GOAL OF THE WORK

The purpose of this laboratory work is:

· construction of distribution laws based on the results of the experiment random variable scatter of parameters of non-wire resistors;

· testing the hypothesis about normal law distribution of deviations of element parameters;

· pilot study changes in the parameters of non-wire resistors when exposed to temperature.

WORK DURATION

Laboratory work is carried out during a 4-hour lesson, including 1 hour for a colloquium to assess students' knowledge of the theoretical part.

THEORETICAL PART

Radio-electronic equipment is constantly under the influence of external and internal disturbing random factors, under the influence of which the parameters of the device elements change. Changing the parameters of elements (resistors, capacitors, semiconductor devices, integrated circuits, etc.) is associated with various physical processes, occurring in materials due to external influences and aging. In addition, the parameters of RES elements have a production scatter, which is the result of the influence of random factors during their manufacture. Equipment designed from such elements reacts to all variations by changing its output parameters. To predict the reliability of RES, there is a need to establish laws for the distribution of the random value of the scatter of the parameters of the elements, determined by their production and disturbing external conditions (in particular, ambient temperature).

IN laboratory work Using the goodness-of-fit criteria (Pearson or Kolmogorov), the hypothesis about the normal distribution law of the random variable X - the scatter of the parameters of the elements - is tested.

AGREEMENT CRITERIA USED TO TEST STATISTICAL HYPOTHESES

Goodness-of-fit criteria allow us to assess the probability of the assumption that the sample obtained from the experiment does not contradict the a priori chosen distribution law of the random variable under consideration. The solution to this problem is based on the use of the fundamental position of mathematical statistics, according to which The empirical (statistical) distribution function converges in probability to the prior (comparable theoretical) distribution function when the sample size increases without limit, provided that the sample belongs to the prior distribution in question. At final value samples, the empirical and a priori distribution functions will, generally speaking, differ from each other. Therefore, for the sample X 1 , X 2 ,… x n random variable X a certain numerical measure of discrepancy (goodness-of-fit criterion) () of the empirical distribution function is introduced

, l =1, 2, …, n , (1)

Where

= X 1 , X 2 ,… x n– sample of experimental data

and a priori – distribution function.

The rule for testing the hypothesis about the agreement between the a priori and empirical distributions is formulated as follows: if

then the hypothesis that the prior distribution to which the sample belongs X 1 , X 2 ,…,x n equal to F(X) must be rejected. To determine the threshold value WITH a certain acceptable probability a of rejecting the hypothesis that the sample belongs to the distribution is established F. Probability a is called the significance level of the goodness-of-fit criterion. Then

those. WITH– threshold value criterion is equal to the a-percentage point of the distribution function of the divergence measure.

The event can also occur if the hypothesis put forward about the distribution law is true. However, if a is small enough, then the possibility of such situations can practically be neglected. Commonly specified values ​​for a are a = 0.05 and a = 0.01.

If the distribution law of the divergence measure () does not depend on F, then the rule for rejecting the agreement hypothesis and F

(4)

does not depend on the prior distribution. Such criteria are called nonparametric (see section 3.1.2).

The hypothesis about the nature of the distribution can be tested using the goodness-of-fit test in a different sequence: using the obtained value, it is necessary to determine the probability a n= R{ n). If the resulting value a n < a , то отклонения значимые; если an³ a, then the deviations are not significant. Values ​​of a n, very close to 1 (very good agreement) may indicate poor quality of the sample (for example, elements that give large deviations from the average were thrown out from the original sample without reason).

The goodness-of-fit criteria used in statistics differ from each other in various measures of discrepancy between the statistical and theoretical distribution laws (). Some of them are discussed below.

3.1.1. Agreement criterion c 2

When using the goodness-of-fit criterion c 2 (Pearson's criterion), the measure of discrepancy between the empirical and prior distributions is determined as follows.

Region possible values, on which it is defined F(x) - the prior distribution function is divided into final number non-overlapping intervals – , i = 1, 2,…, L.

Let us introduce the notation: – prior probability sample value hits in the interval

It's obvious that . Let the elements of the observed sample X 1 , X 2 ,…, x n belong to the interval.

It's clear that .

Let us take as a measure of the discrepancy between the empirical and a priori distributions the value

, (5)

where is the experimental hit number of random variable values x in the interval,

L– the number of intervals into which all experimental values ​​of the quantity are divided x,

n– sample size,

p i– probability of hitting a random variable x in the - interval, calculated for the theoretical distribution law (the product determines the number of hits in the - interval for the theoretical law).

As Pearson proved, when n® ¥ the distribution law of quantity (5) tends to - distribution with S = L- 1 degrees of freedom, unless the hypothesis about the distribution is true.

If checked complex hypothesis that the sample belongs to the distribution , where the unknown parameter (scalar or vector) of the distribution is , then from the experiment (based on the resulting sample) the estimate is determined unknown parameter– . In this case, S - the number of degrees of freedom c 2 - distribution is equal to L – r – 1, Where r– number of estimated distribution parameters. .

The rule for testing the hypothesis about whether a sample belongs to a distribution can be formulated as follows: with a sufficiently large n(n> 50) and for a given significance level a, the hypothesis is rejected if

where - a - percentage point - distributions with degrees of freedom.

Kolmogorov criterion

Let us take as a measure of the discrepancy between the a priori and empirical distributions the statistics

().= , (7)

Where - upper limit difference module for all obtained values X.

The distribution of this statistic (random variable) for any n does not depend on

If only a sample X 1 , X 2 ,… x n according to which it was built this last one also belongs - continuous function. However, the exact expression for the distribution function at a finite value n very cumbersome . A.N. Kolmogorov found a fairly simple asymptotic expression (for ) for the functions:

, z> 0. (8) Thus, for large sizes samples (with n> 50), using (8) we get

Criteria for agreement (compliance)

To test the hypothesis about the correspondence of the empirical distribution to the theoretical distribution law, special statistical indicators- agreement criteria (or compliance criteria). These include the criteria of Pearson, Kolmogorov, Romanovsky, Yastremsky, etc. Most agreement criteria are based on the use of deviations of empirical frequencies from theoretical ones. Obviously, the smaller these deviations, the better the theoretical distribution corresponds to the empirical one (or describes it).

Consent criteria - these are criteria for testing hypotheses about the correspondence of the empirical distribution to the theoretical probability distribution. Such criteria are divided into two classes: general and special. General goodness-of-fit tests apply to the most general formulation of a hypothesis, namely the hypothesis that observed results agree with any a priori assumed probability distribution. Special goodness-of-fit tests require special null hypotheses that formulate agreement with a certain form probability distributions.

Consent criteria based on established law distributions make it possible to establish when discrepancies between theoretical and empirical frequencies should be considered insignificant (random), and when - significant (non-random). It follows from this that the agreement criteria make it possible to reject or confirm the correctness of the hypothesis put forward when aligning the series about the nature of the distribution in the empirical series and to answer whether it is possible to accept for a given empirical distribution a model expressed by some theoretical distribution law.

Pearson's χ 2 (chi-square) goodness-of-fit test is one of the main goodness-of-fit tests. Proposed by the English mathematician Karl Pearson (1857-1936) to assess the randomness (significance) of discrepancies between the frequencies of empirical and theoretical distributions:

Where k- the number of groups into which the empirical distribution is divided; fi- empirical frequency of a trait in i-th group; / ts °р - theoretical frequency of the sign in i-th group.

Scheme for applying the criterion y) to assess the consistency of theoretical and empirical distributions comes down to the following.

• 1. The calculated measure of discrepancy % 2 acch is determined.
• 2. The number of degrees of freedom is determined.
• 3. Based on the number of degrees of freedom v, %^bl is determined using a special table
• 4. If % 2 asch >x 2 abl, then for a given level of significance a and the number of degrees of freedom v, the hypothesis about the insignificance (randomness) of the discrepancies is rejected. Otherwise, the hypothesis can be recognized as not contradicting the experimental data obtained and with probability (1 - a) it can be argued that the discrepancies between theoretical and empirical frequencies are random.

Significance level - this is the probability of erroneously rejecting the put forward hypothesis, i.e. the probability that a correct hypothesis will be rejected. IN statistical research Depending on the importance and responsibility of the tasks being solved, the following three levels of significance are used:

• 1) a = 0.1, then P = 0,9;
• 2) a = 0.05, then P = 0,95;
• 3) a = 0.01, then P = 0,99.

Using the goodness-of-fit criterion y), The following conditions must be met.

• 1. The volume of the population under study must satisfy the condition p> 50, while the frequency or group size must be at least 5. If this condition is violated, it is necessary to first combine small frequencies (less than 5).
• 2. The empirical distribution must consist of data obtained as a result of random sampling, i.e. they must be independent.

The disadvantage of the Pearson goodness-of-fit criterion is the loss of some of the original information associated with the need to group observation results into intervals and combine individual intervals with a small number of observations. In this regard, it is recommended to supplement the check of distribution compliance with the criterion y) other criteria. This is especially necessary when the sample size is P ~ 100.

In statistics, the Kolmogorov goodness-of-fit test (also known as the Kolmogorov–Smirnov goodness-of-fit test) is used to determine whether two empirical distributions obey the same law, or to determine whether a resulting distribution obeys an assumed model. The Kolmogorov criterion is based on determining the maximum discrepancy between accumulated frequencies or frequencies of empirical or theoretical distributions. The Kolmogorov criterion is calculated using the following formulas:

Where D And d- accordingly, the maximum difference between the accumulated frequencies (/-/") and between the accumulated frequencies ( rr") empirical and theoretical series of distributions; N- number of units in the aggregate.

Having calculated the value X, a special table is used to determine the probability with which it can be stated that deviations of empirical frequencies from theoretical ones are random. If the sign takes values ​​up to 0.3, then this means that there is a complete coincidence of frequencies. With a large number of observations, the Kolmogorov test is able to detect any deviation from the hypothesis. This means that any difference in the sample distribution from the theoretical one will be detected with its help if there are a sufficiently large number of observations. The practical significance of this property is insignificant, since in most cases it is difficult to count on obtaining large number observations under constant conditions, the theoretical idea of ​​the distribution law to which the sample should obey is always approximate, and the accuracy of statistical tests should not exceed the accuracy of the selected model.

The Romanovsky goodness-of-fit test is based on the use of the Pearson criterion, i.e. already found values ​​x 2 > and the number of degrees of freedom:

where v is the number of degrees of freedom of variation.

The Romanovsky criterion is convenient in the absence of tables for x2. If K r TO? > 3, then they are non-random and the theoretical distribution cannot serve as a model for the empirical distribution being studied.

B. S. Yastremsky used in the criterion of agreement not the number of degrees of freedom, but the number of groups ( k), a special value 0, depending on the number of groups, and a chi-square value. The Yastremsky agreement criterion has the same meaning as the Romanovsky criterion and is expressed by the formula

where x 2 is Pearson's goodness-of-fit test; /e gr - number of groups; 0 - coefficient, for the number of groups less than 20 equal to 0.6.

If 1ph act > 3, the discrepancies between theoretical and empirical distributions are not random, i.e. the empirical distribution does not meet the requirements of a normal distribution. If 1f act

In this section, we will consider one of the issues related to testing the plausibility of hypotheses, namely, the issue of consistency of theoretical and statistical distribution.

Let us assume that this statistical distribution is aligned using some theoretical curve f(x)(Fig. 7.6.1). No matter how well the theoretical curve is selected, some discrepancies are inevitable between it and the statistical distribution. The question naturally arises: are these discrepancies explained only by random circumstances associated with a limited number of observations, or are they significant and are associated with the fact that the curve we selected does not align the given statistical distribution well. To answer this question, so-called “consent criteria” are used.

LAWS OF DISTRIBUTION OF RANDOM VARIABLES

The idea behind applying the consent criteria is as follows.

Based on this statistical material, we have to test the hypothesis N, consisting in the fact that the random variable X obeys some specific distribution law. This law can be specified in one form or another: for example, in the form of a distribution function F(x) or as distribution density f(x), or as a set of probabilities p t , Where p t- the probability that the value X will fall within l something discharge.

Since of these forms the distribution function F(x) is the most general and determines any other, we will formulate a hypothesis N, as consisting in the fact that the quantity X has a distribution function ^(q:).

To accept or reject a hypothesis N, consider some quantity U, characterizing the degree of discrepancy between theoretical and statistical distributions. Magnitude U can be selected different ways; for example, as U you can take the sum of squared deviations of theoretical probabilities p t from the corresponding frequencies R* or the sum of the same squares with some coefficients (“weights”), or maximum deviation statistical function distribution F*(x) from theoretical F(x) etc. Let us assume that the value U chosen in one way or another. Obviously there is some random value. The distribution law of this random variable depends on the distribution law of the random variable X, on which experiments were carried out, and on the number of experiments P. If the hypothesis N is true, then the law of distribution of the quantity U determined by the law of distribution of the quantity X(function F(x)) and number P.

Let us assume that we know this distribution law. As a result of this series of experiments, it was discovered that the measure we chose

CONSENT CRITERIA

discrepancies U took on some meaning A. The question is whether this can be explained by random causes or whether this discrepancy is too large and indicates the presence of a significant difference between the theoretical and statistical distributions and, therefore, the unsuitability of the hypothesis N? To answer this question, assume that the hypothesis N is correct, and under this assumption we calculate the probability that, due to random reasons associated with an insufficient amount of experimental material, the measure of discrepancy U will be no less than the value we observed experimentally And, i.e., we calculate the probability of the event:

If this probability is very small, then the hypothesis N should be rejected as less plausible; if this probability is significant, it should be recognized that the experimental data do not contradict the hypothesis N.

The question arises: how should the measure of divergence £/ be chosen? It turns out that with some methods of choosing it, the law of distribution of the quantity U has very simple properties and with a sufficiently large P practically independent of function F(x). It is precisely these measures of discrepancy that are used in mathematical statistics as consent criteria.

Let's consider one of the most frequently used criteria of agreement - the so-called “criterion y?" Pearson.

Let us assume that hectares of independent experiments have been carried out, each of which contains a random variable X took on a certain meaning. The experimental results are summarized in k categories and presented in the form of a statistical series.

In this note, the χ 2 distribution is used to test the consistency of a data set with a fixed probability distribution. The agreement criterion often O You belonging to a particular category are compared with the frequencies that would be theoretically expected if the data actually had the specified distribution.

Testing using the χ 2 goodness-of-fit criterion is performed in several stages. First, a specific probability distribution is determined and compared with the original data. Secondly, a hypothesis is put forward about the parameters of the selected probability distribution (for example, its mathematical expectation) or their assessment is carried out. Thirdly, based on theoretical distribution The theoretical probability corresponding to each category is determined. Finally, the χ2 test statistic is used to check the consistency of the data and distribution:

Where f 0- observed frequency, f e- theoretical or expected frequency, k- number of categories remaining after merging, R- number of parameters to be estimated.

Download the note in or format, examples in format

Using the χ 2 goodness-of-fit test for the Poisson distribution

To calculate using this formula in Excel, it is convenient to use the =SUMPRODUCT() function (Fig. 1).

To estimate the parameter λ you can use the estimate . Theoretical frequency X successes (X = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and more) corresponding to the parameter λ = 2.9 can be determined using the function =POISSON.DIST(X;;FALSE). Multiplying the Poisson probability by the sample size n, we get the theoretical frequency f e(Fig. 2).

Rice. 2. Actual and theoretical frequencies arrivals per minute

As follows from Fig. 2, the theoretical frequency of nine or more arrivals does not exceed 1.0. To ensure that each category contains a frequency of 1.0 or greater, the category “9 or more” should be combined with the category “8.” That is, nine categories remain (0, 1, 2, 3, 4, 5, 6, 7, 8 and more). Because the expected value Poisson distribution is determined on the basis of sample data, the number of degrees of freedom is equal to k – p – 1 = 9 – 1 – 1 = 7. Using a significance level equal to 0.05 we find critical valueχ 2 statistics, which has 7 degrees of freedom according to the formula =HI2.OBR(1-0.05;7) = 14.067. Decisive rule formulated as follows: hypothesis H 0 is rejected if χ 2 > 14.067, otherwise the hypothesis H 0 does not deviate.

To calculate χ 2 we use formula (1) (Fig. 3).

Rice. 3. Calculation of χ 2 -goodness-of-fit criterion for the Poisson distribution

Since χ 2 = 2.277< 14,067, следует, что гипотезу H 0 cannot be rejected. In other words, we have no reason to assert that the arrival of clients at the bank does not obey the Poisson distribution.

Application of χ 2 -goodness-of-fit test for normal distribution

In previous notes, when testing hypotheses about numerical variables, we assumed that the population under study was normally distributed. To check this assumption, you can use graphical tools, for example, a box plot or a normal distribution graph (for more details, see). At large volumes samples, to test these assumptions, you can use the χ 2 goodness-of-fit test for normal distribution.

Let us consider, as an example, data on the 5-year returns of 158 investment funds (Fig. 4). Suppose you want to believe whether the data is normally distributed. The null and alternative hypotheses are formulated as follows: H 0: 5-year yield obeys normal distribution, H 1: The 5-year yield does not follow a normal distribution. The normal distribution has two parameters - the mathematical expectation μ and standard deviationσ, which can be estimated based on sample data. IN in this case = 10.149 and S = 4,773.

Rice. 4. An ordered array containing data on the five-year average annual return of 158 funds

Data on fund returns can be grouped, for example, into classes (intervals) with a width of 5% (Fig. 5).

Rice. 5. Frequency distribution for five-year average annual returns of 158 funds

Since the normal distribution is continuous, it is necessary to determine the area of ​​the figures bounded by the normal distribution curve and the boundaries of each interval. Additionally, since the normal distribution theoretically ranges from –∞ to +∞, it is necessary to take into account the area of ​​shapes that fall outside the class boundaries. So, the area under the normal curve to the left of the point –10 is equal to the area of ​​the figure lying under the standardized normal curve to the left of the Z value equal to

Z = (–10 – 10.149) / 4.773 = –4.22

The area of ​​the figure lying under the standardized normal curve to the left of the value Z = –4.22 is determined by the formula =NORM.DIST(-10;10.149;4.773;TRUE) and is approximately equal to 0.00001. In order to calculate the area of ​​the figure lying under the normal curve between points –10 and –5, you first need to calculate the area of ​​the figure lying to the left of point –5: =NORM.DIST(-5,10.149,4.773,TRUE) = 0.00075 . So, the area of ​​the figure lying under the normal curve between points –10 and –5 is 0.00075 – 0.00001 = 0.00074. Similarly, you can calculate the area of ​​the figure limited by the boundaries of each class (Fig. 6).

Rice. 6. Areas and expected frequencies for each class of 5-year returns

It can be seen that the theoretical frequencies in the four extreme classes (two minimum and two maximum) are less than 1, so we will combine the classes, as shown in Fig. 7.

Rice. 7. Calculations associated with the use of the χ 2 goodness-of-fit test for the normal distribution

We use the χ 2 criterion for the agreement of data with a normal distribution using formula (1). In our example, after merging, six classes remain. Since the expected value and standard deviation are estimated from sample data, the number of degrees of freedom is k – p – 1 = 6 – 2 – 1 = 3. Using a significance level of 0.05, we find that the critical value of χ 2 statistics, which has three degrees of freedom = CI2.OBR(1-0.05;F3) = 7.815. Calculations associated with the use of the χ 2 goodness-of-fit criterion are shown in Fig. 7.

It can be seen that χ 2 -statistic = 3.964< χ U 2 7,815, следовательно гипотезу H 0 cannot be rejected. In other words, we have no basis to assert that the 5-year returns of investment funds focused on high growth are not subject to a normal distribution.

In several latest notes considered different approaches to the analysis of categorical data. Methods for testing hypotheses about categorical data obtained from the analysis of two or more independent samples are described. In addition to the chi-square tests, nonparametric procedures are considered. The Wilcoxon rank test is described, which is used in situations where the application conditions are not met t-criteria for testing the hypothesis about the equality of mathematical expectations of two independent groups, as well as the Kruskal-Wallis test, which is an alternative to one-factor analysis of variance (Fig. 8).

Rice. 8. Structural scheme methods for testing hypotheses about categorical data

Materials from the book Levin et al. Statistics for Managers are used. – M.: Williams, 2004. – p. 763–769

Definition 51. Criteria that allow you to judge whether the values ​​are consistent X 1 , X 2 ,…, x n random variable X with a hypothesis regarding its distribution function are called consent criteria.

The idea of ​​using consent criteria

Let a hypothesis be tested based on this statistical material N, consisting in the fact that SV X obeys some specific distribution law. This law can be specified either as a distribution function F(x), or in the form of distribution density f(x), or as a set of probabilities p i. Since of all these forms the distribution function F(x) is the most general (exists for both DSV and NSV) and determines any other, we will formulate a hypothesis N, as consisting in the fact that the quantity X has a distribution function F(x).

To accept or reject a hypothesis N, consider some quantity U, characterizing the degree of divergence (deviation) of the theoretical and statistical distributions. MagnitudeU can be selected in various ways: 1) sum of squared deviations of theoretical probabilities p i from the corresponding frequencies, 2) the sum of the same squares with some coefficients (weights), 3) the maximum deviation of the statistical (empirical) distribution function from the theoretical F(x).

Let the value U chosen in one way or another. Obviously, this is some random variable. Law of distribution U depends on the distribution law of the random variable X, on which experiments were carried out, and on the number of experiments n. If the hypothesis N is true, then the law of distribution of the quantity U determined by the law of distribution of the quantity X(function F(x)) and number n.

Let us assume that this distribution law is known. As a result of this series of experiments, it was discovered that the chosen measure of discrepancy U took on some meaning u. Question: can this be explained by random reasons or this discrepancy is too is large and indicates the presence of a significant difference between the theoretical and statistical (empirical) distributions and, therefore, the unsuitability of the hypothesis N? To answer this question, let us assume that the hypothesis N is correct, and under this assumption we calculate the probability that, due to random reasons associated with an insufficient amount of experimental material, the measure of discrepancy U will be no less than the experimentally observed value u, that is, we calculate the probability of the event: .

If this probability is small, then the hypothesis N should be rejected as little plausible, but if this probability is significant, then we conclude that the experimental data do not contradict the hypothesis N.

The question arises: how should the measure of discrepancy (deviation) be chosen? U? It turns out that with some methods of choosing it, the law of distribution of the quantity U has very simple properties and with a sufficiently large n practically independent of function F(x). It is precisely these measures of discrepancy that are used in mathematical statistics as criteria for agreement.

Definition 51/. The criterion of agreement is the criterion for testing the hypothesis about the assumed law of an unknown distribution.

For quantitative data with distributions close to normal, use parametric methods based on indicators such as mathematical expectation and standard deviation. In particular, to determine the reliability of the difference in means for two samples, the Student method (criterion) is used, and in order to judge the differences between three or a large number samples, - test F, or analysis of variance. If we are dealing with non-quantitative data or the samples are too small to be confident that the populations from which they are taken follow a normal distribution, then use nonparametric methods - criterion χ 2(chi-square) or Pearson for qualitative data and signs, ranks, Mann-Whitney, Wilcoxon, etc. tests for ordinal data.

In addition, the choice statistical method depends on whether the samples whose means are compared are independent(i.e., for example, taken from two different groups of subjects) or dependent(i.e., reflecting the results of the same group of subjects before and after exposure or after two different exposures).

pp. 1. Pearson test (- chi-square)

Let it be produced n independent experiments, in each of which the random variable X took a certain value, that is, a sample of observations of the random variable was given X (population) volume n. Let us consider the problem of checking the closeness of the theoretical and empirical distribution functions for discrete distribution, that is, it is necessary to check whether the experimental data are consistent with the hypothesis N 0, stating that the random variable X has a distribution law F(x) at significance level α . Let's call this law “theoretical”.

When obtaining a goodness-of-fit criterion for testing a hypothesis, determine the measure D deviations of the empirical distribution function of a given sample from the estimated (theoretical) distribution function F(x).

The most commonly used measure is the one introduced by Pearson. Let's consider this measure. Let's split the set of random variable values X on r sets - groups S 1 , S 2 ,…, S r, without common points. In practice, such a partition is carried out using ( r- 1) numbers c 1 < c 2 < … < c r-1 . In this case, the end of each interval is excluded from the corresponding set, and the left one is included.

S 1 S 2 S 3 …. S r -1 S r

c 1 c 2 c 3 c r -1

Let p i, , - the probability that SV X belongs to many S i(obviously ). Let n i, , - the number of quantities (variant) from among the observed ones, belonging to many S i(empirical frequencies). Then the relative frequency of SV hits X in many S i at n observations. It's obvious that , .

For the split above, p i there is an increment F(x) on the set S i, and the increment is on the same set. Let us summarize the results of the experiments in a table in the form of a grouped statistical series.

Knowing theoretical law distribution, you can find the theoretical probabilities of a random variable falling into each group: R 1 , R 2 , …, p r. When checking the consistency of the theoretical and empirical (statistical) distributions, we will proceed from the discrepancies between the theoretical probabilities p i and observed frequencies.

For measure D the discrepancies (deviations) of the empirical distribution function from the theoretical one take the sum of the squared deviations of the theoretical probabilities p i from the corresponding frequencies taken with certain "weights" c i: .

Odds c i are introduced because general case deviations related to different groups, cannot be considered equal in importance: the same in absolute value the deviation may be insignificant if the probability itself p i is large, and very noticeable if it is small. Therefore, naturally the “weights” c i take inversely proportional to the probabilities. How to choose this coefficient?

K. Pearson showed that if we put , then for large n quantity distribution law U has very simple properties: it is practically independent of the distribution function F(x) and on the number of experiments n, but depends only on the number of groups r, namely, this law with increasing n approaches the so-called chi-square distribution .

If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:

What will we do with the received material:

If this material was useful to you, you can save it to your page on social networks: