Mann Whitney method of mathematical statistics. Nonparametric Methods for Comparing Two Samples and Their Application in R

Purpose of the criterion

U - the Mann-Whitney test is designed to assess the differences between two samples in terms of level any sign measured starting from the order scale (not lower). It allows you to identify differences between small samples, when n 1, n 2 3 or n 1 = 2, n 2 5, and is more powerful than the Rosenbaum criterion.

This method determines whether the zone of overlapping values ​​between two rows of ordered values ​​is small enough. In this case, the 1st row (group sample) is the row of values ​​in which the values, according to a preliminary estimate, are higher, and the 2nd row is the one where they are presumably lower.

The smaller the crossover area, the more likely the differences are to be significant. These differences are sometimes referred to as differences in location two samples.

The calculated (empirical) value of the criterion U reflects how large the zone of coincidence between the rows is. Therefore, the smaller U emp. the more likely that the differences are significant.

Criteria restrictions

The trait must be measured on an ordinal, interval, or proportional scale.

Samples must be independent.

Each sample must contain at least 3 observations: n 1 ,n 2  3 ; it is allowed that there are 2 observations in one sample, but then there must be at least 5 of them in the second.

Each sample should contain no more than 60 observations: n 1 ,n 2  60. However, already at n 1 ,n 2  20 ranking becomes quite laborious.

Algorithm for calculating the Mann-Whitney criterion.

To calculate the criterion, it is necessary to mentally combine all the values ​​of the 1st sample and the 2nd sample into one common combined sample and arrange them.

It is convenient to make all calculations in a table (table 28), consisting of 4 columns. This table contains the ordered values ​​of the combined sample.

Wherein:

merged sample values ​​are sorted in ascending order;

the values ​​of each of the samples are recorded in their own column: the values ​​of the 1st sample are recorded in column No. 2, the values ​​of the 2nd sample are recorded in column No. 3;

each value is written on a separate line;

the total number of rows in this table is N=n 1 +n 2 , where n 1 is the number of subjects in the 1st sample, n 2 is the number of subjects in the 2nd sample

Table 28

The values ​​of the combined sample are ranked according to the ranking rules, and the ranks R 1 corresponding to the values ​​of the 1st sample are written in column No. 1, the ranks R 2 corresponding to the values ​​of the 2nd sample are written in column No. 4,

The sum of ranks is calculated separately for column No. 1 (for sample 1) and separately for column No. 4 (for sample 2). Be sure to check if the total rank sum matches the calculated rank sum for the pooled sample.

Determine the larger of the two rank sums. Let's denote it as T x.

Determine the calculated value of the criterion U by the formula:

where n 1 - the number of subjects in sample 1,

n 2 - the number of subjects in sample 2,

T x - the larger of the two rank sums,

n x - the number of subjects in the sample with a larger sum of ranks.

Output rule: Determine the critical values ​​of U according to the table of critical values ​​for the Mann-Whitney test.

If U emp. U cr. 0.05, the differences between the samples are not statistically significant.

If U emp. U cr. 0.05, the differences between the samples are statistically significant.

How less value U, the higher the reliability of differences.

Test questions:

What are the conditions for applying the Student's criterion.

What parameters of feature distributions do you need to know in order to calculate Student's t-test?

Formulate a decision rule based on the results of calculations of the Student's criterion.

Why is it necessary to simultaneously evaluate the variability of features in samples when calculating the Student's t-test?

How can two variances be compared?

In what cases is it necessary to introduce the Snedekor correction into the derivation rule for the Student's criterion?

What are the conditions for applying the Rosenbuam criterion.

Formulate a decision rule based on the results of calculations of the Rosenbaum criterion.

List the conditions for applying the Mann-Whitney test.

What is the total pooled sample when calculating the Mann-Whitney test.

Formulate a decision rule based on the results of calculations of the Mann-Whitney criterion.

Independent practical task:

Study the Kruskal-Wallis criteria and the Jonkyer tendencies from textbooks on your own. Make a summary according to the scheme similar to the one used in the lectures.

Materials for studying the topic:

a) basic literature:

Ermolaev O. Yu. Mathematical statistics for psychologists [Text]: textbook / O. Yu. Ermolaev. - 5th ed. - M.: MPSI: Flinta, 2011. - 336 p. - S. 101-124; 169-172.

Nasledov A.D. Mathematical Methods psychological research: Analysis and interpretation of data [Text]: textbook / A. D. Nasledov. - 3rd ed., stereotype. - St. Petersburg: Speech, 2007. - 392 p. - S. 162-167; 173-176; 181-182.

Sidorenko E. V. Methods of mathematical processing in psychology [Text] / E. V. Sidorenko. - St. Petersburg: Speech, 2010. - 350 p.: ill. - S. 39-72.

b) additional literature:

Glass J. Statistical methods in pedagogy and psychology [Text]. / J. Glass, J. Stanley - M., 1976. - 494 p. - S. 265-280.

Kuteinikov A.N. Mathematical methods in psychology [Text]: educational and methodological complex / A. N. Kuteinikov. - St. Petersburg: Speech, 2008. - 172 p.: tab. - S. 81-93.

Sukhodolsky G.V. Fundamentals of mathematical statistics for psychologists [Text]: textbook / G. V. Sukhodolsky. - St. Petersburg: Publishing House of St. Petersburg State University, 1998. - 464 p. - S. 305-323.

Test

Methodology "House"

The "House" technique (N.I. Gutkina) is a task for drawing a picture with the image of a house, the individual details of which consist of elements of capital letters. The methodology is designed for children aged 5-10 years and can be used to determine the child's readiness for school.

Purpose of the study: to determine the child's ability to copy a complex pattern.

The task allows you to identify the child's ability to navigate the pattern, accurately copy it, determine the features of the development of involuntary attention, spatial perception, sensorimotor coordination and fine motor skills hands

materials: sample drawing, sheet of paper, pencil.

Research progress

Before completing the task, the child is given the instruction: “There is a sheet of paper and a pencil in front of you. Draw on this sheet exactly the same picture as here (a sheet with the image of a house is placed in front of the baby). Take your time, be careful, try to make your drawing exactly the same as on the sample. If you draw something wrong, do not erase with an elastic band (make sure that the child does not have an elastic band). It is necessary to draw correctly on top of the wrong drawing or near it. Do you understand the task? Then get to work."

In the course of the task, it is necessary to fix:

1. What hand does the child draw with (right or left).

2. How he works with the sample: how often he looks at it, whether he draws lines over the sample drawing that follow the contours of the picture, whether he compares what he has drawn with the sample or draws from memory.

3. Draws lines quickly or slowly.

4. Whether distracted during work.

5. Statements and questions while drawing.

6. Does he check his drawing with a sample after finishing work?

When the child reports the end of the work, he is invited to check whether everything is correct with him. If he sees inaccuracies in his drawing, he can correct them, but this must be fixed by the experimenter.

Processing and analysis of results

Processing of the experimental material is carried out by the method of scoring, which are awarded for errors. Errors are like this.

1. The absence of any detail of the picture (4 points). The picture may be missing a fence (one or two halves), smoke, a chimney, a roof, shading on the roof, a window, a line depicting the base of the house.

2. Enlargement of individual details of the drawing by more than two times while maintaining the size of the entire drawing relatively correctly (3 points for each enlarged detail).

3. An element of the picture is incorrectly depicted (3 points). Smoke rings, a fence, shading on a roof, a window, a chimney may be depicted incorrectly. Moreover, if the sticks that make up the right (left) part of the fence are drawn incorrectly, then 2 points are awarded not for each wrong stick, but for the entire right (left) part of the fence as a whole. The same applies to smoke rings coming out of the chimney and hatching on the roof of the house: 2 points are awarded not for each incorrect ring, but for all incorrectly copied smoke; not for every wrong line in the shading, but for the entire shading of the roof as a whole.

The right and left parts of the fence are evaluated separately: so, if it is incorrectly drawn right part, and the left one is copied without errors (or vice versa), then the child receives 2 points for the drawn fence; if mistakes were made in both the right and left parts, then 4 points (2 points for each part). If a part of the right (left) side of the fence is copied correctly, and a part is incorrect, then 1 point is awarded for this side of the fence; the same applies to smoke rings and shading on the roof: if only one part of the smoke rings is drawn correctly, then the smoke is estimated at 1 point; if only one part of the hatching on the roof is reproduced correctly, then the entire hatching is worth 1 point. An incorrectly reproduced number of elements in the detail of the drawing is not considered an error, that is, it does not matter how many sticks there are on the fence, smoke rings or lines in the hatching of the roof.

4. Incorrect arrangement of details in the space of the drawing (1 point). Errors of this type include: the location of the fence is not on a common line with the base of the house, but above it, the house seems to be hanging in the air or below the line of the base of the house; displacement of the pipe to the left edge of the roof; a significant shift of the window in any direction from the center; the location of the smoke is more than 30 ° deviation from the horizontal line; the base of the roof corresponds in size to the base of the house, and does not exceed it (in the sample, the roof hangs over the house).

5. Deviation of straight lines by more than 30° from the given direction (1 point): vertical and horizontal lines that make up the house and the roof; fence sticks; changing the angle of inclination of the side lines of the roof (their location at a right or obtuse angle to the base of the roof instead of a sharp one); deviation of the fence base line by more than 30 ° from the horizontal line.

6. Breaks between lines where they should be connected (1 point for each break). In the event that the hatch lines on the roof do not reach the roof line, 1 point is given for the entire hatch as a whole, and not for each incorrect hatch line.

7. Lines overlap each other (1 point for each overlap). If the hatch lines on the roof go beyond the roof lines, 1 point is given for the entire hatch as a whole, and not for each incorrect hatch line.

Good execution of the drawing is estimated at "0" points. Thus, the worse the task is performed, the higher the total score. However, when interpreting the results of the experiment, it is necessary to take into account the age of the child. Five-year-old children almost never get a "0" grade due to insufficient maturity of the brain structures responsible for sensorimotor coordination.

When analyzing children's drawing it is necessary to pay attention to the nature of the lines: very bold or “shaggy” lines may indicate a state of anxiety in the child. But in no case can a conclusion about anxiety be made only on the basis of this figure. Suspicions must be checked by special methods for determining anxiety.

Let's present the received data in the form of Histogram 1.

Histogram 1. Results obtained by the "House" method

Please build me a histogram like this. Delayed children mental development have above average (about 10%) and) average level of development (about 30% and below average (60%)

On average, children with normal development have a high level of development (about 60%), an average level of development (about 20%) and above average 20%. Here, too, you signed incorrectly for me, the teacher crossed out and said unreadable. you should have signed 10% above average and not low like in the 1st red column. In the 2nd red column, sign the average level of development (about 30%) and not low, and in the third red column, below the average 60. And on this histogram, you must build a modified histogram. I carried out corrective work and the number of children allegedly changed: from low level Below the average, most of them began to approach the average of 60% of children, 40% approached the high, these were children with an average value. That is, it is necessary to build an experimental group and CPR: with an average of 60% and 40 high.

And I need to make a table according to the money whitney criterion, I need to change the data again so that the below average level approaches the average and the average level approaches the high. Please write out the table. The number of subjects was 10 people, the norm and 10 spr. It’s just that it’s not very clear to me how you ranked, as I understand it, you adjusted the results (I asked you about this) and put down the ranks and then acted according to the formula ... if not, then explain. The course defense is coming. The calculations will be checked by the Associate Professor of the Department of Psychology himself. Please, help..

Purpose of the Mann-Whitney U-test

Real statistical method was proposed by Frank Wilcoxon (see photo) in 1945. However, in 1947, the method was improved and expanded by H. B. Mann and D. R. Whitney, so the U-test is more commonly referred to by their names.

The criterion is designed to assess the differences between two samples in terms of the level of any trait, quantitatively measured. It allows you to identify differences between small samples when n 1 ,n 2 ≥3 or n 1 =2, n 2 ≥5, and is more powerful than the Rosenbaum test.

Description of the Mann-Whitney U test

There are several ways to use the criterion and several options for tables of critical values ​​corresponding to these methods (Gubler E. V., 1978; Runion R., 1982; Zakharov V. P., 1985; McCall R., 1970; Krauth J., 1988) .

This method determines if the area of ​​overlapping values ​​between two series is small enough. We remember that we call the 1st row (sample, group) the row of values ​​in which the values, according to a preliminary estimate, are higher, and the 2nd row is the one where they are supposedly lower.

The smaller the crossover area, the more likely the differences are to be significant. Sometimes these differences are called differences in the location of two samples (Welkowitz J. et al., 1982).

The empirical value of the U criterion reflects how large the zone of coincidence between the rows is. Therefore, the smaller U emp, the more likely it is that the differences are significant.

Hypotheses U - Mann-Whitney test

U -Mann-Whitney test is used to evaluate the differences between two small samples (n 1 , n 2 ≥3 or n 1 =2, n 2 ≥5) according to the level of the quantitatively measured trait.

Null hypothesis H 0 =(the level of the feature in the second sample is not lower than the level of the feature in the first sample); alternative hypothesis - H 1 = (the level of the feature in the second sample is lower than the level of the feature in the first sample).

Consider the algorithm for applying the Mann-Whitney U-criterion:

1. Transfer all the data of the subjects to individual cards, marking the cards of the 1st sample in one color, and the 2nd - in another.

2. Lay out all the cards in a single row in ascending order of the feature and rank in that order.

3. Re-arrange the cards by color into two groups.

5. Determine the larger of the two rank sums.

6. Calculate empirical value U:

, where is the number of subjects in the sample (i = 1, 2), - the number of subjects in the group with larger amount ranks.

7. Set the significance level α and, using a special table, determine the critical valueUkr(α) . If , then H 0 at the chosen level of significance is accepted.

Consider using the Mann-Whitney U test for our example.

When ranking, we combine two samples into one. Ranks are assigned in ascending order of the value of the measured value, i.e. the lowest rank corresponds to the lowest score. Note that in case of coincidence of scores for several subjects, the rank of such a score should be considered as the arithmetic mean of those positions that these scores occupy when they are arranged in ascending order.

Using the proposed ranking principle, we obtain a table of ranks. Note that the choice of the arithmetic mean as the rank applies to any ranking.

To use the Mann-Whitney test, we calculate the sums of the ranks of the considered samples (see table).

Conducting a study according to the methodology gave the following results:

The results of the calculation of the Mann-Whitney U-test based on the results of the study are presented in Table 1 (ranking), in Figure 1 (significance axis tee):

The sum for the first sample is 72.5, for the second - 137.5. Let us denote the largest of these sums by T x (T x =137.5). Among the volumes n 1 =10 and n 2 =10 samples n x 17.5

The obtained empirical value U emp (17.5) is in the zone of significance, and, therefore, our hypothesis was confirmed.

The critical value of the criterion is found according to a special table. Let the significance level be 0.05.

The hypothesis H0 about the insignificant differences between the scores of the two samples is accepted if< . Otherwise, H0 is rejected and the difference is determined to be significant.

Therefore, the differences in level can be considered significant.

The scheme for using the Mann-Whitney test is as follows

Criterion U Mann - Whitney

Assigning a criterion. The criterion is designed to assess the differences between two samples by level any trait that can be quantified. It allows you to distinguish between small samples when P 1, n 2 > 3 or p L \u003d 2, p 2\u003e 5, and is more powerful than the criterion Q Rosenbaum.

This method determines if the area of ​​overlapping values ​​between two series is small enough. We remember that we call the 1st row (sample, group) the row of values ​​in which the values, according to a preliminary estimate, are higher, and the 2nd row is the one where they are supposedly lower.

The smaller the crossover area, the more likely it is that differences reliable. These differences are sometimes referred to as differences in location two samples. The empirical value of the criterion reflects how large the zone of coincidence between the rows is. So the less t/ 3Mn , especially it is likely that the differences reliable.

Hypotheses.

The level of non-verbal intelligence in the group of physics students is higher than in the group of psychology students.

Graphical representation of a criterionU. Pa fig. 7.25 presents three of the many possible options for the ratio of two series of values.

In option (a), the second row is lower than the first, and the rows almost do not intersect. Overlay area ( S j) too small to obscure differences between rows. There is a chance that the differences between them are significant. We can determine this exactly using the criterion U.

In variant (b), the second row is also lower than the first, but the area of ​​overlapping values ​​for the two rows is quite extensive (5 2). It may not yet reach a critical value, when the differences will have to be recognized as insignificant. But whether this is so can only be determined by exact calculation of the criterion U.

In option (c), the second row is lower than the first, but the overlap is so extensive (5 3) that the differences between the rows are obscured.

Rice. 7.25.

in two samples

Note. The overlap (5 t , S 2 , *$z) indicates the areas of possible overlap. Criteria restrictionsU.

• 1. Each sample must contain at least three observations: n v p 2 > 3; it is allowed that there are two observations in one sample, but then there must be at least 5 of them in the second.
• 2. Each sample should contain no more than 60 observations; p l, p 2 w, n 2 > 20 ranking becomes quite laborious.

Let's return to the results of a survey of students of the physical and psychological faculties of Leningrad University using D. Veksler's method for measuring verbal and non-verbal intelligence. Using the criterion Q Rosenbaum was with high level significance, it was determined that the level of verbal intelligence in the sample of students of the Faculty of Physics is higher. Let us now try to establish whether this result is reproduced when comparing samples according to the level of non-verbal intelligence. The data are given in the table.

2 below the level of the trait in the sample 1 not significantly significant level. The smaller the value u, the higher the significance of the differences.

Now let's do all this work on the material of our example. As a result of work on 1-6 steps of the algorithm, we will build a table (Table 7.4).

Table 7.4

Calculation of rank sums for samples of students of physical and psychological faculties

The total amount of ranks: 165 + 186 = 351. The calculated amount according to the formula (5.1) is as follows:

The equality of the real and estimated amounts is observed. We see that in terms of the level of non-verbal intelligence, a sample of psychology students is more “higher”. It is this sample that accounts for a large rank sum: 186. Now we are ready to formulate statistical hypotheses:

Self 0: a group of psychology students does not outperform a group of physics students in terms of non-verbal intelligence;

Me: a group of psychology students outperforms a group of physics students in terms of non-verbal intelligence.

In accordance with the next step of the algorithm, we determine the empirical value U :

Because in our case p l * p 2, calculate the empirical value U and for the second rank sum (165), substituting into formula (7.4) the corresponding p x.:

According to Appendix 8, we determine the critical values ​​for p l = 14, n 2 = 12:

We remember that the criterion U is one of two exceptions to general rule making a decision about the reliability of differences, namely, we can state significant differences if (/ emp U Kp 0 05 (at temp = 60, and sp > U Kf) about,05).

Hence, H 0 is taken as follows: the group of psychology students does not surpass the group of physics students in terms of the level of non-verbal intelligence.

Let's pay attention to the fact that for this case the Rosenbaum Q-criterion is not applicable, since the scope of variability in the group of physicists is wider than in the group of psychologists: both the highest and the lowest values ​​of non-verbal intelligence fall on the group of physicists (see Table 7.4) .

where
,

7. Determine the critical value -criteria (see appendix, table A3).

8. Compare calculated and critical value -criteria. If the calculated value is greater than or equal to the critical value, then the hypothesis
equality of means in two samples of changes is rejected. In all other cases, it is taken at a given level of significance.

Lecture 4. Criteria for nonparametric distributions

4.1. -Mann-Whitney test

Assigning a criterion. The criterion is intended to assess the difference between two nonparametric samples by level any trait that can be quantified. It allows you to distinguish between small samples when

Description of the criterion

This method determines if the area of ​​overlapping values ​​between two series is small enough. The smaller this area, the more likely it is that the differences are significant. The empirical value of the criterion reflects how large the zone of coincidence between the rows is. So, the less
especially it is likely that the differences reliable.

Hypotheses

The level of the attribute in group 2 is not lower than the level of the attribute in group 1.

The level of the feature in group 2 is lower than the level of the feature in group 1.

Algorithm for calculating the Mann-Whitney criterion

1. Transfer all data of the subjects to individual cards.

2. Mark the cards of the subjects of sample 1 with one color, say, red, and all the cards from sample 2 with another, for example, blue.

3. Lay out all the cards in a single row according to the degree of increase of the attribute, regardless of which sample they belong to, as if there was one large sample.

4. Rank the values ​​on the cards, assigning a lower rank to the lower value.

5. Re-arrange the cards into two groups, focusing on the color designations: red cards in one row, blue in the other.

7. Determine the larger of the two rank sums.

8. Determine the value by the formula

,

where
the number of subjects in sample 1;
the number of subjects in sample 2;
the larger of the two rank sums;
the number of subjects in the group with a higher sum of ranks.

9. Determine critical values . If a
then

hypothesis
accepted. If a
it is rejected. The less

values , the higher the reliability of the differences.

Example. Compare the effectiveness of two teaching methods in two groups. The test results are presented in table 4.

Table 4

We transfer all the data to another table, highlighting the data of the second group, we emphasize and rank the total sample (see the ranking algorithm in the guidelines for the task).

Find the sum of the ranks of two samples and choose the largest of them:

Calculate the empirical value of the criterion according to the formula (3)

Let us determine the critical value of the criterion at the significance level
(See Appendix Table A1)

Conclusion:since the calculated value of the criterion more critical at significance level
and
, the hypothesis about the equality of the means is accepted, the differences in teaching methods will be insignificant.

Methods of mathematical processing in psychology

CHAPTER I. BASIC CONCEPTS USED IN MATHEMATICAL PROCESSING OF PSYCHOLOGICAL DATA

Possibilities and limitations of parametric and non-parametric criteria

Classification of problems and methods for their solution

CHAPTER II. IDENTIFICATION OF DIFFERENCES IN THE LEVEL OF THE INVESTIGATED SIGN

Deciding on the choice of the method of mathematical processing

If the data has already been received, then you are offered the following algorithm for determining the problem and method.

ALGORITHM 2

Making a decision about the task and processing method at the study planning stage

1. Decide which model seems best to you to prove your scientific assumptions.

2. Carefully read the description of the method, examples and tasks for independent decision that are attached to it.

3. If you are convinced that this is what you need, go back to the Criterion Limits section and decide if you can collect data that will meet these limitations (large sample sizes, having several samples that differ monotonously in some way). sign, for example, by age, etc.).

4. Conduct research and then process the data obtained in advance! selected algorithm if you managed to meet the constraints.

5. If the restrictions could not be met, refer to Algorithm 1.

Algorithm for making a decision on choosing a criterion for comparisons

Q - Rosenbaum criterion

Purpose of the criterion. The criterion is used to assess the differences between two samples in terms of the level of any trait, quantitatively measured. Each sample must contain at least 11 subjects.

Example.

The level of verbal and non-verbal intelligence was measured in the alleged participants in a psychological experiment simulating the activities of an air traffic controller using the D. Wexler technique. 26 young men aged 18 to 24 years were examined ( average age 20.5 years). 14 of them were students of the Faculty of Physics, and 12 were students of the Faculty of Psychology of Leningrad University. Can it be argued that one of the groups is superior to the other in terms of verbal intelligence?

Table I Critical values Rosenbaum's Q test

U - Mann-Whitney test

Purpose of the criterion. The criterion is designed to assess the differences between two samples by level any trait that can be quantified. It allows you to distinguish between small samples when n 1 ,n 2 ≥ 3 or n 1 =2, n 2 ≥5, and is more powerful than the Rosenbaum criterion.

Example

The level of verbal intelligence in the sample of students of the Faculty of Physics is higher than that of students of the Faculty of Psychology of Leningrad University. Let us now try to establish whether this result is reproduced when comparing samples according to the level of non-verbal intelligence. Can it be argued that one of the samples is superior to the other in terms of non-verbal intelligence?

Ranking Rules

1. A lower value is assigned a lower rank. The smallest value is assigned a rank of 1. The largest value is assigned a rank corresponding to the number of ranked values. For example, if n=7, then highest value will receive a rank of 7, with the possible exception for those cases provided for in rule 2.

2. If several values ​​are equal, they are assigned a rank, which is the average of the ranks that they would have received if they were not equal.

Let's say the next 2 values ​​are 12 seconds. They should get ranks 4 and 5, but since they are equal, they get an average rank:

3. The total amount of ranks must match the calculated one, which is determined by the formula:

where N- total ranked observations (values). The discrepancy between the actual and calculated amounts of ranks will indicate an error made in the calculation of ranks or their summation. Before proceeding, you must find the error and fix it.

ALGORITHM 4

Mann-Whitney U test calculation.

1. Transfer all data of the subjects to individual cards.

2. Mark the cards of the subjects of sample 1 with one color, say red, and all the cards from sample 2 with another, for example, blue.

3. Lay out all the cards in a single row according to the degree of increase of the attribute, regardless of which sample they belong to, as if we were working with one large sample.

4. Rank the values ​​on the cards, assigning a lower rank to the lower value. There will be as many ranks as we have (n 1 + n 2).

5. Again, decompose the cards into two groups, focusing on color designations: red cards in one row, blue in the other.

7. Determine the larger of the two rank sums.

8. Determine the value of U by the formula:

where n 1 - the number of subjects in sample 1;

n 2- the number of subjects in sample 2;

T x - the larger of the two rank sums;

n x - the number of subjects in the group with a larger sum of ranks.

9. Determine the critical values ​​of U according to Table. II. If U emp ≤ U to p _ 005, the differences are significant. The smaller the U value, the higher the reliability of the differences.

Table II. Critical values ​​of the Mann-Whitney U criterion

for levels statistical significance p≤0.05 and p≤0.01.

Table II. Continuation

Table II. Continuation