Student criterion calculation. Classical methods of statistics: Student's t-test
1. Student's method (t-test)
This method is used to test the hypothesis of the significance 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 of the 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 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* - Fisher's angular transformation
This criterion evaluates the significance of differences between the percentages of two samples in which the trait of interest is registered.
The empirical value φ* is calculated by the formula:
φ*=(φ 1 - φ 2) . , where
φ 1 - the 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 level of significance φ* of the empirical value is determined by a special table. The larger the value of φ*, the more likely it is that the differences are significant.
2.2 Research results and their analysis
2.2.1 Features of adaptation of patients with a chronic disease
To study the degree of adaptation, the method of diagnosing socio-psychological adaptation by K. Rogers and R. Diamond was used.
Based on the analysis of the integral indicator of adaptation, 3 experimental groups of subjects were identified:
1. with high level adaptations - group A.
The value of the adaptation index is from 66 to 72 points. (M=67)
2. with an average level of adaptation - group B.
The value of the adaptation index is from 49 to 65 points. (M=56.6)
3. with a low level of adaptation - group C.
The value of the adaptation index is from 38 to 48 points. (M=41.3)
The significance of differences in the level of adaptation between the experimental groups was tested using 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 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%).
An analysis was made of gender differences in the level of adaptation of patients with chronic diseases. It was found that the majority of women and men are characterized by an average level of adaptation (65% and 63%, respectively) - see tab. No. 1.
Table #1
Gender differences in the adaptation of patients with chronic diseases
(by groups of subjects, %)
| Subjects | Experimental groups | ||
| A (high hell) | B (average hell) | C (Low AD) | |
| Men | 20 | 63 | 17 |
| Women | 19 | 65 | 16 |
The significance of differences was determined using Fisher's φ-test. It was found that in none of the experimental groups the differences in adaptation between men and women are significant. (Group A–φ=0.098, group B - φ=0.161, group C - φ=0.106).
2.2.2 Personality characteristics of patients with chronic diseases with varying degrees adaptation
Let us first consider the features of the self-consciousness of the experimental groups under test.
Group A (high level of adaptation)
The results obtained using the scale "Self-acceptance" showed that most of the subjects in this group have a high and medium level of self-acceptance (33%). The subjects of group A had no low scores on the “Self-acceptance” scale.
Thus, patients with chronic diseases with a high level of adaptation highly appreciate their appearance, their ability to cope with difficult situations, and consider themselves interesting as a person.
In the study of gender differences, it was found that women with a high level of adaptation in the post-labor period more often show a high level of self-acceptance (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" technique was used.
The interpretation of the data obtained with the help of LD was carried out according to 3 factors:
Grade (O)
Activity (A)
For each factor, in accordance with standard norms, 5 levels are distinguished:
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 were not found in any group for all factors, therefore, when interpreting the results, this category is not investigated.
The results on the assessment factor indicate the level of self-esteem; according to the factor of strength about the development of the volitional sides of the personality; according to the activity factor of extraverted personality.
When analyzing the results in group A (with a high level of adaptation), no low values were found for any factor (assessment, strength, activity), which corrects with the data obtained using the "Self-acceptance" school.
Analysis of the results by factors revealed the following features:
The majority of the subjects in this group found the optimal level of self-esteem in terms of strength (58% - average values, 17% - high). There are also very high values (25%).
This indicates that patients with chronic diseases with a high level of adaptation are self-confident, independent, rely on their own strength in difficult situations.
According to the assessment factor in group A, the values of most of the subjects are referred to the optimal level (high values - 50%, average - 25%). There are also very high values (25%). This suggests that the subjects accept themselves as a person, are aware of themselves as carriers of positive, socially desirable characteristics.
According to the activity factor large quantity medium (42%) and high (33%) values. Very high values are found in 28% of the subjects. These results indicate a high activity of patients with chronic diseases with a high level of adaptation, sociability.
Statistical processing using Fisher's φ-criterion (0.05 significance level) revealed significant differences in power and evaluation factors. In the subjects of the experimental group A, the average values prevail in the strength factor, and the scores are high in the factor. On the basis of which 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.
When studying gender differences, significant differences were found in terms of strength and evaluation factors (φ-criterion, p=0.01). See table. 2, 3.
Table number 2
Sex differences in the factor "Strength" LD (%)
|
Subjects | low | the average | high | very high |
| men | 0 | 33 | 17 | 50 |
| women | 0 | 83 | 17 | 0 |
MeaningGroup A women in most cases show an average value for the “Strength” factor (83%), while men have a very high value (50%).
Table #3
Gender differences in group A by the factor "Score" 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 on the "Score" factor, while men have average (50%) and high (33%).
Thus, men with a high level of adaptation highly appreciate their volitional qualities, self-confidence, and women with a high level of adaptation highly appreciate their social qualities, level of achievement.
To study the localization of control over significant events, the “Internality” scale was used.
When analyzing the results, no low values for this factor were found in the subjects of group A. Medium (50%) and high values (50%) are equally present. This indicates that people with a high level of adaptation in the post-work period believe that the majority important events in their lives is the result of their own actions, that they can control them, and thus they feel responsible for these events and for how life in general develops.
Gender differences according to the criterion "Internality" are not statistically significant.
Thus, we can conclude 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 a person, self-confident, independent, evaluate themselves as active and sociable. Men highly appreciate their strong-willed qualities, ability to cope with difficulties, and women highly appreciate their social qualities.
People of this group tend to rely on their own strength, they know how to manage themselves, their actions, consider themselves responsible for how their life develops as a whole.
Group B (medium level of adaptation)
The results obtained using the scale "Self-acceptance" shows that most of the subjects in this group have an average level of self-acceptance (90%). With a high level of self-acceptance - 5%, with a low level - 5%.
No statistically significant gender differences were found.
Analysis of the results according to the method of personal differential was carried out according to the factors of strength, assessment, activity. See table. 4.
Table No. 4
Representation of subjects of group B with different levels of self-esteem (by factors, in %)
An analysis of the results by factors showed that the majority of the subjects in group B had an optimal level of self-esteem in terms of the strength factor (75% - average values, 17% high). There are also low (5%) and very high (2.5%) levels of values.
According to the evaluation factor, an adequate level of self-esteem prevails (62.5% - average level of values, 10% - high). Low rate in 2.5% of the subjects. A large percentage of very high scores (25%).
According to the activity factor, the largest number of optimal values (60% - medium, 22.5% - high). Low values 7.5%, very high - 10%.
When conducting statistical processing using the φ-criterion (p≤0.01), significant differences were revealed in terms of the facts of strength and evaluation at a "very high" level of values. Group B subjects tend to exaggerate their social qualities.
The study of gender differences revealed differences in all 3 factors (φ-criterion p≤0.05).
Sex differences in the "Strength" factor
Among men, no low values were found for this indicator. Whereas women have 10%.
A very high level of values occurs in men of the group in 5% of cases, in women it does not occur. This indicates that men tend to exaggerate their strong-willed qualities, while women tend to underestimate them.
Sex differences in the nature of "Assessment"
Women are more likely than men to have very high values for this factor (33% women, 16% men).
Sex differences in the "Activity" factor
In women of this experimental group, very high values for this factor are found more often than in men (44% of women, 5% of men).
When analyzing the results on the "Internality" scale, it was found that the majority of patients with chronic diseases with an average level of adaptation have an average level of internality index (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 people from this group do not consider themselves capable of controlling the events in their lives, they attribute 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-work period show an optimal level of self-esteem, i.e. they accept themselves as a person, satisfied with themselves. A certain percentage of people in this group have overestimated self-esteem, as well as underestimation, which indicates personal immaturity, the inability to properly evaluate themselves, the results of their activities. The subjects of this group tend to overestimate their social qualities.
In men of this group, there is a tendency to overestimate their volitional qualities, and in women, social ones.
Group C ( low level adaptation)
The results obtained using the "self-acceptance" scale showed that most of the subjects in this group have a low level of self-acceptance (70%). Some of the subjects have an average level of self-acceptance (30%). No high scores were found on 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 gender differences, it was revealed that women with a low level of adaptation more often have an average level of self-acceptance (66%), while men in all cases (100%). Therefore, women with a low level of adaptation will not always have a low level of self-acceptance.
Analysis of the results according to the method "Personal differential" was carried out according to the factors of strength, assessment, activity. See table. 5.
Table 5 Representation of the subjects of groups C with different levels of self-esteem (by factors, in %)
When analyzing the results of group C, there were no very high values for any factor, which correlates with the data obtained using the "Self-acceptance" scale. High values are found only for the assessment factor (10%).
According to the strength factor, most of the subjects have low values (60%). There are also average values (40%).
For assessment and activity factors, the maximum number of average values (80%). Low values for the assessment factor are found in 10% of the subjects, for the activity factor in 20%.
Thus, the subjects of group C are dominated by moderately low self-esteem values. The test subjects of this group rate their volitional qualities especially low.
When studying gender differences, significant differences were revealed in terms of the strength factor (φ-criterion, 0.03) and assessment.
Men with a low level of adaptation rate their volitional qualities low (80%, while only 49% as women;), and women's social qualities (20% - women, men - 0%)
When analyzing the results on the "Internality" scale, it was found that most 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 most patients with chronic diseases with a low level of adaptation tend to attribute more importance to external circumstances, do not consider themselves capable of controlling their own lives.
Thus, we can conclude 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, dissatisfied with their own behavior, level of achievement. Men in this group rate low self-confidence and ability to cope with difficulties, while women rate themselves low as a whole as a person.
People in this group believe that most of the events in their lives are the result of chance or other people's actions.
Comparative analysis of indicators of self-awareness by groups of subjects made it possible to identify significant differences.
The subjects of group A (high level of adaptation) are characterized by a high level of self-acceptance (67%) compared with the subjects of group B (5%), φ*=4.45; p≤0.01) and group C (0%).
In group C (low level of adaptation) there are more low values (70%) than in group B (5%) - φ*=3.57; p ≤ 0.01 and group A (0%).
According to the strength factor (personal differential), the subjects of group A are more likely to have very high (25%) and high (17%) values than the subjects of group C (0% and 0%).
In group C, there are more low values (60%) than in group A (0%).
By the nature of the assessment in group A, high values are more common (50%) than in group C (10%) - φ * = 2.16; p ≤ 0.01.
In group C, low values (10%) are more common than in group A (6%) in terms of the assessment factor and average values (80%) than in group A (25%) - φ * = 2.72; p ≤ 0.01.
According to the activity factor in group A, there are more very high (25%) and high (33%) values than in group C (0%). Group C has more low values (20%) than group A (6%).
The subjects of group A are characterized by a high level of internality (50%) compared with the subjects of group C (10%) - φ*=2.16; p ≤0.01
The subjects of group C are more likely to have a low level of internality (60%) than the subjects of group A (0%) and the subjects of group B (7.5%) - φ*=3.44; p ≤0.01
Thus, the subjects of group A have, in general, more optimal self-esteem for personal well-being and are characterized by more confident, conscious attitude to life.
2.2.3 Features of the motivational-need sphere of the personality of the subjects
To study the motivational-required sphere, the technique of unfinished sentences was used (see appendices). The results are analyzed 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, in the responses of the subjects, the most common expectation of achievements is 29% (“I will learn to drive a car”), interest in the problems of society -21% (“I will be glad if life in the country improves”), the hope for maintaining the same level life -21% ("I hope that I will be as active"), anxiety about relatives -13% ("In the future, my life is the life of my children").
2. The subjects of group A, assessing their past, note that: they realized their plan, they realized themselves - 54% ("Looking back at my life, I think that life was not lived in vain." "What I aspired to, I managed to achieve"), partially implemented their plans - 21% ("The family turned out to be good, but they devoted little time and devote to children"). 17% of patients with chronic diseases of group A admit the fallacy of their goals, aspirations in the past ("Strived for what did not matter")
3. Analysis of the statements of patients with chronic diseases regarding the fact of the disease showed that some of the subjects were very worried -25%, and some took it for granted without any special feelings -21%
The data obtained additionally with the help of a conversation showed that the subjects of group A are characterized by a variety of interests.
Among them are reading (83%), watching TV (83%), walking (75%), sports (50%), talking (33%), meeting friends and relatives (25%). It can be assumed that the presence of a wide range of interests helps chronic patients in 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 always be with me").
When analyzing the results obtained using the "Acceptance of others" scale, it was revealed 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, my relations with them are warm and friendly.
The data obtained through the conversation showed that 67% of patients with chronic diseases of 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% do not have enough communication.
Thus, despite 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 say that the disease is a milestone in life (“Illness for me is a certain border, life “before” and “after”, it made it possible to evaluate my life”).
6. In the study of gender differences, it was found that in relation to the future, men more often than women show interest in the problems of society (67% of men, 33% of women), i.e. show great social activity.
Group B (medium level of adaptation)
1. In statements about the future, in the answers of the subjects, domestic problems are most often found -20% ("I intend to make repairs at home"), hopes for maintaining the same standard of living -19%. ("I will be very happy if I continue to live like this"), health care -14% ("I intend to lead healthy lifestyle life, this is the main thing for life"), concerns about relatives -10%, 9% of patients expect a better future ("I hope that future life will be better than now").
2. Subjects of group B, assessing their past, note that: they partially implemented their plans - (38%); admitted the fallacy of their goals, aspirations - (35%); realized their plans, realized themselves - (15%).
3. Analysis of the statement about the news of the disease showed that the majority of the subjects were relatively calm about this event (65%), part of the group was very worried (25%) and a small percentage were indifferent (10%).
The data obtained through the conversation indicate that the subjects of group B are characterized by a variety of interests (reading, radio, television, cinema, fishing, meeting friends, pets, etc.). What is important for a calm attitude to their status as a chronic patient.
4. In statements related to communication with relatives, the subjects express concern for loved ones (57%) and expect support from loved ones (25%). In some answers, there is a 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 revealed that the subjects of group B have an average level of acceptance of others (78%). Some of the subjects show 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 the desire to accept other people.
During the conversation, it was found that patients in this group regret that now they communicate less, but in general they are satisfied with communication (73%), 15% are completely satisfied with interaction with other people and 12% believe that they lack communication.
5. In statements reflecting the attitude towards the disease, the subjects of group B write that the disease is a burden for loved ones (27.5%), the disease is a frontier or border (17.5%). Fear of death is heard in the answers of 20 subjects, 25% says that the disease is not the end of life.
Group C (low level of adaptation)
1. In statements about the future in the answers of the subjects, the most common is the lack of expectation of any changes - 30% ("In the future, my life will not change"), expectations of difficulties - 22.5% ("In the future, my life will become even more difficult") , domestic problems - 17.5%. Health concerns are reflected in the answers of 15% of the subjects. The category "expectations of achievement" is not found in the responses of subjects with a low level of adaptation.
2. Assessing their past, group C people 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 a part was unlucky").
15% admit the fallacy of their goals, aspirations 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 severely - 60%; 30% reacted relatively calmly and 10% indifferently.
During the conversation, it was found that the subjects of group C are characterized by passive interests (watching TV, knitting, reading), and many note the absence of a favorite pastime. We can say that the lack of interests makes it difficult to adapt to the disease, since it is not compensated by significant activities.
1. In statements related to communication with relatives, the subjects expect support from relatives (50%) and express fear of being left alone 30%. The answers of 20% of the subjects sound concern for relatives.
2. When analyzing the results obtained using the "Acceptance of others" scale, it was revealed that the subjects of group C have low (60%) and medium; (40%) the level of acceptance of others, which indicates that the people of this group are restrained in communicating with others, feel hostility towards those who surround them.
An 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 are satisfied, but dissatisfied with the fact that the social circle has narrowed (30%).
Giving their definition of the disease, patients of group C write that the disease is the end of life (40%), they express fear of death (20%), the disease is characterized as a milestone 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*=1.604; p≤0.05 and in group C (0%). In group B, a large percentage of answers are related to everyday problems (20%) than in group A (4%) φ*=1.59; p≤0.05.
2. In statements about the past, the subjects of group A more often (54%) than the subjects of group B (15%) note that they have carried out their plan, realized themselves (φ * = 2.42; p≤0.01), and more often than in group C (5%) φ*=2.802; p≤0.01.
The subjects of group C more often (30%) than the subjects of group A (0%) and than the subjects of group B (6%) - φ*=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*=3.306; p≤0.01).
In group B, a larger (38%) percentage of subjects noted that they were able to partially realize themselves than in group C (10%), φ*=1.934; p≤0.02.
3. In statements related to retirement, the subjects of group C more often (60%) than the subjects of groups A (25%) and the subjects of group B (25%) write that they were worried (φ*=1.693; p≤0.04 ).
4. Describing relationships with relatives, the subjects of group A more often (75%) than the subjects of group C (20%) express concern for loved ones (φ*=2.725; p≤0.01).
Patients with chronic diseases in group C are more likely (30%) than in group A (0%) to express fear of being alone.
5. In the answers of the subjects of group A, the definition of the disease as a summing up is more common (17%) than in group C (0%) and in group B (2.5%) - φ*=1.61; p≤0.05.
Significant differences were found on the Acceptance of Others scale. Sick people in group A more often (58%) show a high level of acceptance of others than in group C (0%) and group B (10%) φ*=3.302; p≤0.01.
The subjects of group C more often (60%) show a low level of acceptance of others than in group A (0%) and group B (12.5%) - φ*=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.
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When can the Student's t-test be used?
To apply the Student's t-test, it is necessary that the original data have normal distribution . In the case of applying a two-sample test for independent samples, it is also necessary to satisfy the condition equality (homoscedasticity) of variances.
If these conditions are not met, when comparing sample means, similar methods should be used. nonparametric statistics, among which the most famous are Mann-Whitney U-test(as a two-sample test for independent samples), and sign criterion and Wilcoxon test(used in cases of dependent samples).
To compare means, Student's t-test is calculated using the following formula:
where M 1- arithmetic mean of the first compared population (group), M 2- arithmetic mean of the second compared population (group), m 1 - mean error first arithmetic mean, m2- the average error of the second arithmetic mean.
How to interpret the value of Student's t-test?
The resulting value of 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). Finding the number of degrees of freedom f according to the following formula:
f \u003d (n 1 + n 2) - 2
After that, we define critical value Student's t-test for the required significance level (for example, p=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 of the criterion:
If the calculated value of Student's t-test equal or greater critical, found in the table, we conclude about statistical significance differences between compared values.
If the value of the calculated Student's t-test smaller tabular, which means that the differences between the compared values are not statistically significant.
Student's t-test example
To study the effectiveness of a new iron preparation, two groups of patients with anemia were selected. In the first group, patients received new drug and the second group received a placebo. After that, the level of hemoglobin in 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 differences, we use Student's t-test, calculated as the difference between the means divided by the sum of squared errors:
After performing the calculations, the value of the t-test was equal to 4.51. We find the number of degrees of freedom as (34 + 40) - 2 = 72. We compare the obtained value of Student's t-test 4.51 with the critical value at p=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).
The Fisher distribution is the distribution of a random variable
where random variables X 1 and X 2 are independent and have chi distributions - the square with the number of degrees of freedom k 1 and k2 respectively. At the same time, a couple (k 1 , k 2) is 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 k2 is 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 work.
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 table of critical values.
Form start
| Number of degrees of freedom, f | Student's t-test value at p=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 |
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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 STUDENT.TEST function syntax has the following arguments:
Array1 Required. First set of data.
Array2 Required. Second set of data.
Tails Required. The number of distribution tails. If tails = 1, STUDENT.TEST returns a one-tailed distribution. If tails = 2, STUDENT.TEST returns a two-tailed distribution.
Type Required. The type of t-test being performed.
Options
Remarks
If the arguments "array1" and "array2" have a different number of data points, and "type" = 1 (pair), then the STUDENT.TEST function returns the #N/A error value.
The "tails" and "type" arguments are truncated to integer values.
If tails or type is not a number, STUDENT.TEST returns the #VALUE! error value.
If tails is any value other than 1 or 2, STUDENT.TEST returns the #NUM! error value.
The STUDENT.TEST function uses the given arguments "array1" and "array2" to calculate a non-negative t-statistic. If tails = 1, STUDENT.TEST returns the probability of a higher t-statistic based on the assumption that "array1" and "array2" are samples that belong to a population with the same mean. The value returned by STUDENT.TEST when tails = 2 is twice the value returned when tails = 1 and corresponds to the probability of a higher absolute value of the t-statistic, based on the assumption that "array1" and "array2" are samples belonging to the 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 sheet. To display formula results, select them and press F2 followed by ENTER. Change the width of the columns, if necessary, to see all the data.
/-Student's criterion refers to parametric, therefore, its use is possible only when the results of the experiment are presented in the form of measurements on the last two scales - interval and ratios. Let us illustrate the possibilities of the Student's criterion on a specific example.
Suppose you need to find out the effectiveness of shooting training in a certain technique. 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 method you propose will be more effective. The result of the experiment is a control firing 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 significance of differences according to Student's /-test?
1. Calculate the arithmetic mean values of X for each group separately using the following formula:
where xt- the value of an individual measurement; i is the total number of measurements in the group.
Putting in the formula the actual values from the table. 6, we get:
Comparison of the arithmetic mean values proves that in the experimental group this value (X, = 35) is higher than in the control group. (Hk= 27). However, for the final statement that those involved in the experimental group learned to shoot better, one should be convinced of the statistical significance of the differences (/) between the calculated arithmetic mean values.
2. In both groups, calculate the standard deviation (5) using the following formula:
:de Ximax-- the highest indicator; ximm-- the smallest indicator; To-- tabular coefficient. Order of Evaluation standard deviation(5): -- define Xitrax in both groups; -- define Ximia in these groups; -- determine the number of measurements in each group (l); -- find in a special table (Appendix 12) the value of the coefficient TO, which corresponds to the number of measurements in the group (8). To do this, in the leftmost column under the index (i) 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.
3. Calculate the standard error of the arithmetic mean (m) using 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. Using a special table (Appendix 13), determine the significance of differences. For this, the resulting value (t) compared with the boundary value at 5% significance level (t0fi5) FOR the number of degrees of freedom/= 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 experimental t greater than the boundary value (/0)o5)> m0 the differences between the arithmetic means of the two groups are considered credible at a 50% significance level, and vice versa, in the case when the obtained t less limit value t0<05, it is believed that the differences unreliable and the difference in the arithmetic mean of the groups is random. The boundary value at the 5% significance level (Г0>05) is determined as follows:
calculate the number of degrees of freedom/= 8 + 8 - 2 = 14;
find the table (Appendix 13) boundary value tofi5 at /= 14.
In our example table value tQ<05 = 2.15, compare it with the calculated G, which is equal to 1.7, i.e. less than the limit value (2.15). Therefore, the differences between the arithmetic mean values obtained in the experiment are considered unreliable, which means that there is not enough reason to say that one method of teaching shooting turned out to be more effective than another. In this case, we can write: / = 1.7 at /> 0.05, which means that in the case of 100 similar experiments, the probability (R) obtaining similar results when the arithmetic mean values of the experimental groups are 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 comparatively large numbers measurements, it is conditionally assumed that if the difference between the arithmetic means is equal to or more than three of its errors, the differences are considered significant. In this case, the reliability of differences is determined by the following equation:
As mentioned at the beginning of this section, Student's /-test can only be applied when measurements are made on a scale of intervals and ratios. However, in pedagogical research often there is a need to determine the Reliability of differences between the results obtained on the Name or Order Scale. In such cases, use nonparametric criteria. Unlike parametric criteria, nonparametric criteria do not require calculation certain parameters the results obtained (arithmetic mean, standard deviation, etc.), which is the main reason for their names. Let us now consider two non-parametric criteria for determining the significance of differences between independent results obtained on a scale of order and names.
where f is the degree of freedom, which is defined as
Example . Two groups of students were trained in two various methods. At the end of the training, they were given a test 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
Calculate the sample mean, variance and standard deviation:
Determine the value of t p by the formula t p = 0.45
According to table 1 (see Appendix), we find the critical value t k for the significance level p = 0.01
Conclusion: since the calculated value of the criterion is less than the critical value of 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 using the formula
, where 
3. Determine the critical value of the t-test according to Table 1 of the Appendix.
4. Compare the calculated and critical values of the t-criterion. If the calculated value is greater than or equal to the critical value, then the hypothesis of equality of the means in the two change samples is rejected (But). In all other cases, it is taken at a given level of significance.
U- criterionManna- Whitney
Purpose of the criterion
The criterion is designed to assess the differences between two non-parametric samples in terms of the level of any trait, quantitatively measured. It allows you to identify differences between small samples when n< 30.
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 U criterion reflects how large the zone of coincidence between the rows is. Therefore, the smaller U, the more likely it is that the differences are significant.
Hypotheses
BUT: The level of the feature in group 2 is not lower than the level of the feature in group 1.
HI: The level of the trait in group 2 is lower than the level of the trait in group 1.
Algorithm for calculating the Mann-Whitney criterion (u)
Transfer all the data of the subjects to individual cards.
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.
Lay out all the cards in a single row according to the degree of growth of the attribute, 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 rand 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 2 (see Appendix).
If U emp.> U kr0.05, 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 two teaching methods in 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 with an underline and do the ranking of the total sample (see the ranking algorithm in the guidelines for task 3).
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Values | ||||||||||||||||||
Find the sum of the ranks of two samples and choose the largest of them: T x = 113
Let's calculate the empirical value of the criterion according to the formula 2: U p = 30.
Let us determine the critical value of the criterion from Table 2 of the Appendix at a significance level p = 0.05: U k = 19.
Conclusion: since the calculated value of the criterionUis greater than the critical level at the significance level p = 0.05 and 30 > 19, then the hypothesis of the equality of the means is accepted and the differences in teaching methods are insignificant.
