The problem of ensuring high product quality is closely related to the problem of measurement quality. There is clearly a direct connection between them: where the quality of measurements does not meet the requirements of the technological process, it is impossible to achieve a high level of product quality. Therefore, product quality largely depends on the successful solution of issues related to the accuracy of measuring the quality parameters of materials and components and maintaining specified technological conditions. In other words, technical quality control carried out by measuring the parameters of technological processes, the measurement results of which are necessary for regulating the process.

Hence, quality of measurements is a set of properties of the measurement state that provide measurement results with the required accuracy characteristics, obtained in the required form over a certain period of time.

Basic properties of the measurement state:

Accuracy of measurement results;

Reproducibility of measurement results;

Convergence of measurement results;

Quick results;

Unity of measurements.

At the same time, under reproducibility of measurement results is understood as the closeness of measurement results of the same quantity, obtained in different places, by different methods, by different means, by different operators, at different times, but under the same measurement conditions (temperature, pressure, humidity, etc.).

Convergence of measurement results- this is the closeness of the results of measurements of the same quantity, carried out repeatedly using the same means, by the same method under the same conditions and with the same care.

Any measurement or quantification of something is carried out using appropriate scales.

Scale- this is an ordered series of marks corresponding to the relationship between successive values ​​of the measured quantities. Measurement scale is an agreed-upon sequence of values ​​of quantities of the same name of different sizes.

In metrology, a measurement scale is a means of adequately comparing and determining the numerical values ​​of individual properties and qualities of various objects. In practice, five types of scales are used: a name scale, an order scale, an interval scale, a ratio scale and an absolute value scale.

Name scale (nominal scale). This is the simplest of all scales. In it, numbers act as labels and serve to detect and distinguish the objects being studied. The numbers that make up the naming scale are allowed to be swapped. There are no more-less relationships in this scale, so some believe that the use of a naming scale should not be considered a measurement. When using a naming scale, only certain mathematical operations can be performed. For example, its numbers cannot be added and subtracted, but you can count how many times (how often) a particular number appears.

Order scale. The places occupied by quantities on the order scale are called ranks, and the scale itself is called rank, or non-metric. In such a scale, its constituent numbers are ordered by rank (i.e., occupied places), but the intervals between them cannot be accurately measured. Unlike the naming scale, the order scale allows not only to establish the fact of equality or inequality of measured objects, but also to determine the nature of inequality in the form of judgments: “more-less”, “better-worse”, etc.

Using order scales, you can measure qualitative indicators that do not have a strict quantitative measure. These scales are used especially widely in the humanities: pedagogy, psychology, sociology. A greater number of mathematical operations can be applied to the ranks of the order scale than to the numbers of the name scale.

Interval scale. This is a scale in which numbers are not only ordered by rank, but also separated by certain intervals. The feature that distinguishes it from the ratio scale described below is that the zero point is chosen arbitrarily. Examples include calendar time (the beginning of chronology in different calendars was set for random reasons, temperature, potential energy of a lifted load, electric field potential, etc.).

The results of measurements on an interval scale can be processed by all mathematical methods, except for calculating ratios. Interval scale data provides an answer to the question “how much more?”, but does not allow us to state that one value of a measured value is so many times greater or less than another. For example, if the temperature increased from 10 to 20°C, then it cannot be said that it has become twice as warm.

Relationship scale. This scale differs from the interval scale only in that it strictly defines the position of the zero point. Thanks to this, the ratio scale does not impose any restrictions on the mathematical apparatus used to process observational results.

The ratio scale also measures those quantities that are formed as differences between numbers measured on the interval scale. Thus, calendar time is counted on a scale of intervals, and time intervals - on a scale of ratios.

When using a ratio scale (and only in this case!), the measurement of any quantity is reduced to the experimental determination of the ratio of this quantity to another similar one, taken as a unit. By measuring the length of an object, we find out how many times this length is greater than the length of another body taken as a unit of length (a meter ruler in this case), etc. If we limit ourselves only to the use of ratio scales, then we can give another (narrower, more specific) definition of measurement: to measure a quantity means to experimentally find its relation to the corresponding unit of measurement.

Absolute value scale. In many cases, the magnitude of something is directly measured. For example, the number of defects in a product, the number of units produced, how many students are present at a lecture, the number of years lived, etc. are directly calculated. and so on. For such measurements, the measuring scale is marked

absolute quantitative values ​​of what is being measured. Such a scale of absolute values ​​has the same properties as a scale of ratios, with the only difference being that the values ​​indicated on this scale have absolute rather than relative values.

Measurement results on a scale of absolute values ​​have the greatest reliability, information content and sensitivity to measurement inaccuracies.

Scales of intervals, ratios and absolute values ​​are called metric, since in their construction some measures are used, i.e. dimensions accepted as units of measurement.

measuring instrument, which is an ordered series of marks along with the numbering associated with them or a technical mark on the scale of a measuring device. The scales can be located in a circle, arc or straight line. Indications are taken with the naked eye at distances between divisions of up to 0.7 mm, with smaller ones - using a magnifying glass or microscope; additional scales - verniers - are used for fractional assessment of divisions.

It should be noted that the term “scale” in metrological practice has at least two different meanings. Firstly, a scale or, more precisely, a measurement scale (scale of a physical quantity) is the agreed-upon procedure for determining and designating all possible manifestations (values) of a specific property (quantity). Secondly, the reading devices of analog measuring instruments are called a scale; this value is used in this article.

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Scale elements

• A scale mark is a sign on a scale (dash, tooth, dot, etc.) corresponding to a certain value of a physical quantity.
• Numerical scale mark - a scale mark that has a number attached to it.
• Zero mark - scale mark corresponding to the zero value of the measured value.
• Scale division is the gap between two adjacent scale marks.
• The length of a scale division is the distance between the axes (or centers) of two adjacent scale marks, measured along an imaginary line passing through the midpoints of the shortest scale marks.
• The value of a scale division is the difference in value values ​​corresponding to two adjacent scale marks.
• Scale length is the length of a line passing through the centers of all the shortest scale marks and limited by the start and end marks. A line can be real or imaginary, curved or straight.

Scale division interval (scale division) - the distance between the axes of symmetry of two adjacent strokes (expressed in linear or angular units)

Types of measuring instrument scales

• Single sided scale- scale with a zero mark located at the beginning or end of the scale
• Double sided scale- a scale with a zero mark located between the start and end marks. There are symmetrical (start and end marks correspond to the same values ​​of the measured quantity) and asymmetrical double-sided scales (start and end marks correspond to different values).

Properties of scales

• The initial scale value is the smallest value of the measured quantity that can be counted on the scale of the measuring instrument. In many cases, the scale starts at zero, but there may be other values ​​- for example, for a medical thermometer it is 34.3 °C.
• The final value of the scale is the largest value of the measured quantity that can be counted on the scale of the measuring instrument.
• The nature of the scale is functional dependence a = f(x) between linear (or angular) distance a any mark from the initial scale mark, expressed in fractions of the entire scale length, and the value x measured value corresponding to this mark:
  • Uniform scale - a scale on which the marks are applied evenly.
  • An uneven scale is a scale on which the marks are applied unevenly.
  • Logarithmic or hyperbolic scale - a scale with tapering divisions, characterized by the fact that the mark corresponding to the half-sum of the initial and final values ​​is located between 65 and 100% of the scale length. It should be noted that the expression “logarithmic scale” is also used in relation to another meaning of the concept “scale” (see: Scale of a physical quantity, Logarithmic scale).
  • Power scale - a scale with expanding or contracting divisions, but not falling under the definition of a logarithmic (hyperbolic) scale.

A measurement scale is a set of values ​​that allow you to quantitatively or qualitatively display the properties of a measured object. 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 scales measurements these properties. The measurement scale for a quantitative property is a scale for a physical quantity. Physical quantity scale is an ordered sequence of PV values, adopted by agreement based on the results of accurate measurements.

Types of measurement scales

In practical activities, it is necessary to measure various quantities that characterize the properties of bodies, substances, phenomena and processes. Some properties of measuring scales in metrology appear only qualitatively, others - quantitatively.

Scale – an ordered numerical or symbolic series of values, reflecting acceptable variations in the values ​​of the measured quantity.

In accordance with the logical structure of the manifestation of properties, there are five main types of measurement scales: naming scales, order scales, interval scales, ratio scales, absolute scales.

Nominal scale (name scale)

Figure - Example of a nominal scale (color atlas)

Such measurement scales in metrology are used to classify empirical objects, the properties of which are manifested only in relation to equivalence; these properties cannot be considered physical quantities, therefore scales of this type are not PV scales. Nominal scales, or, as they are also called, naming scales, are also called measurement scales, or classification scales. This is the simplest type of scale, based on assigning numbers to the qualitative properties of objects, playing the role of names.

In nominal 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 nominal scales are the widely used color atlases intended for color identification.

Scale of order (ranks)

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 application 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. An example of an order scale is the Engler viscosity scale, the 12-point Beaufort scale for sea wind strength.

Figure - Example of an order scale (Beaufort scale)

Order measurement 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 the required value should be considered assessment. Assessment on order scales is ambiguous and very conditional, as evidenced by the example considered.

Interval (difference) scale

These measurement scales in metrology are a further development of 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. An example of an interval scale is 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.

Figure - Example of an interval scale (Celsius and Fahrenheit temperature 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 scale of Q value intervals is described by the equation Q = Q o + q[Q], where q is the numerical value of the quantity; Q o - the beginning of the scale; [Q] - unit of the quantity under consideration. Such a scale is completely determined by specifying the starting point Q of the scale and the unit of the given value [Q].

There are practically two ways to set the scale. In the first of them, two values ​​of Q o and Q 1 are selected, values ​​that are relatively simply implemented physically. These values ​​are called reference points, or main rappers, and the interval (Q 1 ~ Q o) - main interval. Point Q o is taken as the origin, and the value (Q 1 -Q o)/n= as unit Q. In this case, n is chosen such that [Q] is an integer value.

Figure - Example of a ratio scale

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 1c = 9192631770 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 under study is taken as the reference point.

Relationship scale

The scale of relations describes 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). An example of a ratio scale is 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, this type of measurement 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 physical quantities.

Figure - Example of an absolute scale (Kelvin temperature scale)

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 q 2 = q 1 /.

Absolute scales

Absolute scales are scales that have all the features 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. An example of an absolute scale can be scales with 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). Metric and absolute scales are classified as linear. The practical implementation of measurement scales in metrology is carried out by standardizing both the scales and units of measurement themselves, and, if necessary, the methods and conditions for their unambiguous reproduction.

Making a measuring scale with your own hands

Video about how to make a dial gauge yourself using the example of making an ammeter scale.

S. Stevens proposed a classification of four types of measurement scales: nominal, ordinal, interval and ratio scale.

Nominal scale(scale of names, nominative scale) consists of assigning a certain designation or symbol (numeric, alphabetic, etc.) to any property or feature. In essence, this is a classification of properties, grouping of objects, combining them into classes, provided that objects belonging to the same class are identical (or similar) to each other with respect to some attribute or property, while objects that differ in this attribute are fall into different classes.

Example: a) classification of taste qualities: A - sweet, B - bitter, C - sour; b) colors of the visible spectrum: red, green, blue, etc.; c) nationality: A is Belarusian, B is Russian, C is Ukrainian; d) dividing people into four types of temperament: sanguine, phlegmatic, melancholic, choleric.

The nominal scale determines that different properties or characteristics are qualitatively different from each other. Habitual operations with numbers - ordering, addition-subtraction, division - when measured on a nominative scale lose their meaning. So, for traits measured on this scale, it cannot be said that some of them are greater and some are less, some are better and some are worse. That is, when comparing objects, we can only conclude whether they belong to the same or different classes, whether they are identical or not in terms of the measured property.

It should be emphasized that the symbols assigned to objects in the nominative scale are conditional and any substitutions or rearrangements of letter (numeric) designations are allowed.

The simplest case of a nominative scale is a dichotomous scale. When measuring on this scale, the characteristics being measured can be encoded by two symbols or numbers, for example 0 and 1 or 3 and 5, or the letters A and B, as well as any two symbols that differ from each other. A trait measured on a dichotomous scale is called an alternative trait.

IN dichotomous scale all objects, signs or properties being studied are divided into two non-overlapping classes, and the researcher raises the question of whether the characteristic of interest “appeared” in the subject or not. For example, in a particular study, the sign of “left-handedness” appeared in 8 subjects out of 20, that is, 8 subjects can be given the number 1, corresponding to the sign of “left-handedness,” and the rest, the number 0, corresponding to the sign of “right-handedness.”

Example: a) classification by gender: 1 - male, 0 - female;
b) answers to the questionnaire: 1 - yes, 0 - no; c) family composition: A - complete family, B - single-parent family.

In the nominative scale, you can calculate the frequency of occurrence of a characteristic, that is, the number of subjects, phenomena, etc., falling into a given class and possessing a given property. Let's say we find out the number of boys and girls in a class. To do this, we code boys, for example, with the number 1, and girls with the number 0. After this, we count the total number of numbers (codes) 1 and 0. This is the calculation of the frequency of the attribute.

The unit of measurement with which we operate is the number of observations (subjects, reactions, choices, etc.), or frequency. More precisely, the unit of measurement is one observation. The total number of observations (subjects, reactions, choices, etc.) is taken as 100%, and then the percentage of, for example, boys and girls in a class can be calculated.

A small number of statistical methods can be applied to measurement results obtained on a nominative scale. Such data can be processed, for example, using the method %, binomial criterion m, Fisher's angular transformation φ, etc.

Ordinal scale(rank scale) is a scale that classifies according to the principle of “more - less”, “higher - lower”, “stronger - weaker”. Measurement on this scale involves assigning numbers to objects depending on the degree of expression of the property being measured. If in the previous scale it was unimportant in what order the measured traits were located, then in the ordinal scale all traits are arranged by rank - from the largest (tall, strong, smart, etc.) to the smallest (low, weak, stupid, etc.) . p.) or vice versa. A typical and very well-known example of an ordinal scale is school grades: from 5 to 1 points or from 0 to 10 points.

The ordinal scale must have at least three classes, for example, “positive reaction - neutral reaction - negative reaction” or “high - medium - low”, etc., so that the measured characteristics can be placed in order.

There are many ways to obtain a measurement on an ordinal scale. But the essence remains general: when comparing subjects with each other, we can say whether a property is more or less expressed, but we cannot say how much more or how much less it is expressed, much less how many times more or less. When measured on a rank scale, therefore, of all the properties of numbers, what is taken into account is that they are different, and that one number is greater than another.

Example: a) places taken by students in the competition (1, 2, 3); b) the student’s rank based on average academic score (1, 2, 3, 4, 5, 6, etc.); c) answers to the test: 1 - never, 2 - sometimes, 3 - often, 4 - always.

In an ordinal scale, we do not know the true distance between classes, but only that they form a sequence. From classes you can simply move on to numbers, if you consider 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 for statistical processing of the obtained data.

When coding ordinal variables, any numbers (codes) can be assigned to them, but order must be preserved in these codes (numbers), or, in other words, each subsequent number must be greater (or less) than the previous one. For example, it is necessary to encode the level of anxiety in five gradations: lowest - 1, low - 2, medium - 3, high - 4, highest - 5. You can use other encoding methods (for example, 14, 23, 34, 45, 56, respectively), but the originally proposed encoding method is the most familiar and therefore the most preferred. Numbers in rank scales indicate only the order of the signs, and operations with numbers in this scale are operations with ranks.

When ranking, two circumstances must be taken into account:
1. Set for yourself and remember the ranking order. You can assign rank 1 to the one that has the 1st place in terms of the severity of this characteristic (for example, “the strongest”). Or you can assign rank 1 to the one with the least severity of the trait, and then increase the rank as the level of the trait increases. There are no strict selection rules here, but it is important to remember in which direction the ranking was made. 2. Follow the ranking rule for related ranks when two or more subjects have the same expression of the property being measured. In this case, such subjects are assigned the same average rank. For example, if you rank subjects by “place in group” and two have the same highest raw scores, then both are assigned an average rank of 1.5: (1+2)/2=1.5. The next subject following this pair is assigned rank 3, etc. This rule is based on the convention of maintaining the same rank sum for related or unrelated ranks. In accordance with this rule, the sum of all assigned ranks for a group of N must be equal to N(N+1)/2, regardless of the presence or absence of connections in the ranks.

The ordinal scale uses a wide variety of statistical techniques. The most commonly used correlation coefficients for measurements obtained on this scale are Spearman and Kendall, and a variety of difference tests are used for data obtained on this scale.

Interval scale(interval scale) is a scale that classifies according to the principle “more by a certain number of units, less by a certain number of units.” Each of the possible values ​​of the attribute is located at an equal distance from the other. The main concept of this scale is the interval, which can be defined as the proportion or part of the property being measured between two adjacent positions on the scale. The size of the interval is a fixed and constant value in all areas of the scale. To measure using an interval scale, special units of measurement are established (in psychology, for example, walls and stenines). An object is assigned a number of units of measurement proportional to the severity of the property being measured. An important feature of the interval scale is that it does not have a natural reference point (zero is arbitrary and does not indicate the absence of the property being measured). Consequently, using this scale, we can judge how much more or how much less the property is expressed when comparing objects, but we cannot judge how many times more or less the property is expressed.

Example: a) temperature measurement on the Celsius scale (°C); b) intelligence tests (conventional unit of measurement IQ); c) Cattell's 16-factor questionnaire (raw scores converted into walls).

A fairly large number of statistical methods are applicable to experimental data obtained on this scale.

Relationship scale - This is a scale that classifies objects or subjects in proportion to the degree of expression of the property being measured. In ratio scales, classes are designated by numbers that are proportional to each other: 2 is to 4 as 4 is to 8. This assumes an absolute zero reference point, so when comparing objects we can say not only how much more or less a property is expressed, but also about how many times (by what percentage, etc.) more or less it is expressed. By measuring the time taken to solve a problem by a pair of subjects, we can say not only who solved the problem faster and by how many seconds (minutes), but also how many times faster.

It should be noted that, despite the familiarity and routineness of the absolute scale, it is not often used in psychology. The capabilities of the human psyche are so great that it is difficult to imagine absolute zero in any measurable psychological variable.

Example: a) measuring reaction time (usually in milliseconds); b) measurement of absolute sensitivity thresholds.

It is useful to characterize the listed scales based on their differentiating ability (power). In this regard, the scales, as power increases, are arranged as follows: nominal, ordinal, interval, ratio scale. Thus, non-metric scales are obviously less powerful - they reflect less information about the difference between objects (subjects) according to the measured property, and, on the contrary, metric scales are more powerful, since they differentiate subjects better. Therefore, if the researcher has a choice, it is necessary to use a more powerful scale. Another thing is that more often there is no such choice, and you have to use an available measuring scale.

Determining on what scale a phenomenon is measured (a sign is presented) is a key point in data analysis: the choice of method and interpretation of the results depends on this.

Usually, the identification of the nominative scale, its differentiation from the rank scale, and even more so from the metric scale, does not cause problems.

Example: Let’s consider the survey question “How confident are you in your abilities?” for an answer to which subjects choose one of the proposed options:

1) absolutely sure;

2) I find it difficult to answer;

3) completely unsure.

If a researcher is interested in the extent to which subjects are confident or unconfident in their abilities, then it is logical to assume that the sign is presented on an ordinal scale. If the researcher is interested in how the answers are distributed among the options or what characterizes each of the three corresponding groups, then it is more reasonable to consider this attribute as nominal.

It is much more difficult to determine the difference between the ordinal and metric scales. The problem stems from the fact that measurements in psychology are usually indirect. We directly measure some observable phenomena or events: the number of answers to questions or tasks solved in the allotted time, or the time it takes to solve a set of tasks, etc. But at the same time, we make judgments about some hidden, latent property that is inaccessible to direct observation: aggressiveness, sociability, ability, etc.

The number of problems solved in the allotted time is, of course, a measurement on a metric scale. But this quantity itself interests us only to the extent that it reflects some ability we are studying. Do equal differences in the solved problems correspond to equal differences in the expression of the property (ability) being studied? If the answer is “yes,” the scale is metric (interval or equal ratios), if “no,” the scale is ordinal.

In such situations, it is easiest to agree that the characteristic is presented on an ordinal scale. But at the same time, we significantly limit ourselves in the choice of methods for subsequent analysis. Moreover, the transition to a less powerful scale dooms us to the loss of some empirical information that is valuable to us. The consequence of this may be a decrease in the statistical reliability of the study results. Therefore, the researcher still strives to find evidence that the scale used is more powerful.

Tasks:

Determine on what scale each of the dimensions below is represented; names, order, intervals, relationships.

1. Ordering the subjects by time of solving the test problem.

2. Preference for pets: dogs, cats, rats, none.

3. Military rank (private, corporal, sergeant, lieutenant, captain) as a measure of promotion.

4. Number of aggressive reactions per day.

5. Academic status (assistant, associate professor, professor) as an indication of belonging to the corresponding category.

6. The subject's ordering of 18 instrumental values ​​(according to Rokeach) according to the degree of their significance for him.

7. Hair color (blonde, brunette, brown-haired, red).

8. Time to solve the problem.

9. Student status in the group (star, preferred, accepted, not accepted).

Bibliography

1. Ermolaev, O.Yu. Mathematical statistics for psychologists /
O.Yu. Ermolaev. - M.: MPSI: Flint. - 2002. – 325 p.

2. Nasledov, A.D. Mathematical methods in psychological research. Analysis and interpretation of data / A.D. Nasledov. - SPb.: Speech. - 2004.

3. Sidorenko, E.V. Methods of mathematical processing in psychology. – St. Petersburg: LLC “Rech” - 2004. – 350 p.

4. Burlachuk, L.F., Morozov S.M. Dictionary – reference book for psychodiagnostics / L.F. Burlachuk, S.M. Morozov - St. Petersburg: Peter Kom. - 1999. – 528 p.

5. Sukhodolsky, G.V. Mathematical methods in psychology / G.V. Sukhodolsky. - Kharkov: Publishing House Humanitarian Center. - 2006. – 512 p.

6. Tarasov, S.G. Fundamentals of the application of mathematical methods in psychology. / S.G. Tarasov. - St. Petersburg: Publishing house: St. Petersburg. un-ta. - 1999. – 326 p.

7. Glinsky, V.V., Ionin, V.G. Statistical data analysis /
V.V. Glinsky, V.G. Ionin. - M.: Owl. - 2008. – 265 p.

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 the 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 a further development of 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 a 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 under study 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 features 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 – a measurement in which the desired value of a quantity is found on the basis of a known relationship between this quantity and 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 between 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. The differential method can also provide high measurement accuracy if the known value is reproduced with high accuracy and the difference between it and the unknown value is small.

Example: measurement of DC voltage U x using a discrete voltage divider R U and a voltmeter V

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.

Measuring transformations 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. During the formation of 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- a 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.

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 the 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 distinctive feature is that they can be predicted, detected and, thanks to this, almost completely eliminated by introducing an appropriate correction.

It should be noted that recently the above definition of systematic error has been subject to justified 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.

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 records 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 the true value of the 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 the 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.

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, determined under normal conditions of its use. For each measuring instrument, the operating conditions are specified in the regulatory and technical documents - a set of influencing quantities (ambient temperature, humidity, pressure, voltage and frequency of the supply network, etc.), under 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.