Análise estatística da distribuição de PoissonResumo
The general properties of POISSON distributions and their relations to the binomial distribuitions are discussed. Two methods of statistical analysis are dealt with in detail: X2-test. In order to carry out the X2-test, the mean frequency and the theoretical frequencies for all classes are calculated. Than the observed and the calculated frequencies are compared, using the well nown formula: f(obs) - f(esp) 2; i(esp). When the expected frequencies are small, one must not forget that the value of X2 may only be calculated, if the expected frequencies are biger than 5. If smaller values should occur, the frequencies of neighboroughing classes must ge pooled. As a second test reintroduced by BRIEGER, consists in comparing the observed and expected error standard of the series. The observed error is calculated by the general formula: δ + Σ f . VK n-1 where n represents the number of cases. The theoretical error of a POISSON series with mean frequency m is always ± Vm. These two values may be compared either by dividing the observed by the theoretical error and using BRIEGER's tables for # or by dividing the respective variances and using SNEDECOR's tables for F. The degree of freedom for the observed error is one less the number of cases studied, and that of the theoretical error is always infinite. In carrying out these tests, one important point must never be overlloked. The values for the first class, even if no concrete cases of the type were observed, must always be zero, an dthe value of the subsequent classes must be 1, 2, 3, etc.. This is easily seen in some of the classical experiments. For instance in BORKEWITZ example of accidents in Prussian armee corps, the classes are: no, one, two, etc., accidents. When counting the frequency of bacteria, these values are: no, one, two, etc., bacteria or cultures of bacteria. Ins studies of plant diseases equally the frequencies are : no, one, two, etc., plants deseased. Howewer more complicated cases may occur. For instance, when analising the degree of polyembriony, frequently the case of "no polyembryony" corresponds to the occurrence of one embryo per each seed. Thus the classes are not: no, one, etc., embryo per seed, but they are: no additional embryo, one additional embryo, etc., per seed with at least one embryo. Another interestin case was found by BRIEGER in genetic studies on the number os rows in maize. Here the minimum number is of course not: no rows, but: no additional beyond eight rows. The next class is not: nine rows, but: 10 rows, since the row number varies always in pairs of rows. Thus the value of successive classes are: no additional pair of rows beyond 8, one additional pair (or 10 rows), two additional pairs (or 12 rows) etc.. The application of the methods is finally shown on the hand of three examples : the number of seeds per fruit in the oranges M Natal" and "Coco" and in "Calamondin". As shown in the text and the tables, the agreement with a POISSON series is very satisfactory in the first two cases. In the third case BRIEGER's error test indicated a significant reduction of variability, and the X2 test showed that there were two many fruits with 4 or 5 seeds and too few with more or with less seeds. Howewer the fact that no fruit was found without seed, may be taken to indicate that in Calamondin fruits are not fully parthenocarpic and may develop only with one seed at the least. Thus a new analysis was carried out, on another class basis. As value for the first class the following value was accepted: no additional seed beyond the indispensable minimum number of one seed, and for the later classes the values were: one, two, etc., additional seeds. Using this new basis for all calculations, a complete agreement of the observed and expected frequencies, of the correspondig POISSON series was obtained, thus proving that our hypothesis of the impossibility of obtaining fruits without any seed was correct for Calamondin while the other two oranges were completely parthenocarpic and fruits without seeds did occur.
Análise estatística da distribuição de Poisson (*)
J. T. A. Gurgel
Docente-livre Seção de Genética Escola Superior de Agricultura"Luiz de Queiroz", Universidade de S. Paulo
ABSTRACT
The general properties of POISSON distributions and their relations to the binomial distribuitions are discussed. Two methods of statistical analysis are dealt with in detail:
X2-test. In order to carry out the X2-test, the mean frequency and the theoretical frequencies for all classes are calculated. Than the observed and the calculated frequencies are compared, using the well nown formula: f(obs) - f(esp) 2; i(esp). When the expected frequencies are small, one must not forget that the value of X2 may only be calculated, if the expected frequencies are biger than 5. If smaller values should occur, the frequencies of neighboroughing classes must ge pooled.
As a second test reintroduced by BRIEGER, consists in comparing the observed and expected error standard of the series. The observed error is calculated by the general formula:
δ + Σ f . VK
n-1
where n represents the number of cases.
The theoretical error of a POISSON series with mean frequency m is always ± Vm.
These two values may be compared either by dividing the observed by the theoretical error and using BRIEGER's tables for # or by dividing the respective variances and using SNEDECOR's tables for F. The degree of freedom for the observed error is one less the number of cases studied, and that of the theoretical error is always infinite.
In carrying out these tests, one important point must never be overlloked. The values for the first class, even if no concrete cases of the type were observed, must always be zero, an dthe value of the subsequent classes must be 1, 2, 3, etc..
This is easily seen in some of the classical experiments. For instance in BORKEWITZ example of accidents in Prussian armee corps, the classes are: no, one, two, etc., accidents. When counting the frequency of bacteria, these values are: no, one, two, etc., bacteria or cultures of bacteria. Ins studies of plant diseases equally the frequencies are : no, one, two, etc., plants deseased.
Howewer more complicated cases may occur. For instance, when analising the degree of polyembriony, frequently the case of "no polyembryony" corresponds to the occurrence of one embryo per each seed. Thus the classes are not: no, one, etc., embryo per seed, but they are: no additional embryo, one additional embryo, etc., per seed with at least one embryo.
Another interestin case was found by BRIEGER in genetic studies on the number os rows in maize. Here the minimum number is of course not: no rows, but: no additional beyond eight rows. The next class is not: nine rows, but: 10 rows, since the row number varies always in pairs of rows. Thus the value of successive classes are: no additional pair of rows beyond 8, one additional pair (or 10 rows), two additional pairs (or 12 rows) etc..
The application of the methods is finally shown on the hand of three examples : the number of seeds per fruit in the oranges MNatal" and "Coco" and in "Calamondin". As shown in the text and the tables, the agreement with a POISSON series is very satisfactory in the first two cases. In the third case BRIEGER's error test indicated a significant reduction of variability, and the X2 test showed that there were two many fruits with 4 or 5 seeds and too few with more or with less seeds. Howewer the fact that no fruit was found without seed, may be taken to indicate that in Calamondin fruits are not fully parthenocarpic and may develop only with one seed at the least. Thus a new analysis was carried out, on another class basis. As value for the first class the following value was accepted: no additional seed beyond the indispensable minimum number of one seed, and for the later classes the values were: one, two, etc., additional seeds. Using this new basis for all calculations, a complete agreement of the observed and expected frequencies, of the correspondig POISSON series was obtained, thus proving that our hypothesis of the impossibility of obtaining fruits without any seed was correct for Calamondin while the other two oranges were completely parthenocarpic and fruits without seeds did occur.
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BIBLIOGRAFIA
2 - FISHER, Ame - Frequency curves, l.a edition - The Macmillan Company - New York - 1922.
Referências bibliográficas
- 1 - BRIEGER, F. G., A. RODRIGUES LIMA e R. FORSTER - Comportamento de variedades e progenies de fumo na resistência ao "vira-cabeça". Bragantia 2: 275-294.-1942.
- 3 - FISHER, Arne - The Mathematical Theory of Probabilities - l.a edition - The Macmillan Company - New York - 1915.
- 4 - FISHER, R. A. - Statistical Methods of Research Workers - 5.a edition - Oliver and Boyd - London - 1934.
- 5 - MOREIRA, S. e J. T. A. GURGEL - A fertilidade do pólen e sua correlação com o número de sementes, em espécies e formas do gênero Citrus - Bragantia 1: 669-712 - 1941.
- 6 - MOREIRA, S., e J. T. A. GURGEL - Poliembrionia em Citrus - Bragantia (em impressão) - 1945.
- 7 - RIDER, P. R. - An Introduction to Modem Statistical Methods - l.a edition - John Wiley and Sons, Inc. - London - 1939.
- 8 - SNEDECOR, G. W. - Statistical Methods - Collegiate Press, Inc., Ames - Iowa - 1938.
- 9 - TIPPET, L. H. C. - The Methods of Statistics - 2.a edition - Williams and Norgate - London - 1939.
- 10 - TRELOAR, A. E. - Elements of Statistical Reasoning - l.a edition - John Wiley and Sons, Inc. - London - 1939.
- 11 - YULE, E. U. and M. E. KENDALL - An Introduction to the Theory of Statistics - ll.a edition - Charles Griffin and Company - London - 1937.
Datas de Publicação
-
Publicação nesta coleção
25 Fev 2013 -
Data do Fascículo
1945
Histórico
-
Recebido
02 Ago 1945
