The model of the logarithmic distribution of the subdivision of the English mathematician Fisher was the first attempt to describe the relationship between the number of species and the number of individuals of these species. With particular success, this model was corroborated in entomological studies. The tabula was first established by Fischer as a theoretical model for describing the distribution of species in collections. This model and statistics of variability was attributed to the report of L. R. Taylor and his co-authors.

The subdivision of the frequencies of the views for the logarithmic subdivision is described by the following sequence:

de  X– number of species represented by one individual, х 2/2 – number of species represented by two individuals together.

The logarithmic model has two parameters  and x. Tse means what is obligatory for the selection N I saw a lot S there is only one possible difference between the frequencies of the sights for їх performance diversity, so like Xє functions Nі S. What is the greater vibirka, vytyagnuta z tsієї spіlnoti, tim is more important X And yet there are fewer individuals, as if there are a lot of species, represented by one individual in the selection. Two parameters Sі N(Zhalna kіlkіst osobin) pov'yazanі mіzh yourself fallow
, de - index of rіznomanіtnostі, which can be taken from equal:

,

de sum of all features N, what to lie S types:

The model of a logarithmic distribution, which is characterized by a small number of rich species and a great frequency of "rare" ones, can describe with the greatest clarity such a complexity, the structure of which is characterized by one or more environmental factors.

As shown by recent studies conducted by Megarran in Ireland, such a series of subdivisions of different types of growths of the terrestrial layer in coniferous cultures in the minds of low lighting.

5.3.3. log-normal distribution

For greater sleep, the characteristic log-normal distribution has changed the appearance, but sound the model on the big, wide, and different view. Such a distribution is typical for systems, if the value of a variable is determined by a great number of factors.

Tsya model vpershe bula zastosovana to razpodіlu raznomanіtnyh vidіv Preston. On the basis of various empirical materials, he showed that the frequency of sightings in great varieties of subdivision is consistent with the logarithmically normal law. According to the method developed by him, the frequency classes are grouped according to the number of features, placed in the gaps, as if they are surrounded by numbers of geometric progression. Preston put on the whole great number of views on the scale of the logarithm on the substituent 2 (log 2) and named the classes that appeared as octaves. But for the description of the model, you can twist the basis of the logarithm. On the graph, the subdivision of the frequencies of the species after subtracting in this way the classes of numbers are given in the home curve of the normal subdivision, the zirzanoї zlіva, in the region of the frequencies of the rare species.

Please call to sign up at the form:

, de

S R – theoretical number of views in an octave, rotations in R octaves in a modal octave; S mo- the number of views in the modal octave;  - Standard deviation of the theoretical log-normal curve, expressed in terms of octaves.

Rice. 5.3.2. log-normal rozpodіl

The log-normal distribution is described by a symmetrical “normal”, that is, a ring-like curve (Fig. 5.3.2.). However, if the data, as if it were not shown, was taken from the circumscribed selection, then the left part of the curve (to see it as a normal, unchecked one) will be expressed indistinctly. Preston called such a point of narrowing the curve of evil "the line of dependence". The line of dependence can break to the left if the vibration is increased. An arrow is pointed at the little one. Most of the vibes are expressed only part of the crooked right-handed mode. Only for the greatness of the number of tributes, chosen in the great biogeographic territories, are being squandered. S- The shape of the curve indicates the folding nature of the differentiation and the overlapping of the niches. The greater number of species in natural, open ecosystems is in the minds of the mind for resources, and not in the minds of direct competition; impersonal adaptation gives the possibility of extending our lives without competitive blame for the place of residence. Tsya model nayimovіrnіsha for undamaged spіlnot.

Vipadkovy change Y can be logarithmically normal rozpodіl іz parameters μ and σ, so vipadkovy zmіnna X = lnY can be normal rozpodіl іz these same parameters μ and σ. Knowing the nature of the ligament between changes X and Y, we can easily induce a graph of the width of the fluctuation of the drop-down change with a logarithmically normal distribution (Figure 4.2).

Figure 4.2 - The curves of the logarithmically normal distribution of the sub-line with different values ​​of the parameters μ and σ

If there is a change in X, the function of the width of the movement, as it is determined by the formula (4.6), and if X = lnY, then:

Possible stars for y > 0:

The value is prominent, that the value is changing, that it corresponds to a logarithmically normal distribution, it can take on more than positive values. As shown in figure 4.2, the curves of the function f(y) may have left-sided asymmetry, as the greater the value of the parameters μ and σ. The skin curve can have one maximum and is assigned for all positive values.

The calculation of the mathematical scaling and the variance of the vertical change with the logarithmically normal distribution does not become particularly difficult:

The way to substantiate and introduce new changes in integrals 4.15 and 4.16 is taken:

In addition, in order to calculate the flexibility of the change in Y with logarithmically normal distribution of the value f(y, μ, σ), in order to calculate the value in the interval (a, b), we should take the integral:

However, in practice it is easier to speed up the fact that the logarithm of the vertical change Y may be normal. The immediacy of what a ≤ Y ≤ b is equivalent to the imaginaryness of what
lna ≤ lnY ≤ lnb.

Let's calculate the magnitude of the fact that vipadkova is changing with logarithmically spread under μ = 1, σ = 0.5, if the value is in the interval (2, 5). Maemo:

From the table of logarithms, ln2 = 0.6932 and ln5 = 1.6094 are known.

Setting lnY = X, we can write:

Moreover, the vipadkova change of X is subordered to the normal distribution from the average values ​​μ = 1 and the standard deviations σ = 0.5. Now it’s easy to calculate the possibility of shukan according to the tables of the integral function of the normal distribution:

Food for self-control

1 Designation of a rectangular rosette.

2 Graph of the thickness of the ymovirnosti of the varicose vein with a straight-cut rhizome

3 The main meaning of a rectangular rosette.

4 Mathematical refinement of the variance of the varicose veins in a straight-cut rosacea.

5 The role of the normal distribution of mathematical statistics.

6 What is such a normal division and how is it connected with the binomial?

7 Graph of the thickening of the fluctuation of the varicose veins with a normal rosacea.

8 What statistical parameters can be used to restore the normal distribution?

9 Why is normal rozpodіl unperturbable?

10 Equation of a normal curve.

11 What is the normalization of vidhilennya?

12 Alignment of the curve of the normal distribution under the normalized form.

13 What values ​​of μ and σ characterize the normal size of normalized form?

14 How often do these samples fit within the boundaries of ±1?, ±2?, ±3?

15 What does the table of the normal integral of dynamics show?

16 Equation of a logarithmically normal curve.

17 Graph of the thickening of the fluctuation of the varicose vein with a logarithmically normal distribution.

18 What kind of transformation is necessary to work out, so that from the logarithmically normal distribution subtract from the normal distribution?

19 What statistical parameters are used logarithmically to set the normal distribution?

TOPIC 5 Changed the parameters of the selection

5.1 t

5.2 F-rozpodil Fischer-Snedekor

5.3 χ 2 -rozpodil

5.1 t

The law of normal distribution appears beyond the sign n > 20–30. Prote eksperimentator often carry out a large number of vimirivs, restoring their vysnovki on small vibrators. With a small number of warnings, the results sound close and rarely show great inspiration. It is easy to explain this by the law of normal rozpodіlu, zgіdno z yakim ymovіrnіst the appearance of small vіdhilen more, nіzh vіdhilen znachnyh. So, the flexibility of respiration, which exceeds the absolute value of ±2σ, is usually 0.05, or one regression per 20 reversals, and resilience ± 3σ is 0.01, or one regression per 100.

If you want to carry out a sexual inspection, for example, in 4-6 replications, then it’s natural to consider that there will be no average yield readings on parallel plots of great crops. For this reason, it is standard care, bolstered for a small amount, in the majority of cases there will be less, less for all general marriages. Also, in vypadki, it is not possible to rely on the criteria of normal rozpodіla in vysnovki.

Since the beginning of the 20th century, in mathematical statistics, having become a new one directly, which can be called the statistics of small vibes. The most practical value for experimental work is little revealed in 1908. English statistician and chemist V. Gosset t-rozpodіl, scho having taken the name of Student's rozpodіlu (Eng. Student - student, pseudonym V. Gosset).

Rozpodіl t Student's for vibirkovyh mean values ​​are equal:

The numeral of the formula means the vibrancy of the vibratory average in the whole of the average marriage, and the banner:

- є pokanik, scho assessing the value of the standard pardon of the average vibirkovo sukupnostі.

In this way, the value of t is reduced to the vibrancy of the middle marriage, which is expressed in the pardons of the vibrka, taken as a unit.

The maxima of the frequency of the normal and t-rose subdivision are varied, but the shape of the curve of the t-disposition is more likely to lie down depending on the number of steps of the will. With even small values ​​of the degrees of freedom, it looks like a flat-topped curve, moreover, the area, bordered by a curve, is larger, lower with a normal spread, and with an increase in the number of guards (n> 30), the spread of t approaches to normal and passes at ∞.n = new at ∞.

Figure 1.1 shows the differential and integral t-Student's subdivisions at 10 degrees of freedom.

Figure 5.1 - Differential (levoruch) and integral (right-hand) t-student

Razpodil t-Student's maє important value when working with small selections: allows you to designate a confidence interval, which inclines the average sukupnіst , that pervert that chi іnsu hypothesis about general marriage. With whom you can’t, you need to know the parameters of marriage і Sufficient for mother їх estimates μ і σ for the singing obligation of vibrating n.

5.1.1 Behrens-Fischer problem

The revision of the hypothesis about the general middle two groups with normal distribution and uneven variances in mathematical statistics is called the Behrens-Fischer problem and can only be approximated. Why is the importance of evenness of variances in divided groups so important? Without going into the details of the problem, it is significant that the more the variance and the overall selections differ, the stronger the difference between the "counted t-test" and the "Student's t-test". With this difference, the value may be like the t-criterion itself, so such a parameter can be divided as the number of steps of freedom. In its own right, the number of steps of freedom is indicated by the value of the reached (critical) level of significance (p< ...) определяемого для вычисленного значения t-критерия.

Nehtuvannya sledniki, osnovnymi more minds admissibility vikoristannya t-Student's t-criterion, bring to the bottom of the creation of the results of re-verification of hypotheses about the equivalence of the middle. In the robots, de transmitting gypothesis about the ribed cucumber of the middle was led behind the pre-person of the T-Cheriter of Student, I am a lot of criterias of the transfer of normally, є PIDSTAVI VIKORITSISTYA VICHORITS.

Another frequent pardon is the application of the Student's t-test to reconsider hypotheses about the equality of three and more group averages. In this way, it is necessary to establish the so-called global linear model, implemented in the procedure of single-factor analysis of variance with fixing effects.

Let's take a look at the specificity of the victory over Student's t-test. Most of the time, the t-test is victorious in two domains. For the first time, it’s necessary to stop for a re-verification of the hypothesis about the equality of the general middle two independent, unconnected selections (the so-called two-rank t-criterion). In this case, there is a control group and a confirmed group that consists of different objects, the number of which in groups can be different. Another type of winner wins the title of a pair of t-criteria, if one and the same group of objects generates numerical material for revising hypotheses about the middle ones. To that qi vibirka is called fallow, po'yazanimi. For example, they are killed instead of leukocytes in healthy creatures, and then in the creatures themselves, after a prominence with a singing dose of vipromination. In both types, there may be some signs of normality in the skin of the normal groups. The dominance of Student's t-test in the most important way works to show two important aspects.

In another way, it’s not worth talking about those that authors don’t have any alternatives to this criterion, otherwise they can’t stink on their own. It can be said without further ado that in this hour of thoughtlessly applying the Student's t-criterion in more biological robots to bring more shkodi, less coriste.

5.2 F-rozpodil Fischer-Snedekor

As a normally separated marriage, take two independent selections with a total of n 1 and n 2 and adjust the dispersion і with steps of freedom ν 1 \u003d n -1 and ν 2 \u003d n 2 -1, then you can signify the dispersion of dispersions:

The variances should be taken in such a way that the variance of the number book is large, so F ≥ 1.

Rozpodіl F to fallow only in the number of steps of freedom ν 1 і ν 2 (law F-rozpodіlu vіdkriv R.A. Fisher). If two equal selections are independent independent ones from the global population with the general average, then in fact the value of F does not exceed the difference between the two and does not outweigh the theoretical value of the criterion F (F fact< F теор). Если генеральные параметры сравниваемых групп различны, то F факт >Theor. The theoretical value of F for 5% and 1% significance level is given in the table, tabulating only the right critical points for F ≥ 1, so it is generally accepted that a larger variance is reduced to a smaller one.

The curves that detract from the function are subdivided for all possible values ​​of F, especially with a small number of guards, may have an asymmetric shape - a long tail of large values ​​and a large concentration of small values ​​of F (Figure 5.2).

Figure 5.2 - Differential (levoruch) and integral (right-handed)
F-rozpodil Fischer-Snedekor

It is significant that Student's t-rozpodіl є we will call the F-rozpodіlu drop when the number of freedom steps is ν 1 = 1 і ν 2 = ν, which is equal to the number of freedom steps for the t rozpodіlu. In this way, such a spіvіdnoshnja mizh F and t:

5.3 χ 2 -rozpodil

A lot of factual differences are given to the models of theoretical differences (normal, binomial, Poisson) Prote, in practice, there are differences that are strongly influenced by the normal. To assess the degree of diversity or the degree of difference between the numbers of the actual and the theoretical roses, statistical criteria are used, for example, the criterion χ 2 . This criterion zastosovuetsya on the basis of the task of statistical analysis, for example, for the verification of hypotheses: about the independence of two principles, which form the basis for grouping the results in a watchfulness of one totality; about the homogeneity of groups of some deyaky significant characteristics; about the theoretical and experimental curves of numbers. Criterion 2 can be called as a criterion of the right, as well as a criterion of independence, a criterion of homogeneity. The law of rozpodіlu χ 2 (хі-square) according to K. Pirson. The curve was subdivided, taken from the chi-square function:

de f - actual F - theoretical frequencies of the number of objects in the sample. Її kind of heavily deposited in the number of steps of freedom. For a small number of steps of freedom, the curve is asymmetric (Figure 5.3), but for larger numbers, the asymmetry changes and at ν = the curve becomes normal Gaussian.

Rozpodіl χ 2, so very yak і t-rozpodіl, okremy vpadok
F - split for ν 1 = ν і ν 2 = ∞.

Figure 5.3 - Differential (levoruch) and integral (right-hand)
χ 2 -rozpodil

Food for self-control

1 In some cases, it is better to have Student's t-rump than the normal rump?

2 What values ​​should be estimated for the Student's t-score?

3 What is the essence of the Behrens-Fischer problem?

4 How is the F-rozpodil numerically expressed for two independent selections from the totality of the change?

5 What are the characteristic values ​​of variant fallows to deposit F-rozpodil?

6 On what basis can the value of criterion 2 be considered in the statistical analysis of experimental data?

TOPIC 6 Fundamentals of mathematical statistics

6.1 Averages

6.2 Arithmetic mean

6.3 Geometric mean

6.4 Harmonic mid

The logarithmically normal function was known to be widely used in the analysis of the reliability of objects in technology, biology, economics and other. For example, the function can be successfully installed to describe the details of bearings, electronic accessories and other types.

The negative value of the decimal parameter of the distribution is logarithmically normal, so the logarithm of the distribution is logarithmically normal. The number of rozpodіlu for different values ​​is shown in fig. 4.3.

Rice. 4.3.

The area of ​​distribution is described by fallow

de Mх i σ – parameters that are evaluated for the results P try to see:

(4.4)

For the logarithmically normal to the law, the function of supremacy

(4.5)

The mobility of the silent robot can be calculated from the tables for the normal distribution (div. Table P6.1 supplement 6) fallow according to the value of the quantile

Mathematical refinement to the point of view

Mean-square variation and coefficient of variation are positively positive

Yakscho v x ≤ 0.3, then consider what ν x = σ 1%.

Often stop recording fallows for the logarithmically normal law of the distribution of tens of logarithms. Vіdpovіdno up to the law

Estimated parameters lg x 0 and σ are assigned to the results of testing:

Mathematical refinement M x, mean square deviation σ x is the coefficient of variation ν x direction to vіdmovi vіdpovіdno equal

Stock 4.6

Calculate the possibility of a wireless robotic gearbox with a pull t= 103 years, so the resource is distributed log-normally with parameters lg t 0 = 3.6; σ = 0.3.

Solution

We know the value of the quantile and signifi cantly the possibility of a silent work:

Suggestion: R(t) = 0,0228.

Rozpodil Weybulla

The Weibull subdivision function is a two-parameter subdivision distribution. Describing the law by her is universal; The author of this law, V. Weibull, who, while describing this analysis, was experimentally aware of the differences in the volume of steel, between springiness. Weibull's law adequately describes the direction of bearings, elements of electronic equipment, which is used to evaluate the reliability of parts and components of machines, including cars, as well as to assess the reliability of machines in the process and their profit. The area of ​​distribution is described by fallow

de - the parameter of the shape of the curve under the curve; λ is the parameter to the scale of the curve under the curve.

The graph of the function of the thickening of the rozpodіlu is shown in fig. 4.4.

Rice. 4.4.

Weibull subdivision function

The function of supremacy for whose law I have rozpodіlu

Mathematical grading of the magnitude of the fall X one

de G( x) - Gamma function.

For uninterrupted values X

For integer values X calculate the gamma function using the formula

also correct formulas

Dispersion of the fall rate is more expensive

Widespread zastosuvannya pіd hаlіzu аnd rozrahunkіv naіynostі virobіv rozpodіl podіl Weybull's law is clarified by the fact that this law, zagalnyuyuuchi exponential rozpodіl, to avenge the dodatkovy parameter α.

By selecting the proper order of parameters a and λ, it is possible to improve the visibility of the rozrahunkov values ​​by matching it with the exponential law, which is one-parameter (parameter λ).

So, for virobiv, if defects can be attached, but if the trivaly hour does not vicorate (and therefore, it’s more old), it’s not safe, and it’s most significant in the cob period, and then it’s fast falling. The reliability function of such a virobe is well described by the Weibull law with the parameter α< 1.

On the other hand, although it is better to control when prepared and may not have any adherence defects, but it recognizes old age, then the reliability function is described by the Weibull law with the parameter α > 1. At α = 3.3, the Weibull distribution is close to normal.

In times, though all the same, the middle is negative or zero terms, then it is possible to add a constant to the skin term to the series, for example, . For one of the powers of mathematical refinement, the operation does not change the main statistical characteristics are low. This operation allows you to go to the log-normal distribution in this direction.

As a result of stopping the logarithm operation (36) up to the last row, the number of rows between data changes. Tse can be improved from rice. 9.16: obviously what.

The function of rozpodіlu a new row of dovnyuvatime

(37)

Ale todi

(38)

(39)

I for example,

(40)

Formulas (37) - (40) give a link between lognormal and outward roses.

Poisson's law of distribution (the law of distribution of rare phenomena)

All rose to dosit great number of trials to practice the normal law of the rose. However, even in the middle of the data, there are rіdkіsnі, vinyatkovі results, then raspodіl tsikh rіdkіsnyh yavishch, in that hour, if the main mass of the normal law, the law of the other law - the law Poisson rose. It is primal to that law that it is possible to break zero. In what direction bіnomіnalny rozpodіl Poisson go over

(41)

De maє the very same sense that the normal rozpodіli.

Law Poisson rose, which is set by the formula (41), indicates the possibility of the appearance of podia, which will appear after approximately equal intervals of an hour, for the mind, that all podії will appear independently one of the same and with a certain intensity, let it be a small one, but a constant one. The number of trials with whom is great, and the ability to appear ochіkuvanoї podії is even small and more expensive. The parameter also characterizes the intensity of the appearance of scoring in the sequence of sampling.

At this time, we will try to break the expectation.

A characteristic feature of this type of distribution will be the following mathematical expressions:

butt 5. 150 shots were picked up at the polygon. Some of them knew the presence of a rare element:

Indicate the law of rozpodіlu shukany element.

Solution. For the correct nutrition in the head of the department, the next step is to reconsider the viconnance of equality (45), which is a characteristic sign Poisson rose. For simplicity, calculate not hundreds of parts, but numbers that are 100 times larger, tobto.

At the link with the team, what is laid down, what has been put in place of the shukan element according to the law Poisson rose. Now, koristuyuchis spіvvіdnoyu (42) calculable through the theoretical, parіvnyаєmo yogo z vihіdnoyu frequency і

Vipadkovy change Y can be logarithmically normal rozpodіl іz parameters μ and σ, so vipadkovy zmіnna X = lnY can be normal rozpodіl іz these same parameters μ and σ. Knowing the nature of the ligament between changes X and Y, we can easily induce a graph of the width of the fluctuation of the drop-down change with a logarithmically normal distribution (Figure 4.2).

Figure 4.2 - The curves of the logarithmically normal distribution of the sub-line with different values ​​of the parameters μ and σ

If there is a change in X, the function of the width of the movement, as it is determined by the formula (4.6), and if X = lnY, then:

Possible stars for y > 0:

The value is prominent, that the value is changing, that it corresponds to a logarithmically normal distribution, it can take on more than positive values. As shown in figure 4.2, the curves of the function f(y) may have left-sided asymmetry, as the greater the value of the parameters μ and σ. The skin curve can have one maximum and is assigned for all positive values.

The calculation of the mathematical scaling and the variance of the vertical change with the logarithmically normal distribution does not become particularly difficult:

The way to substantiate and introduce new changes in integrals 4.15 and 4.16 is taken:

In addition, in order to calculate the flexibility of the change in Y with logarithmically normal distribution of the value f(y, μ, σ), in order to calculate the value in the interval (a, b), we should take the integral:

However, in practice it is easier to speed up the fact that the logarithm of the vertical change Y may be normal. The immediacy of what a ≤ Y ≤ b is equivalent to the imaginaryness of what
lna ≤ lnY ≤ lnb.

Let's calculate the magnitude of the fact that vipadkova is changing with logarithmically spread under μ = 1, σ = 0.5, if the value is in the interval (2, 5). Maemo:

From the table of logarithms, ln2 = 0.6932 and ln5 = 1.6094 are known.

Setting lnY = X, we can write:

Moreover, the vipadkova change of X is subordered to the normal distribution from the average values ​​μ = 1 and the standard deviations σ = 0.5. Now it’s easy to calculate the possibility of shukan according to the tables of the integral function of the normal distribution:

Food for self-control

1 Designation of a rectangular rosette.

2 Graph of the thickness of the ymovirnosti of the varicose vein with a straight-cut rhizome

3 The main meaning of a rectangular rosette.

4 Mathematical refinement of the variance of the varicose veins in a straight-cut rosacea.

5 The role of the normal distribution of mathematical statistics.

6 What is such a normal division and how is it connected with the binomial?

7 Graph of the thickening of the fluctuation of the varicose veins with a normal rosacea.

8 What statistical parameters can be used to restore the normal distribution?

9 Why is normal rozpodіl unperturbable?

10 Equation of a normal curve.

11 What is the normalization of vidhilennya?

12 Alignment of the curve of the normal distribution under the normalized form.

13 What values ​​of μ and σ characterize the normal size of normalized form?

14 How often do these samples fit within the boundaries of ±1?, ±2?, ±3?

15 What does the table of the normal integral of dynamics show?

16 Equation of a logarithmically normal curve.

17 Graph of the thickening of the fluctuation of the varicose vein with a logarithmically normal distribution.

18 What kind of transformation is necessary to work out, so that from the logarithmically normal distribution subtract from the normal distribution?

19 What statistical parameters are used logarithmically to set the normal distribution?

TOPIC 5 Changed the parameters of the selection

5.1 t

5.2 F-rozpodil Fischer-Snedekor

5.3 χ 2 -rozpodil

5.1 t

The law of normal distribution appears beyond the sign n > 20–30. Prote eksperimentator often carry out a large number of vimirivs, restoring their vysnovki on small vibrators. With a small number of warnings, the results sound close and rarely show great inspiration. It is easy to explain this by the law of normal rozpodіlu, zgіdno z yakim ymovіrnіst the appearance of small vіdhilen more, nіzh vіdhilen znachnyh. So, the flexibility of respiration, which exceeds the absolute value of ±2σ, is usually 0.05, or one regression per 20 reversals, and resilience ± 3σ is 0.01, or one regression per 100.

If you want to carry out a sexual inspection, for example, in 4-6 replications, then it’s natural to consider that there will be no average yield readings on parallel plots of great crops. For this reason, it is standard care, bolstered for a small amount, in the majority of cases there will be less, less for all general marriages. Also, in vypadki, it is not possible to rely on the criteria of normal rozpodіla in vysnovki.

Since the beginning of the 20th century, in mathematical statistics, having become a new one directly, which can be called the statistics of small vibes. The most practical value for experimental work is little revealed in 1908. English statistician and chemist V. Gosset t-rozpodіl, scho having taken the name of Student's rozpodіlu (Eng. Student - student, pseudonym V. Gosset).

Rozpodіl t Student's for vibirkovyh mean values ​​are equal:

The numeral of the formula means the vibrancy of the vibratory average in the whole of the average marriage, and the banner:

- є pokanik, scho assessing the value of the standard pardon of the average vibirkovo sukupnostі.

In this way, the value of t is reduced to the vibrancy of the middle marriage, which is expressed in the pardons of the vibrka, taken as a unit.

The maxima of the frequency of the normal and t-rose subdivision are varied, but the shape of the curve of the t-disposition is more likely to lie down depending on the number of steps of the will. With even small values ​​of the degrees of freedom, it looks like a flat-topped curve, moreover, the area, bordered by a curve, is larger, lower with a normal spread, and with an increase in the number of guards (n> 30), the spread of t approaches to normal and passes at ∞.n = new at ∞.

Figure 1.1 shows the differential and integral t-Student's subdivisions at 10 degrees of freedom.

Figure 5.1 - Differential (levoruch) and integral (right-hand) t-student

Razpodil t-Student's maє important value when working with small selections: allows you to designate a confidence interval, which inclines the average sukupnіst , that pervert that chi іnsu hypothesis about general marriage. With whom you can’t, you need to know the parameters of marriage і Sufficient for mother їх estimates μ і σ for the singing obligation of vibrating n.

5.1.1 Behrens-Fischer problem

The revision of the hypothesis about the general middle two groups with normal distribution and uneven variances in mathematical statistics is called the Behrens-Fischer problem and can only be approximated. Why is the importance of evenness of variances in divided groups so important? Without going into the details of the problem, it is significant that the more the variance and the overall selections differ, the stronger the difference between the "counted t-test" and the "Student's t-test". With this difference, the value may be like the t-criterion itself, so such a parameter can be divided as the number of steps of freedom. In its own right, the number of steps of freedom is indicated by the value of the reached (critical) level of significance (p< ...) определяемого для вычисленного значения t-критерия.

Nehtuvannya sledniki, osnovnymi more minds admissibility vikoristannya t-Student's t-criterion, bring to the bottom of the creation of the results of re-verification of hypotheses about the equivalence of the middle. In the robots, de transmitting gypothesis about the ribed cucumber of the middle was led behind the pre-person of the T-Cheriter of Student, I am a lot of criterias of the transfer of normally, є PIDSTAVI VIKORITSISTYA VICHORITS.

Another frequent pardon is the application of the Student's t-test to reconsider hypotheses about the equality of three and more group averages. In this way, it is necessary to establish the so-called global linear model, implemented in the procedure of single-factor analysis of variance with fixing effects.

Let's take a look at the specificity of the victory over Student's t-test. Most of the time, the t-test is victorious in two domains. For the first time, it’s necessary to stop for a re-verification of the hypothesis about the equality of the general middle two independent, unconnected selections (the so-called two-rank t-criterion). In this case, there is a control group and a confirmed group that consists of different objects, the number of which in groups can be different. Another type of winner wins the title of a pair of t-criteria, if one and the same group of objects generates numerical material for revising hypotheses about the middle ones. To that qi vibirka is called fallow, po'yazanimi. For example, they are killed instead of leukocytes in healthy creatures, and then in the creatures themselves, after a prominence with a singing dose of vipromination. In both types, there may be some signs of normality in the skin of the normal groups. The dominance of Student's t-test in the most important way works to show two important aspects.

In another way, it’s not worth talking about those that authors don’t have any alternatives to this criterion, otherwise they can’t stink on their own. It can be said without further ado that in this hour of thoughtlessly applying the Student's t-criterion in more biological robots to bring more shkodi, less coriste.

5.2 F-rozpodil Fischer-Snedekor

As a normally separated marriage, take two independent selections with a total of n 1 and n 2 and adjust the dispersion і with steps of freedom ν 1 \u003d n -1 and ν 2 \u003d n 2 -1, then you can signify the dispersion of dispersions:

The variances should be taken in such a way that the variance of the number book is large, so F ≥ 1.

Rozpodіl F to fallow only in the number of steps of freedom ν 1 і ν 2 (law F-rozpodіlu vіdkriv R.A. Fisher). If two equal selections are independent independent ones from the global population with the general average, then in fact the value of F does not exceed the difference between the two and does not outweigh the theoretical value of the criterion F (F fact< F теор). Если генеральные параметры сравниваемых групп различны, то F факт >Theor. The theoretical value of F for 5% and 1% significance level is given in the table, tabulating only the right critical points for F ≥ 1, so it is generally accepted that a larger variance is reduced to a smaller one.

The curves that detract from the function are subdivided for all possible values ​​of F, especially with a small number of guards, may have an asymmetric shape - a long tail of large values ​​and a large concentration of small values ​​of F (Figure 5.2).

Figure 5.2 - Differential (levoruch) and integral (right-handed)
F-rozpodil Fischer-Snedekor

It is significant that Student's t-rozpodіl є we will call the F-rozpodіlu drop when the number of freedom steps is ν 1 = 1 і ν 2 = ν, which is equal to the number of freedom steps for the t rozpodіlu. In this way, such a spіvіdnoshnja mizh F and t:

5.3 χ 2 -rozpodil

A lot of factual differences are given to the models of theoretical differences (normal, binomial, Poisson) Prote, in practice, there are differences that are strongly influenced by the normal. To assess the degree of diversity or the degree of difference between the numbers of the actual and the theoretical roses, statistical criteria are used, for example, the criterion χ 2 . This criterion zastosovuetsya on the basis of the task of statistical analysis, for example, for the verification of hypotheses: about the independence of two principles, which form the basis for grouping the results in a watchfulness of one totality; about the homogeneity of groups of some deyaky significant characteristics; about the theoretical and experimental curves of numbers. Criterion 2 can be called as a criterion of the right, as well as a criterion of independence, a criterion of homogeneity. The law of rozpodіlu χ 2 (хі-square) according to K. Pirson. The curve was subdivided, taken from the chi-square function:

de f - actual F - theoretical frequencies of the number of objects in the sample. Її kind of heavily deposited in the number of steps of freedom. For a small number of steps of freedom, the curve is asymmetric (Figure 5.3), but for larger numbers, the asymmetry changes and at ν = the curve becomes normal Gaussian.

Rozpodіl χ 2, so very yak і t-rozpodіl, okremy vpadok
F - split for ν 1 = ν і ν 2 = ∞.

Figure 5.3 - Differential (levoruch) and integral (right-hand)
χ 2 -rozpodil

Food for self-control

1 In some cases, it is better to have Student's t-rump than the normal rump?

2 What values ​​should be estimated for the Student's t-score?

3 What is the essence of the Behrens-Fischer problem?

4 How is the F-rozpodil numerically expressed for two independent selections from the totality of the change?

5 What are the characteristic values ​​of variant fallows to deposit F-rozpodil?

6 On what basis can the value of criterion 2 be considered in the statistical analysis of experimental data?

TOPIC 6 Fundamentals of mathematical statistics

6.1 Averages

6.2 Arithmetic mean

6.3 Geometric mean

6.4 Harmonic mid