinctive feature of methodological errors is that they cannot be indicated in the regulatory and technical documentation for the measuring instrument used, since they do not depend on it, but must be determined by the operator in each specific case. In this regard, the operator must clearly distinguish between the quantity actually measured and the quantity to be measured.
Subjective (personal) error measurements are due to the operator’s error in reading readings on SI scales and diagrams of recording instruments. They are caused by the condition of the operator, his position during work, imperfection of the sensory organs, and the ergonomic properties of the SI. Characteristics of personal error are determined on the basis of the normalized nominal value of the scale division of the measuring instrument (or the chart paper of the recording instrument), taking into account the ability of the “average operator” to interpolate within the scale division.
By dependence of the absolute error on the values of the measured quantity errors are distinguished (Fig. 4.4):
additive, independent of the measured value;
multiplicative, which are directly proportional to the measured value;
nonlinear, having a nonlinear dependence on the measured value.
These errors are used mainly to describe the metrological characteristics of SI. The division of errors into additive, multiplicative and nonlinear is very important when addressing the issue of normalization and mathematical description of SI errors.
Examples of additive errors are from a constant load on a scale, from inaccurate zeroing of the instrument needle before measurement, from thermo-EMF in DC circuits. The causes of multiplicative errors can be: a change in the gain of the amplifier, a change in the rigidity of the pressure gauge sensor membrane or the device spring, a change in the reference voltage in a digital voltmeter.
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Rice. (1).4. Additive (a), multiplicative (b) and nonlinear (c) errors
By influence of external conditions distinguish between main and additional SI errors. Basic is called the SI error, defined in normal conditions its application. For each measuring instrument, the operating conditions are specified in the regulatory and technical documents - a set of influencing quantities (temperature environment, humidity, pressure, voltage and frequency of the supply network, etc.), at which its error is normalized. Additional, is called the SI error that arises due to the deviation of any of the influencing quantities.
IN depending on the influence of the nature of changes in the measured quantities SI errors are divided into static and dynamic. Static error- this is the error of the SI used to measure PV, taken as constant. Dynamic is called the SI error, which additionally arises when measuring the PV variable and is caused by the discrepancy between its response to the rate (frequency) of change of the measured signal.
Measurement theory is a theory about the classification of variables according to the nature of the information contained in the numbers - the values of these variables. Origin variable size imposes restrictions on the many actions that can be performed with this value. In other words, for each variable there is class of admissible transformations (KDP), which correctly apply to all values of this quantity.
The classification of quantities by measurability was proposed by S.S. Stevens in 1946. Each group of quantities that have common permissible transformations is called a measurement scale.
Measurement scales
Nominal scale
In the naming scale, all one-to-one transformations are valid. In this scale, numbers are used as marks, only to distinguish objects. The name scale measures, for example, telephone numbers, car numbers, passport numbers, and student ID numbers. The gender of people is also measured in a scale of names, the measurement result takes two values - male, female. Obviously, it makes no sense to add up phone numbers or multiply series of passports.
KDP: bijective transformations.
Ordinal scale
In an ordinal scale, numbers are used not only to differentiate objects, but also to establish order between objects. The simplest example is student assessments. Note that in secondary school grades of 2, 3, 4, 5 are used, and in higher school exactly the same meaning is expressed verbally - unsatisfactory, satisfactory, good, excellent. This emphasizes the “non-numerical” nature of student assessments. In the ordinal scale, all strictly monotonic transformations are admissible.
KDP: all strictly monotonic transformations.
Interval scale
Using an interval scale, the magnitude of potential energy or the coordinate of a point on a straight line is measured. In these cases, neither the natural origin nor the natural unit of measurement can be marked on the scale. The researcher must set the starting point and choose the unit of measurement himself. Acceptable transformations in the interval scale are linear increasing transformations, i.e. linear functions. The temperature scales Celsius and Fahrenheit are related precisely by this relationship: °C = 5/9 (°F - 32), where °C is the temperature (in degrees) on the Celsius scale, and °F is the temperature on the Fahrenheit scale.
KDP: all transformations of the form
Relationship scale
In ratio scales there is a natural reference point - zero, but there is no natural unit of measurement. The ratio scale measures most physical units: body weight, length, charge, as well as prices in economics. Acceptable transformations to the ratio scale are similar (changing only the scale). In other words, linear increasing transformations without a dummy term. Examples of using such transformations: converting prices from one currency to another at a fixed rate, converting weight from kilograms to pounds.
KDP: all transformations of the form
Difference scale
A difference scale has a natural unit of measurement, but no natural reference point. Time is measured on the scale of differences, if the year (or day - from noon to noon) is taken as a natural unit of measurement, and on the scale of intervals in the general case. On modern level knowledge of the natural beginning of time cannot be indicated. Acceptable transformations to the difference scale are shifts.
KDP: all transformations of the form
Absolute scale
Only for the absolute scale the measurement results are numbers in the usual sense of the word. An example is the number of people in the room. For an absolute scale, only an identity transformation is allowed.
KDP:
Hierarchy of measurement scales
All scales are also divided into 2 large groups: quality And quantitative. Qualitative scales include nominal and ordinal, and quantitative scales include all others. This division shows the difference in the nature of the scales: for example, it is impossible to say that a school grade of 2 is as much worse than a grade of 4 as 3 is worse than a grade of 5, so ordinal scales are classified as qualitative. At the same time, for bodies of different masses, a similar statement is correct: a body weighing 5 kg is as much heavier than a body weighing 3 kg as a body weighing 4 kg is heavier than a body weighing 2 kg. Thus, ratio scales are quantitative scales.
Order scales allow not only dividing objects into classes, but also ordering classes in ascending (descending) order of the characteristic being studied: it is known about objects assigned to one of the classes, but only that they are identical to each other, but also that they have a measurable property to a greater or lesser extent than objects from other classes. But at the same time, ordinal scales cannot answer the question of how much (how many times) this property is expressed more strongly in objects from one class than in objects from another class. Examples of order scales include level of education, military and academic ranks, type of settlement (large - medium - small city - village), some natural scientific scales (hardness of minerals, strength of a storm). Thus, we can say that a 6-point storm is certainly stronger than a 4-point storm, but it is impossible to determine how much stronger it is; a university graduate has a higher educational level than a high school graduate, but the difference in level of education cannot be directly measured. Ordered classes are often numbered in ascending (descending) order of the characteristic being measured. However, due to the fact that differences in the value of a feature cannot be accurately measured, arithmetic is not applied to order scales, as well as to nominal scales. The exception is rating scales, when using which the object receives (or gives) ratings based on a certain number of points. Such scales include, for example, school grades, for which it is considered quite acceptable to calculate, for example, the average grade on the matriculation certificate. Strictly speaking, such scales are a special case of an order scale, since it is impossible to determine how much more the knowledge of an “excellent” student is than the knowledge of a “C” student, but due to some theoretical considerations they are often treated as scales of a higher rank - interval scales . Another special case of the order scale is the rank scale, usually used in cases where a characteristic is obviously not amenable to objective measurement (for example, beauty or the degree of hostility), or when the order of objects is more important than the exact magnitude of the differences between them (places occupied in sports competitions). In such cases, the expert is sometimes asked to rank a certain list of objects, qualities, motives, etc., according to a certain criterion.
The numbers assigned to objects on this scale will indicate the degree of expression of the measured property in these objects, but, at the same time, equal differences in numbers will not mean equal differences in the quantities of measured properties. Depending on the desire of the researcher, a larger number can mean a greater degree of expression of the property being measured (as in a scale of mineral hardness) or less (as in a table of results of sports competitions), but in any case, the order relationship remains between the numbers and the objects corresponding to them. The order scale is specified by positive numbers, and there can be as many numbers in this scale as there are measured objects. Examples of order scales in psychology: rating of subjects on any basis, results expert assessment subjects, etc.
If it is possible to establish the order of psychological objects in accordance with the severity of some property, then an ordinal scale is used.
An ordinal scale is formed if one binary relation is implemented on a set - order (the relations “more” and “less”). Constructing an order scale is a more complex procedure than creating a naming scale. It allows you to record the rank, or place, of each value of a variable in relation to other values. This rank can be the result of establishing an order between some stimuli or their attributes by the subject himself (the primary indicator of ranking methods, or rating procedures), but it can also be established by the experimenter as a secondary indicator (for example, when ranking the frequencies of positive responses of subjects to questions related to to different topics).
Equivalence classes, identified using a naming scale, can be ordered according to some basis. There is a scale of strict order (strict order) and a scale of weak order (weak order). In the first case, the relations “more than” and “less than” are implemented on the elements of the set, and in the second case, “not more than or equal to” and “less than or equal to.”
Values can be replaced by squares, logarithms, normalized, etc. With such transformations of the values of quantities determined on the order scale, the place of objects on the scale does not change, i.e. no inversions occur.
Stevens also expressed the view that the results of most psychological measurements, at best, correspond only to order scales.
Order scales are widely used in the psychology of cognitive processes, experimental psychosemantics, and social psychology: ranking, evaluation, including pedagogical ones, provide ordinal scales. A classic example of the use of ordinal scales is in testing personality traits as well as abilities. Most experts in the field of intelligence testing believe that the procedure for measuring this property allows the use of an interval scale and even a ratio scale.
Be that as it may, this scale allows you to introduce a linear ordering of objects on a certain axis of the attribute. This introduces the most important concept - a measurable property, or a linear property, while the naming scale uses a “degenerate” version of the interpretation of the concept “property”: a “point” property (there is a property - there is no property).
The ordinal (rank) scale must contain at least three classes (groups): for example, answers to a questionnaire: “yes”, “I don’t know”, “no”; or -- low, medium, high; etc., so that the measured characteristics can be arranged in order. That is why this scale is called an ordinal, or rank, scale.
From classes it is easy to move on to numbers, if we assume that the lowest class receives a rank (code or number) 1, the middle - 2, the highest - 3 (or vice versa). The greater the number of classes of partitions of the entire experimental set, the wider the possibilities statistical processing obtained data and testing statistical hypotheses.
When encoding ordinal variables, any numbers (codes) can be assigned to them, but the order must be preserved in these codes (digits), or, in other words, each subsequent digit must be greater (or less) than the previous one.
A wider range of statistical measures (in addition to those that are valid for a naming scale) can be used to interpret data obtained through an ordinal scale.
You can use the median as a characteristic of central tendency, and percentiles as a characteristic of dispersion. To establish a relationship between two measurements, ordinal correlation (Kandell's t- and Spearman's p-correlation) is acceptable.
Numerical values on an ordinal scale cannot be added, subtracted, divided, or multiplied. (2, 3).
Name scaleused to describe the belonging of objects to certain classes. This is the weakest quality scale. All objects of the same class are assigned the same number, and objects of different classes are assigned different numbers. In this regard, the naming scale is often called classification scale . It preserves equivalence relations and differences between objects and is used for indexing product ranges (product specifications), documents and types of information in automated control systems, numbering departments in an organization, etc. There are a large number of options for assigning numbers to classes of equivalent objects. Consequently, the concept of uniqueness of a mapping f consists for the scale of names in interambiguity valid conversion. This means that if there are two options for assigning numerical values to classes, then they must be related to each other one-to-one, which makes it possible to establish a connection between the numerical options for describing equivalence classes. Thus, the naming scale is unique up to one-to-one conversion. This means that in this scale there are no concepts of scale and origin.
The name “nominal” is explained by the fact that such a sign gives only nothing associated names objects. These values are either the same or different for different objects; no more subtle relationships between values are recorded. Nominal type scales allow only the distinction of objects based on checking the fulfillment of the equality relation on the set of these elements.
The nominal type of scales corresponds to the simplest type of measurements, in which scale values are used only as names of objects, therefore nominal type scales are often also called naming scales.
Examples of measurements in the nominal type of scales include car numbers, telephone numbers, city codes, persons, objects, etc. The only purpose of such measurements is to identify differences between objects of different classes. If each class consists of one object, a naming scale is used to distinguish objects.
Figure 3.5 shows the measurement on a nominal scale of objects representing three sets of elements A, B, C.
Fig.3.5. Measuring objects on a nominal scale
Here the empirical system is represented by four elements: a A, b B, (c, d) C, belonging to the corresponding sets. The sign system is represented by a digital scale of names, including elements 1,2,...,n and preserving the relation of equality. A homomorphic mapping associates each element from the empirical system with a specific element of the sign system. Two features of nominal scales should be noted.
Firstly, elements cud the same value of the measurement scale is assigned (see Fig. 3.5). This means that these elements do not differ when measured.
Secondly, when measured in the name scale, symbols 1,2,3,...,n , used as scale values are not numbers, but numbers that serve only to designate and differentiate objects. Thus, the number 2 is not twice or one more than the number 1, unlike the numbers 2 and 1.
Any processing of measurement results on a nominal scale must take these features into account. Otherwise, erroneous conclusions may be made regarding the assessment of systems that do not correspond to reality.
Order scale
The scale is called rank (scale of order), if the set of admissible transformations consists of all monotonically increasing admissible transformations of scale values. Consequently, the order scale is unique up to a monotonic transformation.
A transformation that satisfies the condition: if , then and 
for any scale values from the domain of definition. The ordinal type of scales allows not only the distinction of objects, like the nominal type, but is also used to order objects according to measured properties. The numbers on the scale determine the order in which objects appear and do not make it possible to say by how much or how many times one object is preferable to another. This scale also lacks the concepts of scale and origin.
Order scale measurement can be used, for example, in the following situations:
· it is necessary to arrange objects in time or space. This is a situation when one is not interested in comparing the degree of expression of any of their qualities, but only in the relative spatial or temporal arrangement of these objects;
· you need to arrange objects according to some quality, but it does not require accurate measurement;
· any quality is measurable in principle, but cannot currently be measured for practical or theoretical reasons.
An example of an order scale would be a scale of mineral hardness proposed in 1811 by the German scientist F. Mohs and still common in geological field work. Other examples of order scales include scales of wind strength, earthquake strength, grades of goods in trade, various sociological scales, etc.
Any scale obtained from an order scale by an arbitrary monotonically increasing transformation of scale values will also be an exact order scale for the original empirical system with relations.
Somewhat more “strong” than ordinal scales are hyperorder scales. Acceptable for these scales are hypermonotonic transformations, i.e. transformations such that for any 
:
only when belong to the domain of definition and 
.
Thus, when measuring on hyperorder scales, the ordering of the differences in numerical estimates is preserved.
Interval scale
Interval scale used to display the magnitude of the difference between the properties of objects. An example of the use of this scale is to measure temperature in degrees Fahrenheit or Celsius. In expert evaluation, an interval scale is used to assess the usefulness of objects. The main property of the interval scale is the equality of intervals. An interval scale can have arbitrary reference points and scale. Consequently, the interval scale is unique up to a linear transformation. In this scale, the ratio of the difference between numbers in two number systems is determined by the scale of measurement.
One of the most important types of scales is interval type. The type of interval scales contains scales that are unique up to a set of positive linear admissible transformations of the form
,
wherea>0; b – any value. The main property of these scales is that the ratios of intervals in equivalent scales remain unchanged:
This is where the name of this type of scale comes from. An example of interval scales is temperature scales. In this case, the function of the permissible conversion of degrees Celsius to degrees Fahrenheit has the form
,
conversely, the permissible conversion function from degrees Fahrenheit to degrees Celsius is:
.
Another example of a measurement on an interval scale can be the “date of event” attribute, since to measure time on a specific scale it is necessary to fix the scale and origin. The Gregorian and Muslim calendars are two specifications of interval scales.
Thus, when moving to equivalent scales using linear transformations in interval scales, a change occurs in both the origin (parameter b ), and the measurement scale (parameter a ).
Interval scales, like nominal and ordinal scales, preserve the distinction and ordering of the objects being measured. However, in addition to this, they also preserve the relation of distances between pairs of objects. Record
means that the distance between and in TO times greater than the distance between x 3 and x 4 and in any equivalent scale this value (the ratio of the differences in numerical estimates) will be preserved. In this case, the relations between the estimates themselves are not preserved.
In sociological research Interval scales usually measure time and spatial characteristics objects. For example, dates of events, length of service, age, time for completing tasks, differences in marks on a graphic scale, etc. However, directly identifying the measured variables with the property being studied is not so simple.
As another example, consider a mental ability test that measures the time it takes to solve a problem. Although physical time is measured on an interval scale, time used as a measure of mental ability belongs to an order scale. In order to construct a more advanced scale, it is necessary to explore the richer structure of this property.
Common mistake: properties measured on an interval scale are taken as indicators of other properties that are monotonically related to the data. When used to measure related properties, the original interval scales become merely order scales. Ignoring this fact often leads to incorrect results.
The following two types of interval scales are most widely used when conducting sociological measurements.
Based on Likert scales the degree of agreement or disagreement of respondents with certain statements is studied. This scale is symmetrical in nature and measures the intensity of the respondents' feelings. For example, it contains the following gradations: completely agree (1); somewhat agree (2); I am neutral (3); somewhat disagree (4); completely disagree (5). The points assigned to the answers to the questionnaire questions contained in certain gradations are indicated in brackets.
Using a Likert scale, the opinion (attitude) of employees of an organization can be studied towards various management aspects: the work motivation system, the psychological climate in the team, the policy of innovation, etc.
There are various options for modifying the Likert scale, for example, introducing a different number of gradations (5-9).
Semantic differential scale(semantic differentiation) contains a series of bipolar definitions that characterize various properties of the object being studied. This scale was developed by the American scientist Charles Osgood to measure the meaning of concepts and words, and primarily to differentiate the emotional side of the object of measurement when studying social attitudes. In this way, a person’s reaction in relation to the object being studied was determined.
For example, when assessing the moral climate in a team, when developing a questionnaire, indicators characterizing it are first selected (relationships between employees, relationships between managers, relationships between managers and subordinates, etc.). Then, for each indicator (questionnaire question), a scale is compiled, which is a continuum formed by a pair of antonymous adjectives. The continuum contains seven gradations of relationship intensity. For example, on a question characterizing relationships between employees, the scale has the following gradations:
Very good (+3);
Good (+2);
Rather good (+1);
Neither good nor bad (0)
Rather bad (-1);
Bad (-2);
Very bad (-3).
Each respondent expresses his attitude to the problem under study using the entire set of scales. This type of scale is also often used to determine the image of a brand, store, etc.
Relationship scale
Relationship scale (similarity) is called a scale if the set of admissible transformations consists of similarity transformations
wherea>0 are real numbers. It is easy to verify that in ratio scales the ratios of numerical estimates of objects remain unchanged. Indeed, let objects in one scale correspond to scale values and , and in the other and . Then we have:
This relationship explains the name of the ratio scales. Examples of measurements in ratio scales are measurements of the mass and length of objects. It is known that a wide variety of numerical estimates are used to establish mass. So, when measuring in kilograms, we get one numerical value, when measuring in pounds - another, etc. However, it can be noted that no matter what system of units the mass is measured in, the ratio of the masses of any objects is the same and does not change when moving from one numerical system to another, equivalent one. Measuring distances and lengths of objects has the same property.
As can be seen from the examples considered, relationship scales reflect the relationships between the properties of objects, i.e. how many times a property of one object exceeds the same property of another object.
Ratio scales form a subset of interval scales by fixing the zero value of the parameter b: b = 0. Such fixation means setting the zero point of reference for scale values for all ratio scales. The transition from one scale of relations to another scale equivalent to it is carried out using similarity (stretching) transformations, i.e. changing the measurement scale. Ratio scales, being a special case of interval scales, when choosing a zero reference point, preserve not only the relations of the properties of objects, but also the relations of distances between pairs of objects.
Difference scale
Difference scales are defined as scales that are unique up to shift transformations
b – real numbers. This means that when moving from one number system to another, only the origin changes. Difference scales are used in cases where it is necessary to measure how much one object is superior to another object in a certain property. In difference scales, the differences in numerical estimates of properties remain unchanged. Indeed, if -
assessments of objects and on the same scale, and 
And 
-
on another scale, we have:
Examples of measurements on difference scales include measurements of the increase in enterprise production (in absolute units) in the current year compared to the previous year, an increase in the number of institutions, the amount of equipment purchased per year, etc.
Another example of a measurement on a difference scale is chronology (in years). The transition from one chronology to another is carried out by changing the starting point.
Like ratio scales, difference scales are a special case of interval scales obtained by fixing a parameter a (a= 1), i.e. choosing a measurement scale unit. The starting point in difference scales can be arbitrary. Difference scales, like interval scales, preserve the relations of intervals between assessments of pairs of objects, but, unlike the ratio scale, they do not preserve the relations of assessments of the properties of objects.
Absolute scale
Absolute scale –in which the only valid transformations are identity transformations: . This means that there is only one mapping of empirical objects into a numerical system. Hence the name of the scale, since for it the uniqueness of measurement is understood in a literal absolute sense.
Absolute scales are used, for example, to measure the number of objects, objects, events, decisions, etc. The following are used as scale values when measuring the number of objects: natural numbers, when objects are represented by whole units, and real numbers, if in addition to whole units there are also parts of objects.
Absolute scales are a special case of all previously considered types of scales, therefore they preserve any relationship between the numbers of estimates of the measured properties of objects: difference, order, ratio of intervals, ratio and difference of values, etc.
In addition to these, there are intermediate types of scales, such as, for example power scale() and its variety logarithmic scale ().
Figure 3.6 shows the relationship between the main types of scales in the form of a hierarchical structure of the main scales.
Fig.3.6. Hierarchical structure of the main scales
Here the arrows indicate the inclusion of sets of admissible transformations of more “strong” to less “strong” types of scales. Moreover, the scale is “stronger” the less freedom in choice . Some scales are isomorphic, i.e. equivalent. For example, the interval scale and the power scale are equivalent. The logarithmic scale is equivalent to the difference scale and the ratio scale.
Naming and order scales are quality scales. The scale of names describes the difference or equivalence of objects, and the scale of order describes the qualitative superiority, difference of objects. In these scales there is no concept of origin and scale of measurement.
Interval, ratio, difference and absolute scales are quantitative scales. In these scales there are concepts of origin and scale, which are chosen arbitrarily. Quantitative scales allow you to measure how much (interval and difference scales) or how many times (ratio and absolute scales) one object differs from another according to a selected indicator.
The choice of a particular scale for measurement is determined by the nature of the relationships between the objects of the empirical system, the availability of information about these relationships and the goals of decision-making. The use of quantitative scales requires significantly more complete information about objects compared to the use of qualitative scales.
Attention should be paid to the correct alignment of the selected measurement scale with the goals of the solution. For example, if the goal of the decision is to organize objects, then there is no need to measure the quantitative characteristics of objects, it is enough to determine only quality characteristics. A typical example of such a solution is the determination of the best enterprises. To solve this problem, as a rule, it is not necessary to determine how much or how many times one object is better than another, i.e. There is no need to use quantitative scales for this measurement.
Measuring scales
The term "scale" comes from the Latin word "Scala", which means ladder.
A measurement scale is an agreed-upon procedure for determining and designating all possible manifestations of a particular property (for example, size values). There are five main types of measurement scales: names, order, intervals (differences), ratios and absolute scales.
Name scale .
These are the simplest scales that reflect quality properties. Their elements are characterized only by relations of equivalence (equality) and similarity of specific qualitative manifestations of the property.
These scales do not have a zero and a unit of measurement; they do not have comparison relations of the “more-less” type. Arithmetic operations cannot be performed on the naming scale.
Measurement comes down to comparing the measured object with the reference ones and selecting one of them (or two neighboring ones) that matches what is being measured. Measurements in naming scales are performed quite often. The results of a qualitative analysis (determination of blood group) are measurements on a scale of names.
Order scale .
Order scale. Comparison of one size with another based on the principle of “which is larger” or “which is better” is made on a scale of order. These scales do not have units of measurement. More detailed information on how much more or how many times better is sometimes not required. By ranking people by height, using a scale of order, one can draw a conclusion about who is taller, but it is impossible to say how much taller or how much.
Arranging dimensions in ascending or descending order to obtain measurement information on a scale of order is called ranking. On the order scale, sizes that remain unknown are compared with each other. The result of the comparison is a ranked series.
Measurements on the order scale are the most imperfect, the least informative. They do not answer the question of how much or how many times one size is larger than another. Only certain logical operations can be performed on the order scale. For example, if the first size is larger than the second, and the second is larger than the third, then the first is larger than the third. If two sizes are smaller than the third, then their difference is less than the third.
These scale properties are called transitivity properties. At the same time, no arithmetic operations can be performed on the order scale.
Order scale measurements are widely used in control. Here, the verified size Q 1 is compared with the control Q 2. The result of the measurement is a decision about whether the product is suitable or unsuitable according to the controlled size.
A classic example is the assessment of mineral hardness based on the Mohs scale. The Mohs scale of relative hardness of minerals consists of 10 hardness standards: talc -1; gypsum - 2; calcite - 3; fluorite - 4; apatite - 5; ortho-eye - 6; quartz - 7; topaz - 8; corundum - 9; diamond - 10. Relative hardness is determined by scratching the surface of the test object with a standard. Typically, an order scale is used when there is no method that allows assessment in established units of measurement.
Reference scales.
To facilitate measurements on the order scale, you can
fix some reference points as “reference points”. Such a scale is called a reference scale.
Points on reference scales can be assigned numbers called points.
0 1 2 3 4 5 6 7 8
The following are measured using reference scales:
earthquake intensity on the 12-point international scale MSK – 64 (Table 1);
wind force on the Beaufort scale (Table 2);
the strength of sea waves;
film sensitivity;
degree of ice hummocking;
hardness of minerals, etc.
For example, to assess the speed (strength) of wind in points based on its effect on ground objects or on sea waves, a conventional F scale was compiled by Beaufort in 1805. The relationship between points and wind speed at a height of 10 m was adopted in 1946 by the international agreement.
The disadvantage of reference scales is the uncertainty of the intervals between reference points. Therefore, scores cannot be added, subtracted, multiplied or divided. Measuring information obtained on the order scale is unsuitable for mathematical processing. It is also impossible to make a correction to the measurement result, because if the dimensions being compared themselves are unknown, then making a correction does not bring clarity.
Table 1
|
Name |
Brief description |
|
|
Unnoticeable |
Marked only by seismic instruments |
|
|
Very weak |
Felt by individuals who are at rest |
|
|
Felt by a small part of the population. |
||
|
Moderate |
Recognized by small rattling and vibrations of objects and window glass, creaking of doors and walls. |
|
|
Quite strong |
General shaking of buildings, vibrations of furniture, cracks of window glass and plaster, awakening of sleepers. |
|
|
It is felt by everyone. Paintings fall from walls, pieces of plaster break off, causing minor damage to buildings. |
||
|
Very strong |
Cracks in the walls of stone houses. Anti-seismic and wooden buildings remain unharmed |
|
|
Destructive. |
Cracks in steep slopes and wet soil. Monuments move or fall. Houses are heavily damaged. |
|
|
Devastating |
Severe damage and destruction of stone houses. |
|
|
Destroying |
Large cracks in the soil. Landslides and collapses. Destruction of stone buildings, bending of railway rails. |
|
|
Catastrophe |
Wide cracks in the ground. Numerous landslides and collapses. Stone houses are completely destroyed. |
|
|
Strong disaster |
The change in the soil reaches enormous proportions. Numerous collapses, landslides, cracks. The emergence of waterfalls and dams on lakes. River flow deviation. Not a single structure can withstand. |
Table 2
|
Name of the wind |
Action |
|
|
Smoke goes vertically |
||
|
The smoke goes slightly oblique |
||
|
You can feel it on your face, the leaves are rustling. |
||
|
Flags are flying |
||
|
Moderate |
Dust rises |
|
|
Causes waves on the water |
||
|
Whistling in the shrouds, wires humming |
||
|
Foam forms on the waves |
||
|
Very strong |
It's hard to go against the wind |
|
|
Tears off tiles |
||
|
Severe storm |
Uproots trees |
|
|
Fierce Storm |
Great destruction. |
|
|
Devastating effect |
Interval scale .
More advanced in this regard are interval scales composed of strictly defined intervals. The interval scale shows the difference between sizes. It is generally accepted to measure time on a scale divided into intervals equal to the period of the Earth’s revolution around the Sun (chronology). These intervals (years) are in turn divided into smaller ones (days), equal to the period of rotation of the Earth around its axis. The day is in turn divided into hours, hours into minutes, minutes into seconds. This scale is called an interval scale
The interval scale defines mathematical operations such as addition and subtraction . Signed intervals can be added to and subtracted from each other. Thanks to this, you can determine how much one size is larger or smaller than another.
Due to the uncertainty of the starting point on the interval scale, it is impossible to determine how many times one size is larger or smaller than another.
Sometimes interval scales are sometimes obtained by proportionally dividing the interval between fiducial points. Thus, on the Celsius temperature scale, the temperature of ice melting is taken as the starting point. All other temperatures are compared with it. For ease of use of the interval scale, the scale between the melting temperature of ice and the boiling point of water is divided into 100 equal intervals - gradations or degrees. The entire Celsius scale is divided into degrees, both towards positive and negative intervals.
On the Reaumur temperature scale, the same melting temperature of ice is taken as the starting point, but the interval between this temperature and the boiling point of water is divided into 80 equal parts. Thus, a different temperature gradation is used: the Réaumur temperature is higher than the Celsius temperature.
On the Fahrenheit temperature scale, the same interval is divided into 180 parts. Therefore, a degree Fahrenheit is less than a degree Celsius. In addition, the beginning of the intervals on the Fahrenheit scale is shifted by 32 0 towards low temperatures.
Dividing the scale into torn parts - gradations - establishes a scale on it and allows you to express the measurement result in a numerical measure.
Relationship scale.
If, as one of the two reference points, we choose one in which the size is not assumed to be equal to zero, but is actually equal to zero, then on such a scale we can count the absolute value of the size and determine how many times one size is larger or smaller than the other. This scale is called the ratio scale. An example is the Kelvin temperature scale. In it, absolute zero temperature is taken as the starting point, at which the thermal motion of molecules stops. The second reference point is the melting temperature of ice. On the Celsius scale, the interval between these reference points is 273.16 0 C. Therefore, on the Kelvin scale, the interval between these points is divided into 273.16 parts. Each such part is called Kelvin and is equal to a degree Celsius, which facilitates the transition from one scale to another.
The relationship scale is the most perfect, the most informative. All mathematical operations are defined on it: addition, subtraction, multiplication and division. It follows that values of any size on the ratio scale can be added, subtracted, multiplied and divided. Therefore, it is possible to determine how much or how many times one size is larger or smaller than another.
Depending on what intervals the scale is divided into, the same size is marked differently. For example, 0.001 km; 1 m; 100 cm; 1000 m – four presentation options of the same size. They are called the values of the quantity being measured.
Thus, value of the measured quantity – it is an expression of its size in certain units of measurement. The abstract number included in it is called numerical value.
The value of the measured quantity Q is determined by its numerical value g and a certain size 
, taken as a unit of measurement:
.
(53)
where Q is the measured value;
- unit of measurement;
g – numeric value.
Absolute scales . They have all the properties of ratio scales. The units of absolute scales are natural and not chosen by convention, but these units are dimensionless (times, percentages, fractions, complete angles, etc.). Units of quantities described as absolute are not derived SI units, since by definition derived units cannot be dimensionless. These are non-systemic units. The steradian and radian are typical units of absolute scales. Absolute scales can be limited or unlimited.
Limited scales are usually scales with a range from zero to one (efficiency, absorption or reflection coefficient, etc.). Examples of unlimited scales are scales that measure gain, attenuation, etc.
These scales are fundamentally nonlinear. Therefore they do not have units of measurement.