Calculation of the student's criterion. Classical methods of statistics: Student's t-test
1. Student's method (t-test)
This method is used to test the hypothesis about the reliability of the difference in means when analyzing quantitative data in elections with a normal distribution.
where x 1 and x 2 are the arithmetic mean values \u200b\u200bof variables in groups 1 and 2,
SΔ is the standard error of the difference.
If n 1 \u003d n 2, then 
where n 1 and n 2 are the number of elements in the first and second samples, δ 1 and δ 2 are the standard deviations for the first and second samples.
If n 1 ≠ n 2 then 
The level of significance is determined by a special table.
2. Criterion φ * - angular Fisher transform
This criterion assesses the reliability of the differences between the percentages of the two samples in which the sign of interest is registered.
The empirical value of φ * is calculated by the formula:
φ * \u003d (φ 1 - φ 2). where
φ 1 - angle corresponding to a large percentage.
φ 2 - the angle corresponding to a smaller percentage.
n 1 - number of observations in sample 1
n 2 - number of observations in sample 2
The significance level φ * of the empirical value is determined according to a special table. The larger the φ * value, the more likely the differences are reliable.
2.2 Research results and their analysis
2.2.1 Peculiarities of adaptation of patients with chronic disease
To study the degree of adaptation, we used the method of diagnosing social and psychological adaptation by K. Rogers and R. Diamond.
Based on the analysis of the integral indicator of adaptation, 3 experimental groups of subjects were identified:
1.with a high level of adaptation - group A.
The value of the adaptation indicator is from 66 to 72 points. (M \u003d 67)
2.with an average level of adaptation - group B.
The value of the adaptation indicator is from 49 to 65 points. (M \u003d 56.6)
3.with a low level of adaptation - group C.
The value of the adaptation indicator is from 38 to 48 points. (M \u003d 41.3)
The significance of differences in the level of adaptation between the experimental groups was tested using the Student's t-test. Differences are statistically significant at p≤0.01 between groups A and B, groups B and C, groups A and C. Thus, it can be concluded that patients with chronic diseases are characterized by varying degrees of adaptation.
Most patients with chronic diseases are characterized by an average degree of adaptation (65%), with a high level of adaptation - 19%, the third group of patients with a low level of adaptation (16%).
The analysis of gender differences in the level of adaptation of patients with chronic diseases was carried out. It was found that the majority of women and men are characterized by an average level of adaptation (65% and 63%, respectively) - see tab. # 1.
Table 1
Sex differences in adaptation of patients with chronic diseases
(by groups of subjects,%)
| Subjects | Experimental groups | ||
| A (high ad.) | B (average ad. Level) | C (low level) | |
| Men | 20 | 63 | 17 |
| Women | 19 | 65 | 16 |
The significance of differences was determined using Fisher's φ-test. It was revealed that in none of the experimental groups the differences in adaptation between men and women are not significant. (Group A – φ \u003d 0.098, group B - φ \u003d 0.161, group C - φ \u003d 0.106).
2.2.2 Features of the personality of patients with chronic diseases with varying degrees of adaptation
First, let us consider the features of self-awareness of the subjects of the experimental groups.
Group A (highly adaptable)
The results obtained using the scale "Self-activity" showed that most of the subjects in this group have a high and medium level of self-activity (33%). The subjects of group A did not have low indices on the scale "Self-activity".
Thus, patients with chronic diseases with a high level of adaptation highly value their appearance, their ability to cope with difficult situations, consider themselves interesting as a person.
In the study of sex differences, it was revealed that women with a high level of adaptation in the post-labor period more often show a high level of self-activity (83%), while men are equally high and medium (50% and 50%).
For a more detailed study of a person's ideas about himself, the “Personal differential” method was used.
The interpretation of the data obtained using the LD was carried out according to 3 factors:
Grade (O)
Activity (A)
For each factor, in accordance with the standard norms, 5 levels are allocated:
Very low (7-13 points)
Low (14-20 points)
Average (21-34 points)
High (35-41 points)
Very high (42-49 points)
Very low values \u200b\u200bwere not found in any group for all factors, therefore, this category is not studied when interpreting the results.
The results for the assessment factor indicate the level of self-esteem; by the factor of strength on the development of the volitional sides of the personality; by the factor of activity of personality extroversion.
When analyzing the results in group A (with a high level of adaptation), low values \u200b\u200bwere not revealed for any factor (assessment, strength, activity), which corrects with the data obtained using the Samostupitie school.
Analysis of the results by factors revealed the following features:
The majority of the subjects in this group had an optimal level of self-esteem for the strength factor (58% - average values, 17% - high). There are also very high values \u200b\u200b(25%).
This indicates that patients with chronic diseases with a high level of adaptation are confident in themselves, independent, rely on their own strength in difficult situations.
According to the assessment factor in group A, the values \u200b\u200bof most of the subjects were attributed to the optimal level (high values \u200b\u200b- 50%, average - 25%). There are also very high values \u200b\u200b(25%). This suggests that the subjects accept themselves as individuals, recognize themselves as carriers of positive, socially desirable characteristics.
By the factor of activity, the largest number of average (42%) and high (33%) values. Very high values \u200b\u200bare found in 28% of the subjects. These results indicate a high activity of patients with chronic diseases with a high level of adaptation, sociability.
When carrying out statistical processing using Fisher's φ-test (0.05 significance level), significant differences were revealed in the factors of strength and assessment. In the subjects of experimental group A, average values \u200b\u200bprevail in terms of strength factor, and high scores in terms of factor. Based on this, it can be concluded that patients with a high level of adaptation rate their socially desirable characteristics, themselves as a person, higher than their volitional qualities.
In the study of sex differences, significant differences were found in the factors of strength and assessment (φ-criterion, p \u003d 0.01). See table. 2, 3.
Table 2
Sex differences by factor "Strength" LD (%)
|
Subjects | low | the average | high | very high |
| men | 0 | 33 | 17 | 50 |
| women | 0 | 83 | 17 | 0 |
ValueWomen in group A in most cases have an average value for the factor "Strength" (83%), while men - very high (50%).
Table 3
Sex differences in group A by the factor "Assessment" of LD (%)
|
Subjects | low | the average | High | very high |
| Men | 0 | 50 | 33 | 17 |
| Women | 0 | 0 | 67 | 33 |
The majority of women in group A have high (67%) and very high (33%) scores for the "Assessment" factor, while men have medium (50%) and high (33%) scores.
Thus, men with a high level of adaptation highly value their volitional qualities, self-confidence, and women with a high level of adaptation highly value their social qualities, the level of achievements.
To study the localization of control over significant events, the Internality scale was used.
The analysis of the results did not reveal low values \u200b\u200bfor this factor in the subjects of group A. Equally, there are medium (50%) and high values \u200b\u200b(50%). This indicates that people with a high level of adaptation in the post-work period believe that most important events in their life are the result of their own actions, that they can control them, and, thus, they feel responsible for these events and for the fact that how life is going in general.
Sex differences according to the Internality criterion are not statistically significant.
Thus, it can be concluded that patients with chronic diseases with a high level of adaptation have an optimal level of self-esteem (medium and high). They accept themselves as individuals, are confident, independent, assess themselves as active and sociable. Men highly value their volitional qualities, ability to cope with difficulties, and women highly value their social qualities.
People of this group tend to rely on their own strength, know how to manage themselves, their actions, consider themselves responsible for how their life as a whole develops.
Group B (medium level of adaptation)
The results obtained using the “Self-activity” scale show that the majority of the subjects in this group have an average level of self-activity (90%). With a high level of self-activity - 5%, with a low level - 5%.
There were no statistically significant sex differences.
The analysis of the results according to the personal differential method was carried out according to the factors of strength, assessment, activity. See table. 4.
Table 4
Representation of subjects in group B with different levels of self-esteem (by factors, in%)
Analysis of the results by factors showed that the majority of subjects in group B had an optimal level of self-esteem in terms of strength factor (75% - average values, 17% high). There are also low (5%) and very high (2.5%) levels of values.
According to the assessment factor, an adequate level of self-esteem prevails (62.5% - the average level of values, 10% - high). Low rate in 2.5% of subjects. A large percentage of very high rates (25%).
According to the factor of activity, the largest number of optimal values \u200b\u200b(60% - medium, 22.5% - high). Low values \u200b\u200bare 7.5%, very high - 10%.
When carrying out statistical processing using the φ-criterion (p≤0.01), significant differences were revealed in the facts of strength and assessment at a "very high" level of values. Subjects in group B tend to overestimate their social qualities.
When studying sex differences, differences were found for all 3 factors (φ-criterion p≤0.05).
Sex differences by factor "Strength"
Among men, no low values \u200b\u200bwere found for this indicator. Whereas women have 10%.
A very high level of values \u200b\u200bis found in men of the group in 5% of cases, in women it does not occur. This indicates that men tend to exaggerate their volitional qualities, and women tend to underestimate.
Sex differences by the nature of "Assessment"
Women more often than men have very high values \u200b\u200bfor this factor (33% women, 16% men).
Sex differences according to the factor "Activity"
Women in this experimental group more often than men have very high values \u200b\u200bfor this factor (44% women, 5% men).
When analyzing the results on the scale "Internality", it was revealed that the majority of patients with chronic diseases with an average level of adaptation have an average level of the indicator of internality (80%). Subjects with a low value on this scale -7.5%, with a high-12.5%.
This suggests that, in general, patients with an average level of adaptation make high demands on themselves, rely on their own strength. But some of the people from this group do not consider themselves capable of controlling events in their lives; they attribute the responsibility for them to circumstances, to other people.
Thus, the majority of patients with chronic diseases with an average level of adaptation in the post-labor period exhibit an optimal level of self-esteem, i.e. they accept themselves as a person, are satisfied with themselves. A certain percentage of people in this group have an overestimated self-esteem, as well as an underestimated one, which indicates a personal lack of maturity, an inability to correctly assess themselves, the results of their activities. The subjects of this group tend to overestimate their social qualities with an exaggerated high.
Men of this group tend to overestimate their volitional qualities, while women tend to overestimate social ones.
Group C (low level of adaptation)
The results obtained using the "self-activity" scale showed that most of the subjects in this group have a low level of self-activity (70%). Some of the subjects had an average level of self-activity (30%). There were no high values \u200b\u200bon this scale.
Thus, patients with chronic diseases with a low level of adaptation sharply assess their appearance, they believe that they have not shown themselves in anything.
In the study of sex differences, it was revealed that women with a low level of adaptation more often have an average level of self-activity (66%), while men in all cases (100%). Consequently, women with a low level of adaptation will not always have a low level of self-acceptance.
The analysis of the results using the "Personal differential" method was carried out according to the factors of strength, assessment, activity. See table. five.
Table 5 Representation of test groups C with different levels of self-esteem (by factors, in%)
When analyzing the results of group C, very high values \u200b\u200bwere not revealed for any factor, which correlates with the data obtained using the "Events" scale. High values \u200b\u200bare found only for the assessment factor (10%).
In terms of the strength factor, most of the subjects have low values \u200b\u200b(60%). There are also average values \u200b\u200b(40%).
By factors of assessment and activity, the maximum number of average values \u200b\u200b(80%). Low values \u200b\u200bfor the assessment factor are found in 10% of the subjects, and for the activity factor in 20%.
Thus, the subjects of group C are dominated by average low values \u200b\u200bof self-esteem. Subjects in this group rate their volitional qualities especially low.
When studying sex differences, significant differences were found in the factor of strength (φ-criterion, 0.03) and assessment.
Men with a low level of adaptation rate their volitional qualities low (80% when as women only 49%;), and women social qualities (20% women, men 0%)
When analyzing the results on the scale "Internality" it was revealed that the majority of patients with chronic diseases have a low level of internality (60%) and an average level of internality (30%). There are 10% of people with a high level of internality in this group.
This indicates that the majority of patients with chronic diseases with a low level of adaptation are inclined to attribute more importance to external circumstances, do not consider themselves able to control their own lives.
Thus, it can be concluded that patients with chronic diseases with a low level of adaptation are characterized by low and medium levels of self-esteem. Often they are critical of themselves, not satisfied with their own behavior, level of achievement. Men in this group rate self-confidence and the ability to cope with difficulties as low, while women rate themselves as a person as a whole.
People in this group believe that most of the events in their life are the result of an accident or the actions of others.
A comparative analysis of the indicators of self-awareness by groups of subjects allowed us to identify significant differences.
The subjects of group A (high level of adaptation) are characterized by a high level of self-acceptance (67%) in comparison with the subjects of group B (5%), φ * \u003d 4.45; p ≤ 0.01) and group C (0%).
In group C (low level of adaptation) there is a greater number of low values \u200b\u200b(70%) than in group B (5%) - φ * \u003d 3.57; p ≤ 0.01 and group A (0%).
According to the factor of strength (personal differential in subjects of group A, very high (25%) and high (17%) values \u200b\u200bare more often found than in subjects of group C (0% and 0%).
In group C, there are more low values \u200b\u200b(60%) than in group A (0%).
By the nature of the assessment in group A, high values \u200b\u200b(50%) are more common than in group C (10%) - φ * \u003d 2.16; p ≤ 0.01.
In group C, low values \u200b\u200b(10%) are more common than in group A (6%) by the assessment factor and average values \u200b\u200b(80%) than in group A (25%) - φ * \u003d 2.72; p ≤ 0.01.
By factor, the activity in group A is higher than very high (25%) and high (33%) values \u200b\u200bthan in group C (0%). Group C has more low values \u200b\u200b(20%) than group A (6%).
The subjects of group A are characterized by a high level of internality (50%) in comparison with the subjects of group C (10%) - φ * \u003d 2.16; p ≤0.01
Subjects in group C more often have a low level of internality (60%) than in subjects in group A (0%) and subjects in group B (7.5%) - φ * \u003d 3.44; p ≤ 0.01
Thus, the subjects of group A have, on the whole, more optimal self-esteem for personal well-being and are characterized by a more confident, conscious attitude to life.
2.2.3 Features of the motivational-need-related sphere of the personality of the subjects
To study the motivational-need sphere, the technique of unfinished sentences was used (see appendices). The analysis of the results is presented in the following categories:
1. Statement about the future (1, 2, 9, 13)
2. Statement about the past (3, 4)
3. Statement about the disease (6, 7)
4. Statements related to relatives (8)
5. Statements that reflected the attitude towards the disease (10)
Group A (high level of adaptation) - see appendix.
1. In statements about the future, the respondents' answers most often include the expectation of achievements-29% ("I will learn to drive a car"), interest in the problems of society -21% ("I would be glad if life in the country improves"), the hope of maintaining the same level life -21% ("I hope that I will be as active"), concern about relatives -13% ("In the future, my life is the life of my children").
2. Subjects of group A, assessing their past, note that: they realized their plans, realized themselves-54% (“Looking back at my life, I think that life was not lived in vain.” “What I was striving for, I managed to achieve it”), partially realized their plans - 21% ("The family turned out to be good, but I devoted little time and I still devote to children"). 17% of patients with chronic diseases of group A admit the erroneousness of their goals, aspirations in the past ("I strove for something that did not matter")
3. An analysis of the statements of patients with chronic diseases regarding the fact of the disease showed that some of the subjects experienced very strong -25%, and some took it for granted without any special worries -21%
The data obtained additionally through the conversation showed that the subjects of group A are characterized by a variety of interests.
Among them, reading (83%), watching TV (83%), walking (75%), sports (50%), conducting conversations (33%), meeting with friends, relatives (25%). It can be assumed that the presence of a wide range of interests helps the chronic patients of this group to calmly perceive the presence of the disease.
4. In statements related to communication with relatives, the subjects express concern for loved ones - 75% ("I would like my children to be healthy") and expect support from relatives - 25% ("I would like my relatives to be always with me").
When analyzing the results obtained using the Acceptance of Others scale, it was found that the subjects of group A have a high (58%) and an average level of acceptance of others (42%), which indicates a high hope for affiliation, a desire for acceptance. In general, I like other people, relations with them are warm, friendly.
The data obtained through the conversation showed that 67% of patients with chronic diseases in this group are completely satisfied with communication, 25% are satisfied, but now they communicate less than before (the circle of communication has narrowed) and 8% lack communication.
Thus, in spite of the narrowing of the circle of communication, patients with a high level of adaptation are satisfied with interaction with other people.
5. Giving their definition of the disease, 33% of the subjects declare that the disease is a boundary in life (“The disease for me is a certain boundary, life“ before ”and“ after ”, it made it possible to evaluate my life”).
6. In the study of sex differences, it was revealed that in relation to the future, men more often than women show interest in the problems of society (67% of men, 33% of women), ie. show great social activity.
Group B (average level of adaptation)
1. In statements about the future, in the answers of subjects, everyday problems are most often encountered -20% ("I intend to make repairs at home"), hopes for maintaining the previous standard of living -19%. ("I will be very glad if I continue to live the same way"), health concerns -14% ("I intend to lead a healthy lifestyle, this is the main thing for life"), worries about relatives -10%, 9% of patients expect from the future better ("I hope that the future life will be better than now").
2. Subjects of group B, assessing their past, note that: they partially realized their plans (38%); admitted that their goals and aspirations were erroneous - (35%); realized their plans, realized themselves- (15%).
3. Analysis of the statements about the news of the disease showed that the majority of the subjects treated this event relatively calmly (65%), part of the group was very worried (25%) and a small percentage was indifferent (10%).
The data obtained through the conversation indicate that the test groups B are characterized by a variety of interests (reading, radio, television, cinema, fishing, meeting friends, pets, etc.). What is important for a calm attitude towards your status as a chronic patient.
4. In statements related to communication with relatives, the subjects express concern for their loved ones (57%) and expect support from their relatives (25%). Some answers contain fear of being left alone (25%) ("I will be very happy if my relatives live with me and do not forget me").
When analyzing the results obtained using the Acceptance of Others scale, it was found that the subjects of group B have an average level of acceptance of others (78%). Some of the subjects showed a high level of acceptance of others (10%), and some low (12%). This indicates that people with an average level of adaptation are characterized by a desire to accept other people.
During the conversation, it was found that patients in this group regret that they are now communicating less, but in general they are satisfied with communication (73%), 15% are completely satisfied with their interaction with other people and 12% believe that they lack communication.
5. In the statements reflecting the attitude to the disease, the subjects of group B write that the disease is a burden for loved ones (27.5%), the disease is a boundary or border (17.5%). The fear of death is heard in the responses of 20 subjects, 25% say that illness is not the end of life.
Group C (low level of adaptation)
1. In statements about the future, the respondents' answers most often include the absence of expectation of any changes - 30% ("My life will not change in the future"), expectations of difficulties - 22.5% ("In the future, my life will become even more difficult") , everyday problems - 17.5%. Health concerns are reflected in the answers of 15% of the subjects. The category "expectation of achievement" does not occur in the responses of subjects with a low level of adaptation.
2. Assessing their past, people of group C note that they did not do what they could - 40% ("Looking back at my life, I think I could have lived it better and more fun"), that there were more failures than achievements - 30% ("Looking back at my life, I think that in life, part of it was unlucky").
15% admit that their goals and aspirations were erroneous in the past, 10% note that they were able to partially realize themselves. And only 5% write that they have realized their plans, realized themselves.
3. Analysis of the statements of patients with chronic diseases relative to the news of the diagnosis showed that the majority experienced this event hard - 60%; 30% reacted relatively calmly and 10% indifferently.
During the conversation, it was found that the test groups C were characterized by passive interests (watching TV, knitting, reading), and many noted the absence of a favorite activity. We can say that the lack of interests complicates the process of adaptation to the disease, since it is not compensated by significant activities.
1. In statements related to communication with relatives, the subjects expect support from their relatives (50%) and express fear of being left alone 30%. In the answers of 20% of the subjects, care for relatives sounds.
2. When analyzing the results obtained using the scale "Acceptance of others" revealed that the subjects of group C have low (60%) and medium; (40%) by the level of acceptance of others, which indicates that people of this group are restrained in communicating with others, feel dislike for those around them.
Analysis of the results of the conversation showed that people with a low level of adaptation are not satisfied with communication with others (70%), or satisfied, but unhappy with the fact that the circle of communication has narrowed (30%).
Giving their definition of the disease, patients of group C write that the disease is the end of life (40%), express fear of death (20%), the disease is characterized as a boundary for 30% of the subjects.
A comparative analysis of the features of the motivational-need sphere made it possible to identify significant differences.
1. In the statements of the subjects of group A (high level of adaptation) about the future, the expectation of achievements is more common (29%) than in group B (9%) φ * \u003d 1.604; p≤0.05 and group C (0%). In group B, a large percentage of answers are associated with everyday problems (20%) than in group A (4%) φ * \u003d 1.59; p≤0.05.
2. In their statements about the past, subjects of group A more often (54%) than subjects of group B (15%) note that they have accomplished their plans, realized themselves (φ * \u003d 2.42; p≤0.01), and more often than in group C (5%) φ * \u003d 2.802; p≤0.01.
Subjects in group C more often (30%) than subjects in group A (0%) and than subjects in group B (6%) - φ * \u003d 2.83; p≤0.01 note that there were more failures than achievements. They also more often (46%) than the subjects of group B (1%) write that they did not do what they could (φ * \u003d 3.306; p≤0.01).
In group B, a large (38%) percentage of subjects noted that they were able to partially realize themselves than in group C (10%), φ * \u003d 1.934; p≤0.02.
3. In statements related to retirement, subjects of group C more often (60%) than subjects of group A (25%) and subjects of group B (25%) write that they experienced (φ * \u003d 1.693; p≤0.04 ).
4. Describing attitudes towards relatives, subjects in group A more often (75%) than subjects in group C (20%) express concern for their loved ones (φ * \u003d 2.725; p≤0.01).
Patients with chronic diseases in group C more often (30%) than in group A (0%) express fear of being alone.
5. In the answers of the subjects of group A, the definition of the disease as summing up the results (17%) is more often heard than in group C (0%) and in group B (2.5%) - φ * \u003d 1.61; p≤0.05.
Significant differences were found on the Accept to Others scale. Sick people of group A more often (58%) reveal a high level of acceptance of others than in group C (0%) and group B (10%) φ * \u003d 3.302; p≤0.01.
Subjects of group C more often (60%) reveal a low level of acceptance of others than in group A (0%) and group B (12.5%) - φ * \u003d 2.967; p≤0.01
Thus, patients with chronic diseases with a high level of adaptation are characterized by a more optimistic attitude towards the future, a positive assessment of the past, and a high level of acceptance of others.
It lies in breaking the ties between old people and young people. A phenomenon such as gerontophobia, or hostile feelings towards old people, can also be called quite common today. Many of the stressors of the elderly and the elderly can be prevented or overcome relatively painlessly precisely by changing the elderly and the aging process in general. Famous American physician and ...
A number of methods have been applied: - theoretical analysis of scientific literature on sociology, deviantology, psychology of individual differences on the problem of studying the influence of psychological characteristics of people who use drugs; - empirical - psychodiagnostic methods; - comparative analysis; - methods of mathematical and statistical processing of research results: Student's t-test ...
Weakened children are becoming more and more, and therefore special preventive measures are needed to prevent somatic and psychosomatic diseases. CHAPTER III. EMPIRICAL STUDY OF THE PECULIARITIES OF PSYCHOLOGICAL ADAPTATION TO THE DET OF EARLY AGE CHILDREN 3.1 Description of the sample The empirical study was conducted from September to December 2008. The base for ...
When can you use Student's t-test?
To apply the Student's t-test, it is necessary that the initial data have normal distribution... In the case of using a two-sample test for independent samples, the condition equality (homoscedasticity) of variances.
If these conditions are not met, similar methods should be used when comparing sample means. nonparametric statistics, among which the most famous are Mann-Whitney U-test(as a two-sample test for independent samples), and sign criterionand wilcoxon test (used in cases of dependent selections).
To compare the mean values, the Student's t-test is calculated using the following formula:
where M 1 - the arithmetic mean of the first compared population (group), M 2 - the arithmetic mean of the second compared population (group), m 1 - mean error of the first arithmetic mean, m 2 is the mean error of the second arithmetic mean.
How to interpret the value of Student's t-test?
The obtained value of the Student's t-test must be correctly interpreted. To do this, we need to know the number of subjects in each group (n 1 and n 2). Find the number of degrees of freedom f by the following formula:
f \u003d (n 1 + n 2) - 2
After that, we determine the critical value of the Student's t-test for the required level of significance (for example, p \u003d 0.05) and for a given number of degrees of freedom f according to the table ( see below).
We compare the critical and calculated values \u200b\u200bof the criterion:
If the calculated value of the Student's t-test equal or morecritical, found from the table, we conclude about the statistical significance of the differences between the compared values.
If the value of the calculated Student's t-test smallertabular, then the differences between the compared values \u200b\u200bare statistically insignificant.
Example of calculating Student's t-test
To study the effectiveness of the new iron preparation, two groups of patients with anemia were selected. In the first group, patients received a new drug for two weeks, and in the second group, they received a placebo. After that, the hemoglobin level in the peripheral blood was measured. In the first group, the average hemoglobin level was 115.4 ± 1.2 g / l, and in the second - 103.7 ± 2.3 g / l (data are presented in the format M ± m), the compared populations have a normal distribution. The number of the first group was 34, and the second - 40 patients. It is necessary to draw a conclusion about the statistical significance of the obtained differences and the effectiveness of the new iron preparation.
Decision: To assess the significance of the differences, we use the Student's t-test, calculated as the difference between the mean values, divided by the sum of the squares of the errors:
After performing the calculations, the t-criterion value turned out to be 4.51. We find the number of degrees of freedom as (34 + 40) - 2 \u003d 72. Compare the obtained value of the Student's t-test 4.51 with the critical value at p \u003d 0.05, indicated in the table: 1.993. Since the calculated value of the criterion is greater than the critical value, we conclude that the observed differences are statistically significant (significance level p<0,05).
Fisher's distribution is the distribution of a random variable
where the random variables X 1 and X 2 independent and have chi-square distributions with the number of degrees of freedom k 1 and k 2 respectively. The pair (k 1, k 2) - a pair of "numbers of degrees of freedom" of the Fisher distribution, namely, k 1 Is the number of degrees of freedom of the numerator, and k 2 - the number of degrees of freedom of the denominator. Distribution of a random variable F named after the great English statistician R. Fisher (1890-1962), who actively used it in his works.
The Fisher distribution is used to test hypotheses about the adequacy of the model in regression analysis, about the equality of variances, and in other problems of applied statistics.
Student's critical values \u200b\u200btable.
Form start
| Number of degrees of freedom, f | Student's t-test value at p \u003d 0.05 |
| 12.706 | |
| 4.303 | |
| 3.182 | |
| 2.776 | |
| 2.571 | |
| 2.447 | |
| 2.365 | |
| 2.306 | |
| 2.262 | |
| 2.228 | |
| 2.201 | |
| 2.179 | |
| 2.160 | |
| 2.145 | |
| 2.131 | |
| 2.120 | |
| 2.110 | |
| 2.101 | |
| 2.093 | |
| 2.086 | |
| 2.080 | |
| 2.074 | |
| 2.069 | |
| 2.064 | |
| 2.060 | |
| 2.056 | |
| 2.052 | |
| 2.048 | |
| 2.045 | |
| 2.042 | |
| 2.040 | |
| 2.037 | |
| 2.035 | |
| 2.032 | |
| 2.030 | |
| 2.028 | |
| 2.026 | |
| 2.024 | |
| 40-41 | 2.021 |
| 42-43 | 2.018 |
| 44-45 | 2.015 |
| 46-47 | 2.013 |
| 48-49 | 2.011 |
| 50-51 | 2.009 |
| 52-53 | 2.007 |
| 54-55 | 2.005 |
| 56-57 | 2.003 |
| 58-59 | 2.002 |
| 60-61 | 2.000 |
| 62-63 | 1.999 |
| 64-65 | 1.998 |
| 66-67 | 1.997 |
| 68-69 | 1.995 |
| 70-71 | 1.994 |
| 72-73 | 1.993 |
| 74-75 | 1.993 |
| 76-77 | 1.992 |
| 78-79 | 1.991 |
| 80-89 | 1.990 |
| 90-99 | 1.987 |
| 100-119 | 1.984 |
| 120-139 | 1.980 |
| 140-159 | 1.977 |
| 160-179 | 1.975 |
| 180-199 | 1.973 |
| 1.972 | |
| ∞ | 1.960 |
Excel for Office 365 Excel for Office 365 for Mac Excel 2019 Excel 2016 Excel 2019 for Mac Excel 2013 Excel 2010 Excel 2016 for Mac Excel for Mac 2011 Excel Online Excel for iPad Excel for iPhone Excel for Android tablets Excel for Android phones Excel Mobile Excel Starter 2010 Less
Returns the probability corresponding to the Student's t-test. The STUDENT.TEST function determines the probability that two samples are drawn from populations that have the same mean.
Syntax
STUDENT.TEST (array1; array2; tails; type)
The arguments for the STUDENT.TEST function are described below.
Array1 Mandatory. First dataset.
Array2 Mandatory. Second set of data.
Tails Mandatory. The number of distribution tails. If tails \u003d 1, STUDENT.TEST returns the one-tailed distribution. If tails \u003d 2, STUDENT.TEST returns a two-tailed distribution.
A type Mandatory. The type of t-test being performed.
Parameters
Remarks
If array1 and array2 have different numbers of data points, and type \u003d 1 (paired), then STUDENT.TEST returns the # N / A error value.
The tails and type arguments are truncated to integer values.
If tails or type is nonnumeric, STUDENT.TEST returns the #VALUE! Error value.
If tails is any value other than 1 and 2, STUDENT.TEST returns the #NUM! Error value.
The STUDENT.TEST function uses the data from the arguments array1 and array2 to calculate a non-negative t-statistic. If tails \u003d 1, STUDENT.TEST returns the likelihood of a higher t-statistic assuming that array1 and array2 are samples from a population with the same mean. The value returned by the STUDENT.TEST function when "tails" \u003d 2 is twice the value returned when "tails" \u003d 1, and corresponds to the probability of a higher absolute value of the t-statistic, assuming that "array1" and "array2" are samples belonging to a general population with the same mean.
Example
Copy the sample data from the following table and paste it into cell A1 of a new Excel worksheet. To display the results of formulas, select them and press F2, and then press Enter. Change the width of the columns as needed to see all the data.
The /-Student's criterion refers to parametric, therefore, its use is possible only when the experimental results are presented in the form of measurements on the last two scales - interval and ratios. Let us illustrate the possibilities of the Student's test with a specific example.
Suppose you need to find out the effectiveness of training in shooting according to a certain method. For this purpose, a comparative pedagogical experiment is carried out, where one group (experimental), consisting of 8 people, is engaged in the proposed experimental methodology, and the other (control) - according to the traditional, generally accepted. The working hypothesis is that the new technique you propose will be more effective. The result of the experiment is a control shooting of five shots, according to the results of which (Table 6) it is necessary to calculate the reliability of the differences and check the correctness of the hypothesis put forward.
Table 6
What needs to be done to calculate the reliability of differences according to the Student's t-test?
1. Calculate the arithmetic mean values \u200b\u200bX for each group separately using the following formula:
where Xt -the value of an individual dimension; i is the total number of measurements in the group.
Having put in the formula the actual values \u200b\u200bfrom the table. 6, we get:
Comparison of the arithmetic mean values \u200b\u200bproves that in the experimental group this value (X, \u003d 35) is higher than in the control group. (Xk\u003d 27). However, for the final statement that the participants in the experimental group learned to shoot better, one should make sure that the differences (/) between the calculated arithmetic mean values \u200b\u200bare statistically significant.
2. In both groups, calculate the standard deviation (5) using the following formula:
: de Ximax- the highest rate; Ximm- the smallest indicator; TO- tabular coefficient. The procedure for calculating the standard deviation (5): - determine Xitraxin both groups; - define Ximiain these groups; - determine the number of measurements in each group (l); - find a special table (Appendix 12) the value of the coefficient TO,which corresponds to the number of measurements in group (8). To do this, in the leftmost column under the index (and) we find the number 0, since the number of measurements in our example is less than 10, and in the top line - the number 8; at the intersection of these lines - 2.85, which corresponds to the value of the coefficient. AH at 8 test --- substitute the obtained values \u200b\u200binto the formula and make the necessary calculations:
3. Calculate the standard error of the arithmetic mean (t) by the formula:
For our example, the first formula is suitable, since p< 30. Вычислим для каждой группы значения:
4. Calculate the average error of the difference using the formula:
5. Determine the significance of the differences using a special table (Appendix 13). For this, the resulting value (t)compared with the cutoff at 5% significance level (t0fi5)FOR the number of degrees of freedom / \u003d pe + pc- 2, where pack pc ~the total number of individual results, respectively, in the experimental and control groups. If it turns out that the experiment obtained tgreater than the boundary value (/ 0) o5)\u003e m0 the differences between the arithmetic means of the two groups are considered reliableat a 50% significance level, and vice versa, in the case when the obtained t lessboundary value t0<05, it is believed that the differences unreliableand the difference in the arithmetic means of the groups is random. The cut-off value at a 5% significance level (G0\u003e 05) is determined as follows:
calculate the number of degrees of freedom / \u003d 8 + 8 - 2 \u003d 14;
find from the table (Appendix 13) the boundary value tofi5at / \u003d 14.
In our example, the table value tQ<05 = 2.15, compare it with the calculated G,which is 1.7, i.e. less than the boundary value (2.15). Consequently, the differences between the arithmetic mean values \u200b\u200bobtained in the experiment are considered unreliablewhich means that there is not enough reason to say that one method of training in shooting was more effective than another. In this case, we can write: / \u003d 1.7 for / »\u003e 0.05, which means that in the case of 100 similar experiments, the probability (R)obtaining similar results when the arithmetic mean values \u200b\u200bof the experimental groups turn out to be higher than the control ones, more than 5% significance level or less than 95 cases out of 100. The final design of the table, taking into account the calculations obtained and with the corresponding parameters, may look like this.
With relatively large numbers of measurements, it is conventionally assumed that if the difference between the arithmetic means is equal to or greater than three of its errors, the differences are considered reliable. In this case, the significance of the differences is determined by the following equation:
As mentioned at the beginning of this section, the Student's t-test can only be applied in cases where measurements are made on a scale of intervals and ratios. However, in pedagogical research, there is often a need to determine the reliability of differences between the results obtained according to the Scale of names or order. In such cases, use nonparametriccriteria. Unlike parametric, nonparametric criteria do not require the calculation of certain parameters of the results obtained (arithmetic mean, standard deviation, etc.), which is why their names are mainly related. Let us now consider two nonparametric criteria for determining the reliability of differences between independent results obtained on the scale of order and names.
where f is the degree of freedom, which is defined as
Example ... Two groups of students were trained using two different methods. At the end of the training, they were tested throughout the course. It is necessary to assess how significant the differences in the acquired knowledge are. The test results are presented in table 4.
Table 4
Let's calculate the sample mean, variance and standard deviation:
Determine the value of t p by the formula t p \u003d 0.45
According to table 1 (see Appendix), we find the critical value of t k for the significance level p \u003d 0.01
Conclusion: since the calculated value of the criterion is less than the critical 0.45<2,88 гипотеза Но подтверждается и существенных различий в методиках обучения нет на уровне значимости 0,01.
Algorithm for calculating Student's t-test for dependent samples of measurements
1. Determine the calculated value of the t-criterion by the formula
where 
3. Determine the critical value of the t-criterion according to table 1 of the appendix.
4. Compare the calculated and critical values \u200b\u200bof the t-test. If the calculated value is greater than or equal to the critical value, then the hypothesis of equality of the mean values \u200b\u200bin two samples of changes is rejected (Ho). In all other cases, it is taken at a given level of significance.
U- criterionManna- Whitney
Criterion assignment
The criterion is intended to assess the differences between two nonparametric samples by the level of any quantitatively measured feature. It allows you to distinguish between small samples when n< 30.
Description of the criterion
This method determines if the overlapping zone between the two rows is small enough. The smaller this area, the more likely the differences are significant. The empirical value of the U criterion reflects how large the area of \u200b\u200bcoincidence between the rows is. Therefore, the smaller U, the more likely the differences are significant.
Hypotheses
BUT: The trait level in group 2 is not lower than the trait level in group 1.
HI: The trait level in group 2 is lower than the trait level in group 1.
Algorithm for calculating the Mann-Whitney test (u)
Transfer all test subjects' data to individual cards.
Mark the cards of subjects from sample 1 with one color, say red, and all cards from sample 2 with another, for example, blue.
Arrange all the cards in a single row according to the degree of growth of the characteristic, regardless of which sample they belong to, as if we were working with one large sample.
where n 1 is the number of subjects in sample 1;
n 2 - the number of subjects in sample 2,
T x - the larger of the two welded sums;
n x - the number of subjects in the group with a larger sum of ranks.
9. Determine the critical values \u200b\u200bof U according to Table 2 (see Appendix).
If U emp.\u003e U cr0.05, then the hypothesis But is accepted. If U emp. ≤ U cr, then it is rejected. The smaller the U value, the higher the reliability of the differences.
Example. Compare the effectiveness of the two teaching methods in the two groups. The test results are presented in table 5.
Table 5
Let's transfer all the data to another table, highlighting the data of the second group by underlining and making the ranking of the total sample (see the ranking algorithm in the guidelines for task 3).
|
The values | ||||||||||||||||||
Find the sum of the ranks of two samples and choose the largest of them: T x \u003d 113
Let us calculate the empirical value of the criterion according to formula 2: U p \u003d 30.
Let us determine the critical value of the criterion according to Table 2 of the Appendix at the significance level p \u003d 0.05: U k \u003d 19.
Output: since the calculated value of the criterionU is more than critical at a significance level of p \u003d 0.05 and 30\u003e 19, then the hypothesis of equality of means is accepted and the differences in teaching methods are insignificant.
