Events C
	expert j = 1
experts a ij	expert j = 2
	expert j = 1
importance a ij	expert j = 2
The total rank of importance a i

The average value for the total ranks of the considered series

The total square deviation S of total events from the mean value a is

called the concordance coefficient. The value of W varies from 0 to 1. At W = 0, there is absolutely no consistency; there is no connection between the assessments of various experts. On the contrary, at W = 1 the agreement of expert opinions is complete.

If the sequence (5.2) has equalities in addition to strict inequalities, i.e., there is a coincidence of ranks, then the formula for calculating the concordance coefficient has the form

When the ranks are repeated, to obtain a normal ranking that has a mean rank equal to

must be attributed to events that have same ranks, a rank equal to the average of the places that these events shared among themselves.

For example, the following ranking of events is obtained:

Ranks a i

Events 2 and 5 shared the second and third places. So they are ranked

events 3, 4 and 6 shared the fourth, fifth, sixth places, and they are assigned the rank

Thus, we get a normal ranking:

Ranks a" i

Example. Consider ranking m=10 events p=3 by experts;N,Q,R. The calculation results are presented in Table. 5.3.

For the extreme values ​​of the concordance coefficient, the following assumptions can be made. If W= 0, then there is no consistency in the estimates, therefore, in order to obtain reliable estimates, it is necessary to clarify the initial data on events and (or) change the composition of the expert group. When W = 1, it is far from always possible to consider the obtained estimates as objective, since sometimes it turns out that all members of the expert group agreed in advance, protecting their common interests.

It is necessary that the found value of W be greater than set value W 3 (W>W 3). You can take W 3 = 0.5, i.e. at W > 0.5, the actions of experts are more coordinated than not coordinated. For W< 0,5 полученные оценки нельзя считать достоверными, и поэтому следует повторить опрос заново. Жесткость данного утверждения опреде­ляется важностью проводимого исследования и возможностью повторной экспертизы. Практика показывает, что очень часто этим требованием пренебрегают.

The calculation of the coefficient W, taking into account the competence of experts, is given in the work.

Using an ordinal scale allows you to assign ranks to objects according to some attribute. Thus, metric values ​​are translated into rank values. At the same time, differences in the degree of manifestation of properties are fixed. In the ranking process, 2 rules should be followed.

Rank order rule. It is necessary to decide who gets the first rank: the object with the highest degree of expression of any quality, or vice versa. Most often it is absolutely indifferent and does not affect the final result. It is traditionally accepted to attribute the first rank to objects with a greater degree of quality (higher value - lower rank). For example, the champion is awarded first place, and not vice versa. Although, here, if the reverse order had been adopted, the results would not have changed from this. So each researcher has the right to determine the order of ranking himself. For example, E.V. Sidorenko recommends assigning a lower rank to a lower value. In some cases, this is more convenient, but more unusual.

For example: there is an unordered sample whose data needs to be ranked. (2, 7, 6, 8, 11, 15, 9). After ordering the sample, we rank it.

Metric data		Alternative option:	Metric data

Separately, the following should be said. There is a group of rarely used non-parametric tests (Wilcoxon T-test, Mann-Whitney U-test, Rosenbaum Q-test, etc.), when working with which you should always assign a lower rank to a lower value.

Linked rank rule. Objects with the same severity of properties are assigned the same rank. This rank is the average of the ranks they would have received had they not been equal. For example, you need to rank a sample containing a series of identical metric data: (4, 5, 9, 2, 6, 5, 9, 7, 5, 12). After ordering the sample, the arithmetic mean of the associated ranks should be calculated.

Metric data	Pre-Ranking	Final ranking

Assignments for independent work.

Rank the sample according to the rule " greater value- lower rank": (111, 104, 115, 107, 95, 104, 104).

Rank the sample according to the rule "lower value - lower rank" (20, 25, 8, 7, 20, 14, 27).

Combine the previous two samples and rank according to the rule "higher value - lower rank"

Indicators of which features from Table I are nominative, which are metric?

Translate the awareness indicators from Table I of the Appendix into a rank scale. Highlight the levels of severity of indicators by converting them into a nominative scale.

Table I Data for processing

students

university profile

awareness

hidden figures

missed

arithmetic

comprehension

exception images

analogy

number series

conclusions

geometric addition

memorization of words

average IQ

extraversion- introversion

neuroticism

average mark

University profile: 0 - student's choice of a humanitarian profile;

1 - student's choice of a mathematical or natural science profile

Quite well approximates R. s. T, and the difference is negligible when . If the hypothesis H 0 is true, according to which component X 1 , ... , X n random vector X are independent random variables, R.'s projection with. To is determined by the formula

where (see ).

There is an internal communication between R. of page. and . As shown in , if the hypothesis H 0 is true, the projection Kendall's correlation coefficient into the family of linear R. s. up to constant factor coincides with the Spearman rank correlation coefficient, namely:

It follows from this equality that the correlation coefficient corr between and is equal to

i.e., for large nP. With. and are asymptotically equivalent (see ).

Lit.: Gaek Ya., Shidak Z., Theory of rank criteria, trans. from English, M., 1971; K e n d a l l M. G., Rank correlation methods, 4ed., L., 1970. M. S. Nikulin.

Mathematical Encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "RANK STATISTICS" is in other dictionaries:

ranking statistics- - [A.S. Goldberg. English Russian Energy Dictionary. 2006] Topics energy in general EN rank statistics … Technical Translator's Handbook

This term has other meanings, see Statistics (meanings). statistics (in narrow sense) is measurable numeric function from the sample, independent of the unknown parameters of the distribution. AT broad sense term (mathematical) ... ... Wikipedia

- (statistics) 1. Set of data and mathematical methods used to study relationships between different variables. It includes methods such as linear regression(linear regression) and rank correlation. 2. Values ​​​​used ... ... Economic dictionary

STATISTICS- 1. A type of activity aimed at obtaining, processing and analyzing information that characterizes the quantitative patterns of life on the island in all its diversity, inextricably linked with its qualitative content. In a narrower sense of the word ... ... Russian sociological encyclopedia

- (non parametric statistics) Statistical technique, which do not allow special functional forms for relationships between variables. Rank correlation two variables is an example. The use of such technical ... ... Economic dictionary- K. m., which received their name. due to the fact that they are based on the “co relation” of variables, they are statistical methods, the beginning of which was laid in the works of Karl Pearson approximately in late XIX in. They are closely related to... Psychological Encyclopedia

Developer Digital Illusions CE Publisher ... Wikipedia

Karl Pearson Karl (Carl) Pearson Date of birth ... Wikipedia

When exhibiting expert assessments or in other ranking cases, there are situations where two or more more qualities are assigned the same ranks. In this case, the ranking rules are:

1. The smallest numerical value is assigned a rank of 1.

2. The highest numerical value is assigned a rank equal to the number of ranked values.

3. If there are several initial numerical values are equal, they are assigned a rank equal to average those ranks that these quantities would receive if they were in order one after the other and were not equal.

Note that both the first and last values ​​of the initial series for ranking can fall under this case.

4. total amount real ranks should coincide with the calculated one, determined by formula (1).

For example, a psychologist received from 11 subjects the following values indicator of non-verbal intelligence: 113, 107, 123, 122, 117, 117, 105, 108, 114, 102, 104. It is necessary to rank these indicators.

No. of examinees p / p	IQ	Conditional ranks	Ranks
		(8)	8,5
		(9)	8,5

Because 5 and 6 subjects have equal intelligence indicators, then they need to put conditional ranks, necessarily going in order one after another - and mark these ranks parentheses- (). But since they should have the same ranks. Then in the ranks column we must put the arithmetic mean of the ranks in brackets, i.e. . Often conditional and real ranks are written in one column.

Let's check the correctness of the ranking according to the formula (1):

Let's sum the real ranks: 6+4+11+10+8.5+8.5+3+5+7+1+2=66.

Because sums match, then the ranking is correct.

The rank scale uses many statistical methods. The Spearman and Kendall correlation coefficients are most often applied to the measurements obtained in this scale, in addition, various criteria for differences are used in relation to the data obtained in this scale.

Interval scale

In the interval scale, each of the possible values ​​of the measured quantities is separated from the nearest one by equal distance. The main concept of this scale is interval, which can be defined as the proportion or part of a measurable property between two adjacent positions on a scale.

Interval size- the value is fixed and constant in all sections of the scale. For measurement by means of a scale of intervals, special units of measurement are established, in psychology this is walls. When working with this scale, the measured property or object is assigned a number, equal to the number units of measure, equivalent to the quantity of the property. An important feature interval scale is that it does not have a natural reference point (zero is arbitrary and does not indicate the absence of a measurable property).

So, in psychology, Ch. Osgood's semantic differential is often used, which is an example of measuring various psychological features personality, social attitudes, value orientations, subjective-personal meaning, various aspects self-esteem.

3 - 2 - 1 0 +1 +2 +3

Absolutely Don't know Absolutely

disagree (not sure) agree

However, as S. Stevens and a number of other researchers emphasize, psychological measurements on the scale of intervals in essence often turn out to be measurements made on the scale of orders. The basis for this assertion is the fact that functionality people change depending on different conditions. When measuring, for example, strength with a dynamometer or attention span with a stopwatch, the measurement results at the beginning and end of the experiment will not be quantified at equal intervals due to the subject's fatigue.

Only measurement according to strictly standardized test methodology, provided that the distribution of values ​​in a representative (see below) sample is sufficiently close to normal (see below), can be considered a measurement on an interval scale. An example of the latter is standardized intelligence tests, where the conventional unit of IQ is equivalent at both low and high intelligence values.

It is also of fundamental importance that the experimental data obtained on this scale can be big number statistical methods.

Relationship scale

The relationship scale is called also scale equal relationship. A feature of this scale is the presence of a firmly fixed zero, which means complete absence any property or feature. The ratio jackal is the most informative scale that allows any mathematical operations and the use of various statistical methods.

The scale of ratios is essentially very close to the interval scale, because if we strictly fix the origin, then any interval scale turns into a scale of ratios.

It is in the scale of ratios that accurate and ultra-precise measurements are made in such sciences as physics, chemistry, microbiology. Measurements on the scale of relations are also made in sciences close to psychology, such as psychophysics, psychophysiology, and psychogenetics.