{"id":843,"date":"2025-07-29T20:42:47","date_gmt":"2025-07-30T00:42:47","guid":{"rendered":"https:\/\/literaciadigital.ufms.br\/?page_id=843"},"modified":"2025-07-29T20:48:37","modified_gmt":"2025-07-30T00:48:37","slug":"15-4","status":"publish","type":"page","link":"https:\/\/literaciadigital.ufms.br\/en\/data8\/15-0\/15-4\/","title":{"rendered":"Cap\u00edtulo 15.4"},"content":{"rendered":"<div style=\"position: relative\">\n<div style=\"float: left;width: 300px;background-color: #f5f5f5;border: 1px solid #ddd;border-radius: 5px;padding: 15px;margin-right: 20px;margin-bottom: 5px;overflow: hidden\">\n<h3 style=\"margin: 0 0 10px 0;padding-bottom: 8px;border-bottom: 1px solid #ddd\">\u00cdndice<\/h3>\n<ol style=\"margin: 0;padding-left: 0;list-style-type: none\">\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/\">1. O que \u00e9 Ci\u00eancia de Dados?<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-1\/\">1.1. Introdu\u00e7\u00e3o<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-1\/1-1\/\">1.1.1. Ferramentas Computacionais<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-1\/1-2\/\">1.1.2. T\u00e9cnicas Estat\u00edsticas<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-2\/\">1.2. Por que Ci\u00eancia de Dados?<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-3\/\">1.3. Tra\u00e7ando os Cl\u00e1ssicos<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-3\/3-1\/\">1.3.1. Personagens Liter\u00e1rios<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/1-0\/1-3\/3-2\/\">1.3.2. Outro Tipo de Personagem<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/2-0\/\">2. Causalidade e Experimentos<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/2-0\/2-1\/\">2.1. John Snow e a Bomba da Broad Street<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/2-0\/2-2\/\">2.2. O &#8220;Grande Experimento&#8221; de Snow<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/2-0\/2-3\/\">2.3. Estabelecendo Causalidade<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/2-0\/2-4\/\">2.4. Randomiza\u00e7\u00e3o<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/2-0\/2-5\/\">2.5. Notas Finais<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/3-0\/\">3. Progamando em Python<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/3-0\/3-1\/\">3.1. Express\u00f5es<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/3-0\/3-2\/\">3.2. Nomes<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/3-0\/3-2\/2-1\/\">3.2.1. Exemplo: Taxas de Crescimento<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/3-0\/3-3\/\">3.3. Chamadas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/3-0\/3-4\/\">3.4. Introdu\u00e7\u00e3o \u00e0s Tabelas<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/4-0\/\">4. Tipos de Dados<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/4-0\/4-1\/\">4.1. N\u00fameros<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/4-0\/4-2\/\">4.2. Strings<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/4-0\/4-2\/2-1\/\">4.2.1. M\u00e9todos de Strings<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/4-0\/4-3\/\">4.3. Compara\u00e7\u00f5es<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/5-0\/\">5. Sequ\u00eancias<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/5-0\/5-1\/\">5.1. Arrays<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/5-0\/5-2\/\">5.2. Ranges<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/5-0\/5-3\/\">5.3. Mais sobre Arrays<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/6-0\/\">6. Tabelas<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/6-0\/6-1\/\">6.1. Ordenando Linhas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/6-0\/6-2\/\">6.2. Selecionando Linhas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/6-0\/6-3\/\">6.3. Exemplo: Tend\u00eancias Populacionais<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/6-0\/6-4\/\">6.4. Examplo: Propor\u00e7\u00f5es de Sexos<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/7-0\/\">7. Visualiza\u00e7\u00e3o<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/7-0\/7-1\/\">7.1. Visualizando Distribui\u00e7\u00f5es<br \/>\nCateg\u00f3ricas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/7-0\/7-2\/\">7.2. Visualizando Distribui\u00e7\u00f5es Num\u00e9ricas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/7-0\/7-3\/\">7.3. Gr\u00e1ficos Sobrepostos<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/8-0\/\">8. Fun\u00e7\u00f5es e Tabelas<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/8-0\/8-1\/\">8.1. Aplicando Fun\u00e7\u00e3o a uma Coluna<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/8-0\/8-2\/\">8.2. Classificando por uma Vari\u00e1vel<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/8-0\/8-3\/\">8.3. Classifica\u00e7\u00e3o Cruzada<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/8-0\/8-4\/\">8.4. Unindo Tabelas por Colunas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/8-0\/8-5\/\">8.5. Compartilhamento de Bicicletas<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/9-0\/\">9. Aleatoriedade<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/9-0\/9-1\/\">9.1. Declara\u00e7\u00f5es Condicionais<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/9-0\/9-2\/\">9.2. Itera\u00e7\u00e3o<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/9-0\/9-3\/\">9.3. Simula\u00e7\u00e3o<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/9-0\/9-4\/\">9.4. O Problema de Monty Hall<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/9-0\/9-5\/\">9.5. Encontrando Probabilidades<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/10-0\/\">10. Amostragem e Distribui\u00e7\u00f5es Emp\u00edricas<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/10-0\/10-1\/\">10.1. Distribui\u00e7\u00f5es Emp\u00edricas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/10-0\/10-2\/\">10.2. Amostragem de uma Popula\u00e7\u00e3o<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/10-0\/10-3\/\">10.3. Distribui\u00e7\u00e3o Emp\u00edrica de uma<br \/>\nEstat\u00edstica<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/10-0\/10-4\/\">10.4. Amostragem Aleat\u00f3ria em Python <\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/11-0\/\">11. Testando Hip\u00f3teses<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/11-0\/11-1\/\">11.1. Avaliando um Modelo<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/11-0\/11-2\/\">11.2. M\u00faltiplas Categorias<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/11-0\/11-3\/\">11.3. Decis\u00f5es e Incertezas<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/11-0\/11-4\/\">11.4. Probabilidades de Erro<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/12-0\/\">12. Comparando Duas Amostras<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/12-0\/12-1\/\">12.1. Teste A\/B<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/12-0\/12-2\/\">12.2. Causalidade<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/12-0\/12-3\/\">12.3. Esvaziar<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/13-0\/\">13. Estima\u00e7\u00e3o<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/13-0\/13-1\/\">13.1. Percentis<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/13-0\/13-2\/\">13.2. O Bootstrap<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/13-0\/13-3\/\">13.3. Intervalos de Confian\u00e7a<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/13-0\/13-4\/\">13.4. Usando Intervalos de Confian\u00e7a<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/\">14. Por que a M\u00e9dia \u00e9 Importante<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-1\/\">14.1. Propriedades da M\u00e9dia<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-2\/\">14.2. Variabilidade<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-3\/\">14.3. O DP e a Curva Normal<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-4\/\">14.4. Teorema Central do Limite<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-5\/\">14.5. Variabilidade da M\u00e9dia da Amostra<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-6\/\">14.6. Escolhendo um Tamanho de Amostra<\/a><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom: 5px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/\">15. Previs\u00e3o<\/a>\n<ul style=\"margin: 5px 0 5px 15px;padding-left: 10px;list-style-type: none;border-left: 1px solid #ddd\">\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-1\/\">15.1. Correla\u00e7\u00e3o<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-2\/\">15.2. Linha de Regress\u00e3o<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-3\/\">15.3. M\u00e9todo dos M\u00ednimos Quadrados<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-4\/\">15.4. Regress\u00e3o de M\u00ednimos Quadrados<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-5\/\">15.5. Diagn\u00f3sticos Visuais<\/a><\/li>\n<li style=\"margin-bottom: 3px\"><a style=\"padding: 2px 0\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-6\/\">15.6. Diagn\u00f3stico Num\u00e9rico<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p><!-- Main Content --><\/p>\n<div style=\"overflow: hidden\">\n<p><!--###########################################################################################################################################################--><\/p>\n<pre><code><span style=\"color: black\">from datascience import *\r\n%matplotlib inline\r\npath_data = '..\/..\/..\/assets\/data\/'\r\nimport matplotlib.pyplot as plots\r\nplots.style.use('fivethirtyeight')\r\nimport numpy as np<\/span><\/code><\/pre>\n<p>&nbsp;<\/p>\n<h1 id=\"regress-o-de-m-nimos-quadrados\" style=\"text-align: center\">Regress\u00e3o de M\u00ednimos Quadrados<\/h1>\n<p style=\"text-align: justify\">Em uma se\u00e7\u00e3o anterior, desenvolvemos f\u00f3rmulas para a inclina\u00e7\u00e3o e a intercepta\u00e7\u00e3o da linha de regress\u00e3o por meio de um diagrama de dispers\u00e3o <em>em formato de bola de futebol<\/em>. Acontece que a inclina\u00e7\u00e3o e a intercepta\u00e7\u00e3o da linha de m\u00ednimos quadrados t\u00eam as mesmas f\u00f3rmulas daquelas que desenvolvemos, <em> independentemente da forma do gr\u00e1fico de dispers\u00e3o<\/em>.<\/p>\n<p style=\"text-align: justify\">Vimos isso no exemplo sobre Little Women, mas vamos confirmar em um exemplo em que o gr\u00e1fico de dispers\u00e3o claramente n\u00e3o tem o formato de um futebol americano. Pelos dados, estamos mais uma vez em d\u00edvida com os ricos <a href=\"http:\/\/www.stat.ufl.edu\/~winner\/datasets.html\">arquivos de dados do Prof. Larry Winner<\/a> da Universidade da Fl\u00f3rida. Um <a href=\"http:\/\/digitalcommons.wku.edu\/ijes\/vol6\/iss2\/10\/\">estudo de 2013<\/a> no International Journal of Exercise Science estudou atletas universit\u00e1rios de arremesso de peso e examinou a rela\u00e7\u00e3o entre for\u00e7a e dist\u00e2ncia de arremesso de peso. A popula\u00e7\u00e3o consiste em 28 atletas universit\u00e1rias. A for\u00e7a foi medida pela maior quantidade (em quilogramas) que o atleta levantou. o &#8220;1RM power clean&#8221; na pr\u00e9-temporada A dist\u00e2ncia (em metros) foi a melhor marca pessoal do atleta.<\/p>\n<pre><code><span style=\"color: black\">def standard_units(any_numbers):\r\n    \"Converta qualquer array de n\u00fameros em unidades padr\u00e3o.\"\r\n    return (any_numbers - np.mean(any_numbers))\/np.std(any_numbers)\r\n\r\ndef correlation(t, x, y):\r\n    return np.mean(standard_units(t.column(x))*standard_units(t.column(y)))\r\n\r\ndef slope(table, x, y):\r\n    r = correlation(table, x, y)\r\n    return r * np.std(table.column(y))\/np.std(table.column(x))\r\n\r\ndef intercept(table, x, y):\r\n    a = slope(table, x, y)\r\n    return np.mean(table.column(y)) - a * np.mean(table.column(x))\r\n\r\ndef fit(table, x, y):\r\n    \"\"\"Retorne a altura da linha de regress\u00e3o em cada valor de x.\"\"\"\r\n    a = slope(table, x, y)\r\n    b = intercept(table, x, y)\r\n    return a * table.column(x) + b<\/span><\/code><\/pre>\n<p style=\"text-align: justify\">\n<pre><code><span style=\"color: black\">shotput = Table.read_table(path_data + 'shotput.csv')<\/span><\/code><\/pre>\n<p style=\"text-align: justify\">\n<pre><code><span style=\"color: black\">shotput<\/span><\/code><\/pre>\n<table style=\"font-family: monospace;border-collapse: collapse;width: auto;margin-left: 1em\" border=\"1\">\n<thead>\n<tr style=\"background-color: #f0f0f0;border-bottom: 2px solid #ddd\">\n<th style=\"text-align: left;padding: 4px 8px\">Weight Lifted<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Shot Put Distance<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">37.5<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6.4<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">51.5<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">10.2<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">61.3<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">12.4<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">61.3<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">13<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">63.6<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">13.2<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">66.1<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">13<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">70<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">12.7<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">92.7<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">13.9<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">90.5<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">15.5<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">90.5<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">15.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">\n<pre><code><span style=\"color: black\">shotput.scatter('Weight Lifted')<\/span><\/code><\/pre>\n<p style=\"text-align: justify\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-844\" src=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-1.png\" alt=\"\" width=\"367\" height=\"342\" srcset=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-1.png 367w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-1-300x280.png 300w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-1-343x320.png 343w\" sizes=\"(max-width: 367px) 100vw, 367px\" \/><\/p>\n<p style=\"text-align: justify\">Esse n\u00e3o \u00e9 um gr\u00e1fico de dispers\u00e3o em forma de bola de futebol. Na verdade, parece ter um leve componente n\u00e3o linear. Mas se insistirmos em usar uma linha reta para fazer nossas previs\u00f5es, ainda existe uma melhor linha reta entre todas as linhas retas.<\/p>\n<p style=\"text-align: justify\">Nossas f\u00f3rmulas para a inclina\u00e7\u00e3o e intercepta\u00e7\u00e3o da linha de regress\u00e3o, derivadas de gr\u00e1ficos de dispers\u00e3o em formato de futebol, fornecem os seguintes valores.<\/p>\n<pre><code><span style=\"color: black\">slope(shotput, 'Weight Lifted', 'Shot Put Distance')<\/span><\/code><\/pre>\n<table style=\"font-family: monospace;border-spacing: 0;border-collapse: collapse;width: auto;margin-left: 1em\">\n<tbody>\n<tr>\n<td style=\"text-align: right;color: #888;padding-right: 0.5em\">Out[1]:<\/td>\n<td style=\"text-align: left\">0.09834382159781997<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">\n<pre><code><span style=\"color: black\">intercept(shotput, 'Weight Lifted', 'Shot Put Distance')<\/span><\/code><\/pre>\n<table style=\"font-family: monospace;border-spacing: 0;border-collapse: collapse;width: auto;margin-left: 1em\">\n<tbody>\n<tr>\n<td style=\"text-align: right;color: #888;padding-right: 0.5em\">Out[2]:<\/td>\n<td style=\"text-align: left\">5.959629098373952<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">\n<p style=\"text-align: justify\">Ainda faz sentido usar essas f\u00f3rmulas mesmo que o gr\u00e1fico de dispers\u00e3o n\u00e3o tenha o formato de uma bola de futebol? Podemos responder isso encontrando a inclina\u00e7\u00e3o e a intercepta\u00e7\u00e3o da linha que minimiza o mse.<\/p>\n<p style=\"text-align: justify\">Definiremos a fun\u00e7\u00e3o <code>shotput_linear_mse<\/code> para obter uma inclina\u00e7\u00e3o arbitr\u00e1ria e interceptar como argumentos e retornar o mse correspondente. Ent\u00e3o <code>minimize<\/code> aplicado a <code>shotput_linear_mse<\/code> retornar\u00e1 a melhor inclina\u00e7\u00e3o e intercepta\u00e7\u00e3o.<\/p>\n<pre><code><span style=\"color: black\">def shotput_linear_mse(any_slope, any_intercept):\r\n    x = shotput.column('Weight Lifted')\r\n    y = shotput.column('Shot Put Distance')\r\n    fitted = any_slope*x + any_intercept\r\n    return np.mean((y - fitted) ** 2)<\/span><\/code><\/pre>\n<p style=\"text-align: justify\">\n<pre><code><span style=\"color: black\">minimize(shotput_linear_mse)<\/span><\/code><\/pre>\n<table style=\"font-family: monospace;border-spacing: 0;border-collapse: collapse;width: auto;margin-left: 1em\">\n<tbody>\n<tr>\n<td style=\"text-align: right;color: #888;padding-right: 0.5em\">Out[3]:<\/td>\n<td style=\"text-align: left\">array([0.09834382, 5.95962911])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">\n<p style=\"text-align: justify\">Esses valores s\u00e3o os mesmos que obtivemos usando nossas f\u00f3rmulas. Para resumir:<\/p>\n<p style=\"text-align: justify\"><strong>Independentemente da forma do gr\u00e1fico de dispers\u00e3o, existe uma linha \u00fanica que minimiza o erro quadr\u00e1tico m\u00e9dio da estimativa. Ela \u00e9 chamada de linha de regress\u00e3o, e sua inclina\u00e7\u00e3o e intercepto s\u00e3o dados por<\/strong><\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;font-family: serif;font-size: 1.6em\"><strong>inclina\u00e7\u00e3o da linha de regress\u00e3o<\/strong> = r \u22c5 <sup>(DP de y)<\/sup>\u2044<sub>(DP de x)<\/sub><\/div>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;font-family: serif;font-size: 1.6em\"><strong>intercepto da linha de regress\u00e3o<\/strong> = m\u00e9dia de y &#8211; inclina\u00e7\u00e3o \u22c5 m\u00e9dia de x<\/div>\n<p>&nbsp;<\/p>\n<pre><code><span style=\"color: black\">fitted = fit(shotput, 'Weight Lifted', 'Shot Put Distance')\r\nshotput.with_column('Best Straight Line', fitted).scatter('Weight Lifted')<\/span><\/code><\/pre>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-845\" src=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-2.png\" alt=\"\" width=\"547\" height=\"345\" srcset=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-2.png 547w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-2-300x189.png 300w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-2-507x320.png 507w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p>\n<h2 id=\"regress-o-n-o-linear\">Regress\u00e3o N\u00e3o Linear<\/h2>\n<p style=\"text-align: justify\">O gr\u00e1fico acima refor\u00e7a nossa observa\u00e7\u00e3o anterior de que o gr\u00e1fico de dispers\u00e3o \u00e9 um pouco curvado. Portanto, \u00e9 melhor ajustar uma curva do que uma linha reta. O <a href=\"http:\/\/digitalcommons.wku.edu\/ijes\/vol6\/iss2\/10\/\">estudo<\/a> postulou uma rela\u00e7\u00e3o quadr\u00e1tica entre o peso levantado e a dist\u00e2ncia do arremesso de peso. Ent\u00e3o, vamos usar fun\u00e7\u00f5es quadr\u00e1ticas como nossos preditores e ver se conseguimos encontrar a melhor.<\/p>\n<p style=\"text-align: justify\">Temos que encontrar a melhor fun\u00e7\u00e3o quadr\u00e1tica entre todas as fun\u00e7\u00f5es quadr\u00e1ticas, em vez da melhor linha reta entre todas as linhas retas. O m\u00e9todo dos m\u00ednimos quadrados nos permite fazer isso.<\/p>\n<p style=\"text-align: justify\">A matem\u00e1tica dessa minimiza\u00e7\u00e3o \u00e9 complicada e n\u00e3o \u00e9 f\u00e1cil de ver apenas examinando o gr\u00e1fico de dispers\u00e3o. Mas a minimiza\u00e7\u00e3o num\u00e9rica \u00e9 t\u00e3o f\u00e1cil quanto com preditores lineares! Podemos obter o melhor preditor quadr\u00e1tico usando novamente <code>minimize<\/code>. Vamos ver como isso funciona.<\/p>\n<p style=\"text-align: justify\">Lembre-se de que uma fun\u00e7\u00e3o quadr\u00e1tica tem a forma<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;font-family: serif;font-size: 2.2em\">f(x) = ax<sup>2<\/sup> + bx + c<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">para constantes <em>a<\/em>, <em>b<\/em> e <em>c<\/em>.<\/p>\n<p style=\"text-align: justify\">Para encontrar a melhor fun\u00e7\u00e3o quadr\u00e1tica para prever a dist\u00e2ncia com base no peso levantado, usando o crit\u00e9rio dos m\u00ednimos quadrados, primeiro escreveremos uma fun\u00e7\u00e3o que recebe as tr\u00eas constantes como argumentos, calcula os valores ajustados usando a fun\u00e7\u00e3o quadr\u00e1tica acima e, em seguida, retorna o erro quadr\u00e1tico m\u00e9dio.<\/p>\n<p style=\"text-align: justify\">A fun\u00e7\u00e3o \u00e9 chamada <code>shotput_quadratic_mse<\/code>. Observe que a defini\u00e7\u00e3o \u00e9 an\u00e1loga \u00e0 de <code>lw_mse<\/code>, exceto que os valores ajustados s\u00e3o baseados em uma fun\u00e7\u00e3o quadr\u00e1tica em vez de linear.<\/p>\n<pre><code><span style=\"color: black\">def shotput_quadratic_mse(a, b, c):\r\n    x = shotput.column('Weight Lifted')\r\n    y = shotput.column('Shot Put Distance')\r\n    fitted = a*(x**2) + b*x + c\r\n    return np.mean((y - fitted) ** 2)<\/span><\/code><\/pre>\n<p style=\"text-align: justify\">\n<p style=\"text-align: justify\">Agora podemos usar <code>minimize<\/code> como antes para encontrar as constantes que minimizam o erro quadr\u00e1tico m\u00e9dio.<\/p>\n<pre><code><span style=\"color: black\">best = minimize(shotput_quadratic_mse)\r\nbest<\/span><\/code><\/pre>\n<table style=\"font-family: monospace;border-spacing: 0;border-collapse: collapse;width: auto;margin-left: 1em\">\n<tbody>\n<tr>\n<td style=\"text-align: right;color: #888;padding-right: 0.5em\">Out[4]:<\/td>\n<td style=\"text-align: left\">array([-1.04004838e-03, 2.82708045e-01, -1.53182115e+00])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">\n<p style=\"text-align: justify\">Nossa previs\u00e3o da dist\u00e2ncia do arremesso de peso para um atleta que levanta <em>x<\/em> quilogramas \u00e9 de cerca de<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;font-family: serif;font-size: 2.2em\">-0.00104x<sup>2<\/sup> + 0.2827x &#8211; 1.5318<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">metros. Por exemplo, se o atleta consegue levantar 100 quilos, a dist\u00e2ncia prevista \u00e9 de 16,33 metros. No gr\u00e1fico de dispers\u00e3o, isso est\u00e1 pr\u00f3ximo ao centro de uma faixa vertical em torno de 100 quilos.<\/p>\n<pre><code><span style=\"color: black\">(-0.00104)*(100**2) + 0.2827*100 - 1.5318<\/span><\/code><\/pre>\n<table style=\"font-family: monospace;border-spacing: 0;border-collapse: collapse;width: auto;margin-left: 1em\">\n<tbody>\n<tr>\n<td style=\"text-align: right;color: #888;padding-right: 0.5em\">Out[5]:<\/td>\n<td style=\"text-align: left\">16.3382<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify\">\n<p style=\"text-align: justify\">Aqui est\u00e3o as previs\u00f5es para todos os valores de <code>Weight Lifted<\/code>. Voc\u00ea pode ver que eles passam pelo centro do gr\u00e1fico de dispers\u00e3o, numa aproxima\u00e7\u00e3o aproximada.<\/p>\n<pre><code><span style=\"color: black\">x = shotput.column(0)\r\nshotput_fit = best.item(0)*(x**2) + best.item(1)*x + best.item(2)<\/span><\/code><\/pre>\n<p style=\"text-align: justify\">\n<pre><code><span style=\"color: black\">shotput.with_column('Best Quadratic Curve', shotput_fit).scatter(0)<\/span><\/code><\/pre>\n<p style=\"text-align: justify\"><img decoding=\"async\" class=\"alignnone size-full wp-image-846\" src=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-3.png\" alt=\"\" width=\"572\" height=\"345\" srcset=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-3.png 572w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-3-300x181.png 300w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/15-4-3-531x320.png 531w\" sizes=\"(max-width: 572px) 100vw, 572px\" \/><\/p>\n<p style=\"text-align: justify\"><strong>Nota:<\/strong> Ajustamos uma quadr\u00e1tica aqui porque foi sugerida no estudo original. Mas \u00e9 importante notar que na extremidade direita do gr\u00e1fico, a curva quadr\u00e1tica parece estar pr\u00f3xima do pico, ap\u00f3s o qual a curva ir\u00e1 come\u00e7ar a descer. Portanto, talvez n\u00e3o queiramos usar este modelo para novos atletas que conseguem levantar pesos muito maiores do que aqueles em nosso conjunto de dados.<\/p>\n<p>&nbsp;<\/p>\n<p><!--###########################################################################################################################################################--><\/p>\n<table width=\"100%\">\n<tbody>\n<tr>\n<td align=\"left\"><a class=\"next-page-link\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-3\/\">\u2190 Cap\u00edtulo 15.3 &#8211; M\u00e9todo dos M\u00ednimos Quadrados<\/a><\/td>\n<td align=\"right\"><a class=\"next-page-link\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/15-0\/15-5\/\">Cap\u00edtulo 15.5 &#8211; Diagn\u00f3sticos Visuais \u2192<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!--###########################################################################################################################################################--><\/p>\n<\/div>\n<\/div>\n<div style=\"clear: both;height: 1px;margin-top: -1px\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u00cdndice 1. O que \u00e9 Ci\u00eancia de Dados? 1.1. Introdu\u00e7\u00e3o 1.1.1. Ferramentas Computacionais 1.1.2. T\u00e9cnicas Estat\u00edsticas 1.2. Por que Ci\u00eancia de Dados? 1.3. Tra\u00e7ando os Cl\u00e1ssicos 1.3.1. Personagens Liter\u00e1rios 1.3.2. Outro Tipo de Personagem 2. Causalidade e Experimentos 2.1. John Snow e a Bomba da Broad Street 2.2. O &#8220;Grande Experimento&#8221; de Snow 2.3. Estabelecendo [&hellip;]<\/p>\n","protected":false},"author":21894,"featured_media":0,"parent":787,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-templates\/full-width.php","meta":{"footnotes":""},"coauthors":[14],"class_list":["post-843","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/843","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/users\/21894"}],"replies":[{"embeddable":true,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/comments?post=843"}],"version-history":[{"count":3,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/843\/revisions"}],"predecessor-version":[{"id":849,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/843\/revisions\/849"}],"up":[{"embeddable":true,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/787"}],"wp:attachment":[{"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/media?parent=843"}],"wp:term":[{"taxonomy":"author","embeddable":true,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/coauthors?post=843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}