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Demonstração de Algoritmos de Estatística em PL/SQL

Esta é uma demonstração de funções que implementam alguns cálculos estatísticos. Os algoritmos que representam as fórmulas estatísticas estão escritos em uma estrutura simples e semelhante a aplicação pura das respectivas fórmulas, para deixar o código o mais limpo possível, e assim, didático.

Até o momento, estas são algumas das fórmulas estatísticas que estão presentes:

  • Média Aritmética

$$\mu = \frac{1}{n} \sum_{i=1}^{n} x_i$$

  • Média Aritmética Ponderada

$$\mu_p = \frac{1}{\sum_{i=1}^{n} w_i} \sum_{i=1}^{n} w_i x_i$$

  • Média Geométrica

$$\mu_g = \sqrt[n]{\prod_{i=1}^{n} x_i}$$

  • Média Geométrica Ponderada

$$\mu_{gp} = \sqrt[\sum_{i=1}^{n} w_i]{\prod_{i=1}^{n} {x_i}^{w_i}}$$

  • Média Harmônica

$$\mu_h = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$

  • Média Harmônica Ponderada

$$\mu_{hp} = \frac{\sum_{i=1}^{n} w_i}{\sum_{i=1}^{n} \frac{w_i}{x_i}}$$

  • Média Quadrática

$$\mu_q = \sqrt{\frac{1}{n} \sum_{i=1}^{n} {x_i}^2}$$

  • Média Quadrática Ponderada

$$\mu_{qp} = \sqrt{\frac{1}{\sum_{i=1}^{n} w_i} \sum_{i=1}^{n} w_i{x_i}^2}$$

  • Média Cúbica

$$\mu_c = \sqrt[3]{\frac{1}{n} \sum_{i=1}^{n} {x_i}^3}$$

  • Média Cúbica Ponderada

$$\mu_{cp} = \sqrt[3]{\frac{1}{\sum_{i=1}^{n} w_i} \sum_{i=1}^{n} w_i {x_i}^3}$$

  • Média Desarmônica

$$\mu_d = \frac{2}{\frac{1}{\frac{\sum_{i=1}^{n} x_i}{n}} + \frac{1}{\frac{{\bigl(\frac{\sum_{i=1}^{n} x_i}{n}\bigl)}^2}{\frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}}}}$$

  • Média Desarmônica Ponderada

$$\mu_{dp} = \frac{2}{\frac{1}{\frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}} + \frac{1}{\frac{{\Bigl(\frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}\Bigl)}^2}{\frac{\sum_{i=1}^{n} w_i}{\sum_{i=1}^{n} \frac{w_i}{x_i}}}}}$$

  • Mediana

  • Moda

  • Desvio Absoluto Médio

$$D_{am} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \mu|$$

  • Desvio Absoluto Mediano

$$D_{am} = Md(|x_i - \tilde{x}|)$$

  • Variância Populacional

$$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$$

  • Desvio Padrão Populacional

$$\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}$$

  • Variância Amostral

$$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$$

  • Desvio Padrão Amostral

$$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$

  • Variância Populacional (para dados agrupados)

$$\sigma^2 = \frac{1}{\sum_{i=1}^{n} w_i} \sum_{i=1}^{n} \bigl((x_i - \mu)^2 w_i\bigl)$$

  • Desvio Padrão Populacional (para dados agrupados)

$$\sigma = \sqrt{\frac{1}{\sum_{i=1}^{n} w_i} \sum_{i=1}^{n} \bigl((x_i - \mu)^2 w_i\bigl)}$$

  • Variância Amostral (para dados agrupados)

$$s^2 = \frac{1}{\sum_{i=1}^{n} w_i - 1} \sum_{i=1}^{n} \bigl((x_i - \bar{x})^2 w_i\bigl)$$

  • Desvio Padrão Amostral (para dados agrupados)

$$s = \sqrt{\frac{1}{\sum_{i=1}^{n} w_i - 1} \sum_{i=1}^{n} \bigl((x_i - \bar{x})^2 w_i\bigl)}$$

  • Coeficiente de Variação

$$CV = \frac{\sigma}{\mu} \times 100$$

  • Coeficiente de Variação (para dados agrupados)

$$CV = \frac{\sigma}{\mu} \times 100$$

  • Covariância Populacional

$$\sigma_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu_x)(y_i - \mu_y)$$

  • Covariância Amostral

$$s_{xy} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$$

  • Coeficiente de Correlação Populacional de Pearson

$$\rho_{xy} = \frac{\sigma_{xy}}{\sigma_x \sigma_y}$$

  • Coeficiente de Correlação Amostral de Pearson

$$r_{xy} = \frac{s_{xy}}{s_x s_y}$$

  • Somatório dos Quadrados

$$SS_x = \sum_{i=1}^{n} {x_i}^2 - \frac{(\sum_{i=1}^{n} x_i)^2}{n}$$

  • Somatório dos Produtos XY

$$SS_{xy} = \sum_{i=1}^{n} x_i y_i - \frac{(\sum_{i=1}^{n} x_i)(\sum_{i=1}^{n} y_i)}{n}$$

  • Coeficiente de Correlação de Pearson

$$r = \frac{SS_{xy}}{\sqrt{SS_x \times SS_y}}$$

  • Z-score Populacional

$$z = \frac{x - \mu}{\sigma}$$

  • Z-score Amostral

$$z = \frac{x - \bar{x}}{s}$$

  • Três Desvios

  • Amplitude

  • Assimetria

$$A = \frac{1}{n} \sum_{i=1}^{n} \Bigl(\frac{x_i - \bar{x}}{s}\Bigl)^3$$

  • Curtose

$$K = \frac{1}{n} \sum_{i=1}^{n} \Bigl(\frac{x_i - \bar{x}}{s}\Bigl)^4 - 3$$

  • Quartis

$$i = \frac{j(n+1)}{4}$$

$$Q_j = x_i + \biggl(\frac{j(n+1)}{4} - i\biggl) (x_{i+1} - x_i)$$

para j = 1, 2 e 3

Além de funções para:

  • Desagrupar dados
  • Agrupar dados
  • Ordenar dados
  • Ordenar frequências

Obs.: As funções possuem versões para dados agrupados ou não agrupados e para dados amostrais ou populacionais. Há também um script de exemplo, com demonstrações das chamadas das funções.

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