{"id":738,"date":"2025-07-29T16:01:32","date_gmt":"2025-07-29T20:01:32","guid":{"rendered":"https:\/\/literaciadigital.ufms.br\/?page_id=738"},"modified":"2025-10-11T10:50:46","modified_gmt":"2025-10-11T14:50:46","slug":"14-2","status":"publish","type":"page","link":"https:\/\/literaciadigital.ufms.br\/en\/data8\/14-0\/14-2\/","title":{"rendered":"Cap\u00edtulo 14.2"},"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=\"variabilidade\" style=\"text-align: center\">Variabilidade<\/h1>\n<p style=\"text-align: justify\">A m\u00e9dia nos diz onde um histograma se equilibra. Mas em quase todos os histogramas que vimos, os valores se espalham dos dois lados da m\u00e9dia. Qu\u00e3o longe da m\u00e9dia eles podem estar? Para responder a essa pergunta, desenvolveremos uma medida de variabilidade em rela\u00e7\u00e3o \u00e0 m\u00e9dia.<\/p>\n<p style=\"text-align: justify\">Come\u00e7aremos descrevendo como calcular a medida. Em seguida, veremos por que \u00e9 uma boa medida para calcular.<\/p>\n<h2 id=\"aproxima-o-do-tamanho-das-devia-es-da-m-dia\" style=\"text-align: justify\">Aproxima\u00e7\u00e3o do Tamanho das Devia\u00e7\u00f5es da M\u00e9dia<\/h2>\n<p style=\"text-align: justify\">Para simplificar, come\u00e7aremos nossos c\u00e1lculos no contexto de uma matriz simples <code>any_numbers<\/code> composta apenas por quatro valores. Como veremos, nosso m\u00e9todo se estender\u00e1 facilmente a qualquer outra matriz de valores.<\/p>\n<pre><code><span style=\"color: black\">any_numbers = make_array(1, 2, 2, 10)<\/span><\/code><\/pre>\n<p style=\"text-align: justify\">O objetivo \u00e9 medir aproximadamente qu\u00e3o distantes os n\u00fameros est\u00e3o de sua m\u00e9dia. Para fazer isso, primeiro precisamos da m\u00e9dia:<\/p>\n<pre><code><span style=\"color: black\"># Passo 1. A m\u00e9dia.\r\n\r\nmean = np.mean(any_numbers)\r\nmean<\/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\">3.75<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A seguir, vamos descobrir a que dist\u00e2ncia cada valor est\u00e1 da m\u00e9dia. Eles s\u00e3o chamados de <em>desvios da m\u00e9dia<\/em>. Um &#8220;desvio da m\u00e9dia&#8221; \u00e9 apenas um valor menos a m\u00e9dia. A tabela <code>calculation_steps<\/code> exibe os resultados .<\/p>\n<pre><code><span style=\"color: black\"># Passo 2. Os desvios da m\u00e9dia.\r\n\r\ndeviations = any_numbers - mean\r\ncalculation_steps = Table().with_columns(\r\n        'Value', any_numbers,\r\n        'Deviation from Average', deviations\r\n        )\r\ncalculation_steps<\/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\">Value<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Deviation from Average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-2.75<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">2<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-1.75<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">2<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-1.75<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">10<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6.25<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Alguns dos desvios s\u00e3o negativos; correspondem a valores abaixo da m\u00e9dia. Desvios positivos correspondem a valores acima da m\u00e9dia.<\/p>\n<p style=\"text-align: justify\">Para calcular aproximadamente o tamanho dos desvios, \u00e9 natural calcular a m\u00e9dia dos desvios. Mas algo interessante acontece quando todos os desvios s\u00e3o somados:<\/p>\n<pre><code><span style=\"color: black\">sum(deviations)<\/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\">0.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Os desvios positivos cancelam exatamente os negativos. Isso \u00e9 verdade para todas as listas de n\u00fameros, n\u00e3o importa como o histograma da lista se pare\u00e7a: <strong>a soma dos desvios da m\u00e9dia \u00e9 zero.<\/strong><\/p>\n<p style=\"text-align: justify\">Como a soma dos desvios \u00e9 0, a m\u00e9dia dos desvios tamb\u00e9m ser\u00e1 0:<\/p>\n<pre><code><span style=\"color: black\">np.mean(deviations)<\/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\">0.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Por causa disso, a m\u00e9dia dos desvios n\u00e3o \u00e9 uma medida \u00fatil do tamanho dos desvios. O que realmente queremos saber \u00e9 aproximadamente qu\u00e3o grandes s\u00e3o os desvios, independentemente de serem positivos ou negativos. para eliminar os sinais dos desvios.<\/p>\n<p style=\"text-align: justify\">Existem duas maneiras consagradas de perder sinais: o valor absoluto e o quadrado. Acontece que tomar o quadrado constr\u00f3i uma medida com propriedades extremamente poderosas, algumas das quais estudaremos neste curso.<\/p>\n<p style=\"text-align: justify\">Ent\u00e3o vamos eliminar os sinais elevando ao quadrado todos os desvios. Depois tiraremos a m\u00e9dia dos quadrados:<\/p>\n<pre><code><span style=\"color: black\"># Etapa 3. Os desvios quadrados da m\u00e9dia\r\n\r\nsquared_deviations = deviations ** 2\r\ncalculation_steps = calculation_steps.with_column(\r\n   'Squared Deviations from Average', squared_deviations\r\n    )\r\ncalculation_steps<\/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\">Value<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Deviation from Average<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Squared Deviations from Average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-2.75<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">7.5625<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">2<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-1.75<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">3.0625<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">2<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-1.75<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">3.0625<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">10<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6.25<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">39.0625<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<pre><code><span style=\"color: black\"># Etapa 4. Vari\u00e2ncia = desvio quadr\u00e1tico m\u00e9dio da m\u00e9dia\r\n\r\nvariance = np.mean(squared_deviations)\r\nvariance<\/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\">13.1875<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Vari\u00e2ncia:<\/strong> O desvio quadr\u00e1tico m\u00e9dio calculado acima \u00e9 chamado de <em>vari\u00e2ncia<\/em> dos valores.<\/p>\n<p style=\"text-align: justify\">Embora a vari\u00e2ncia nos d\u00ea uma ideia de dispers\u00e3o, ela n\u00e3o est\u00e1 na mesma escala que a vari\u00e1vel original, pois suas unidades s\u00e3o o quadrado do original. Isso torna a interpreta\u00e7\u00e3o muito dif\u00edcil.<\/p>\n<p style=\"text-align: justify\">Portanto, voltamos \u00e0 escala original extraindo a raiz quadrada positiva da vari\u00e2ncia:<\/p>\n<pre><code><span style=\"color: black\"># Passo 5.\r\n# Desvio Padr\u00e3o:      raiz do desvio quadr\u00e1tico m\u00e9dio da m\u00e9dia\r\n# Etapas de c\u00e1lculo: 5    4      3       2             1\r\n\r\nsd = variance ** 0.5\r\nsd<\/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\">3.6314597615834874<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<h2 id=\"desvio-padr-o\" style=\"text-align: justify\">Desvio Padr\u00e3o<\/h2>\n<p style=\"text-align: justify\">A quantidade que acabamos de calcular \u00e9 chamada de <em>desvio padr\u00e3o<\/em> da lista, e \u00e9 abreviada como DP. Ele mede aproximadamente qu\u00e3o distantes os n\u00fameros da lista est\u00e3o de sua m\u00e9dia.<\/p>\n<p style=\"text-align: justify\"><strong>Defini\u00e7\u00e3o.<\/strong> O DP de uma lista \u00e9 definido como a <em>raiz quadrada da m\u00e9dia dos quadrados das devia\u00e7\u00f5es da m\u00e9dia<\/em>. Isso \u00e9 complicado. Mas se lermos da direita para a esquerda, temos a sequ\u00eancia de passos no c\u00e1lculo.<\/p>\n<p style=\"text-align: justify\"><strong>C\u00e1lculo.<\/strong> Os cinco passos descritos acima resultam no SD. Voc\u00ea tamb\u00e9m pode usar a fun\u00e7\u00e3o <code>np.std<\/code> para calcular o SD dos valores em uma matriz:<\/p>\n<pre><code><span style=\"color: black\">np.std(any_numbers)<\/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[6]:<\/td>\n<td style=\"text-align: left\">3.6314597615834874<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<h2 id=\"trabalhando-com-o-sd\" style=\"text-align: justify\">Trabalhando com o SD<\/h2>\n<p>Obs: SD \u00e9 a sigla em ingles para Standard Deviation (Desvio Padr\u00e3o)<\/p>\n<p style=\"text-align: justify\">Para ver o que podemos aprender com o SD, vamos passar para um conjunto de dados mais interessante do que <code>any_numbers<\/code>. A tabela <code>nba13<\/code> cont\u00e9m dados sobre os jogadores da National Basketball Association (NBA) em 2013. Para cada jogador, a tabela registra a posi\u00e7\u00e3o em que o jogador normalmente jogava, sua altura em polegadas, seu peso em libras e sua idade em anos.<\/p>\n<pre><code><span style=\"color: black\">nba13 = Table.read_table(path_data + 'nba2013.csv')\r\nnba13<\/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\">Name<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Position<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Height<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Weight<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Age in 2013<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">DeQuan Jones<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">80<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">221<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">23<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Darius Miller<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">80<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">235<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">23<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Trevor Ariza<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">80<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">210<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">28<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">James Jones<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">80<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">215<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">32<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Wesley Johnson<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">79<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">215<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">26<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Klay Thompson<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">79<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">205<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">23<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Thabo Sefolosha<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">79<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">215<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">29<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Chase Budinger<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">79<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">218<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">25<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Kevin Martin<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">79<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">185<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">30<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Evan Fournier<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">79<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">206<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Aqui est\u00e1 um histograma da altura dos jogadores.<\/p>\n<pre><code><span style=\"color: black\">nba13.select('Height').hist(bins=np.arange(68, 88, 1))<\/span><\/code><\/pre>\n<p style=\"text-align: justify\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-739\" src=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-1.png\" alt=\"\" width=\"433\" height=\"284\" srcset=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-1.png 433w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-1-300x197.png 300w\" sizes=\"(max-width: 433px) 100vw, 433px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">N\u00e3o \u00e9 nenhuma surpresa que os jogadores da NBA sejam altos! Sua altura m\u00e9dia \u00e9 pouco mais de 79 polegadas (6&#8217;7&#8243;), cerca de 25 cent\u00edmetros mais alta do que a altura m\u00e9dia dos homens nos Estados Unidos.<\/p>\n<pre><code><span style=\"color: black\">mean_height = np.mean(nba13.column('Height'))\r\nmean_height<\/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[7]:<\/td>\n<td style=\"text-align: left\">79.06534653465347<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A que dist\u00e2ncia est\u00e3o as alturas dos jogadores da m\u00e9dia? Isso \u00e9 medido pelo SD das alturas, que \u00e9 cerca de 3,45 polegadas.<\/p>\n<pre><code><span style=\"color: black\">sd_height = np.std(nba13.column('Height'))\r\nsd_height<\/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[8]:<\/td>\n<td style=\"text-align: left\">3.4505971830275546<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">O imponente centro Hasheem Thabeet do Oklahoma City Thunder era o jogador mais alto, com 87 polegadas de altura.<\/p>\n<pre><code><span style=\"color: black\">nba13.sort('Height', descending=True).show(3)<\/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\">Name<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Position<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Height<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Weight<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Age in 2013<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Hasheem Thabeet<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Center<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">87<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">263<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">26<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Roy Hibbert<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Center<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">86<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">278<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">26<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Alex Len<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Center<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">85<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">255<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Thabeet estava cerca de 20 cent\u00edmetros acima da altura m\u00e9dia.<\/p>\n<pre><code><span style=\"color: black\">87 - mean_height<\/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[9]:<\/td>\n<td style=\"text-align: left\">7.934653465346528<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Isso \u00e9 um desvio da m\u00e9dia e \u00e9 cerca de 2,3 vezes o desvio padr\u00e3o:<\/p>\n<pre><code><span style=\"color: black\">(87 - mean_height)\/sd_height<\/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[10]:<\/td>\n<td style=\"text-align: left\">2.2995015194397923<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Em outras palavras, a altura do jogador mais alto estava cerca de 2,3 SD acima da m\u00e9dia.<\/p>\n<p style=\"text-align: justify\">Com 69 polegadas de altura, Isaiah Thomas foi um dos dois jogadores mais baixos da NBA em 2013. Sua altura estava cerca de 2,9 SD abaixo da m\u00e9dia.<\/p>\n<pre><code><span style=\"color: black\">nba13.sort('Height').show(3)<\/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\">Name<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Position<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Height<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Weight<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Age in 2013<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Isaiah Thomas<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">69<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">185<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">24<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Nate Robinson<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">69<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">180<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">John Lucas III<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">71<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">157<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<pre><code><span style=\"color: black\">(69 - mean_height)\/sd_height<\/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[11]:<\/td>\n<td style=\"text-align: left\">-2.9169868288775844<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">O que observamos \u00e9 que os jogadores mais altos e mais baixos estavam a apenas alguns SDs da altura m\u00e9dia. Este \u00e9 um exemplo de por que o SD \u00e9 uma medida \u00fatil de propaga\u00e7\u00e3o. N\u00e3o importa o formato do histograma, a m\u00e9dia e o SD juntos dizem muito sobre onde o histograma est\u00e1 situado na reta num\u00e9rica.<\/p>\n<h2 id=\"primeira-raz-o-principal-para-medir-a-dispers-o-pelo-dp\" style=\"text-align: justify\">Primeira raz\u00e3o principal para medir a dispers\u00e3o pelo SD<\/h2>\n<p style=\"text-align: justify\"><strong>Declara\u00e7\u00e3o informal.<\/strong> Em todos os conjuntos de dados num\u00e9ricos, a maior parte das entradas est\u00e1 dentro do intervalo &#8220;m\u00e9dia \u00b1 alguns SDs&#8221;.<\/p>\n<p style=\"text-align: justify\">Por enquanto, resista ao desejo de saber exatamente o que palavras vagas como &#8220;maior parte&#8221; e &#8220;alguns&#8221; significam. N\u00f3s as tornaremos precisas mais tarde nesta se\u00e7\u00e3o. Vamos apenas examinar a declara\u00e7\u00e3o no contexto de mais alguns exemplos.<\/p>\n<p style=\"text-align: justify\">J\u00e1 vimos que <em>todas<\/em> as alturas dos jogadores da NBA estavam no intervalo &#8220;m\u00e9dia \u00b1 3 SDs&#8221;.<\/p>\n<p style=\"text-align: justify\">E as idades? Aqui est\u00e1 um histograma da distribui\u00e7\u00e3o, juntamente com a m\u00e9dia e o SD das idades.<\/p>\n<pre><code><span style=\"color: black\">nba13.select('Age in 2013').hist(bins=np.arange(15, 45, 1))<\/span><\/code><\/pre>\n<p style=\"text-align: justify\"><img decoding=\"async\" class=\"alignnone size-full wp-image-740\" src=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-2.png\" alt=\"\" width=\"437\" height=\"284\" srcset=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-2.png 437w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-2-300x195.png 300w\" sizes=\"(max-width: 437px) 100vw, 437px\" \/><\/p>\n<p>&nbsp;<\/p>\n<pre><code><span style=\"color: black\">ages = nba13.column('Age in 2013')\r\nmean_age = np.mean(ages)\r\nsd_age = np.std(ages)\r\nmean_age, sd_age<\/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[12]:<\/td>\n<td style=\"text-align: left\">(26.19009900990099, 4.321200441720307)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A idade m\u00e9dia era de pouco mais de 26 anos e o SD era de cerca de 4,3 anos.<\/p>\n<p style=\"text-align: justify\">A que dist\u00e2ncia estavam as idades da m\u00e9dia? Assim como fizemos com as alturas, vejamos um exemplo.<\/p>\n<p style=\"text-align: justify\">Juwan Howard era o jogador mais velho, aos 40 anos.<\/p>\n<pre><code><span style=\"color: black\">nba13.sort('Age in 2013', descending=True).show(3)<\/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\">Name<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Position<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Height<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Weight<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Age in 2013<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Juwan Howard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Forward<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">81<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">250<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">40<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Steve Nash<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">75<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">178<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">39<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Derek Fisher<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">Guard<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">73<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">210<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">39<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A idade de Howard estava cerca de 3,2 SD acima da m\u00e9dia.<\/p>\n<pre><code><span style=\"color: black\">(40 - mean_age)\/sd_age<\/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[13]:<\/td>\n<td style=\"text-align: left\">3.1958482778922357<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">O que observamos para as alturas e idades \u00e9 verdadeiro em grande generalidade. Para <em>todas<\/em> as listas, a maior parte das entradas est\u00e1 a n\u00e3o mais do que 2 ou 3 desvios padr\u00e3o da m\u00e9dia.<\/p>\n<h2 id=\"limites-de-chebychev-\" style=\"text-align: justify\">Limites de Chebychev<\/h2>\n<p style=\"text-align: justify\">O matem\u00e1tico russo <a href=\"https:\/\/en.wikipedia.org\/wiki\/Pafnuty_Chebyshev\">Pafnuty Chebychev<\/a> (1821-1894) provou um resultado que torna nossas afirma\u00e7\u00f5es aproximadas precisas.<\/p>\n<p style=\"text-align: justify\"><strong>Para todas as listas e todos os n\u00fameros z, a propor\u00e7\u00e3o de entradas que est\u00e3o no intervalo &#8220;m\u00e9dia \u00b1 z SDs&#8221; \u00e9 pelo menos 1 &#8211; <sup>1<\/sup>\u2044<sub>z<sup>2<\/sup><\/sub>.<\/strong><\/p>\n<p style=\"text-align: justify\">\u00c9 importante notar que o resultado fornece um limite inferior, n\u00e3o um valor exato ou uma aproxima\u00e7\u00e3o.<\/p>\n<p style=\"text-align: justify\">O que torna o resultado poderoso \u00e9 que ele \u00e9 verdadeiro para todas as listas \u2013 todas as distribui\u00e7\u00f5es, n\u00e3o importa qu\u00e3o irregulares.<\/p>\n<p style=\"text-align: justify\">Especificamente, ele diz que para cada lista:<\/p>\n<ul style=\"text-align: justify\">\n<li>a propor\u00e7\u00e3o no intervalo &#8220;m\u00e9dia \u00b1 2 SDs&#8221; \u00e9 <strong>pelo menos 1 &#8211; 1\/4 = 0.75<\/strong><\/li>\n<li>a propor\u00e7\u00e3o no intervalo &#8220;m\u00e9dia \u00b1 3 SDs&#8221; \u00e9 <strong>pelo menos 1 &#8211; 1\/9 \u2248 0.89<\/strong><\/li>\n<li>a propor\u00e7\u00e3o no intervalo &#8220;m\u00e9dia \u00b1 4.5 SDs&#8221; \u00e9 <strong>pelo menos 1 &#8211; 1\/4.5<sup>2<\/sup> \u2248 0.95<\/strong><\/li>\n<\/ul>\n<p style=\"text-align: justify\">Como observamos acima, o resultado de Chebychev fornece um limite inferior, n\u00e3o uma resposta exata ou uma aproxima\u00e7\u00e3o. Por exemplo, a percentagem de entradas no intervalo &#8220;m\u00e9dia \u00b1 2 SDs&#8221; pode ser consideravelmente maior que 75%. Mas n\u00e3o pode ser menor.<\/p>\n<h2 id=\"unidades-padr-o\" style=\"text-align: justify\">Unidades padr\u00e3o<\/h2>\n<p style=\"text-align: justify\">Nos c\u00e1lculos acima, a quantidade z mede <em>unidades padr\u00e3o<\/em>, o n\u00famero de desvios padr\u00e3o acima da m\u00e9dia.<\/p>\n<p style=\"text-align: justify\">Alguns valores de unidades padr\u00e3o s\u00e3o negativos, correspondendo a valores originais abaixo da m\u00e9dia. Outros valores de unidades padr\u00e3o s\u00e3o positivos. Mas, independentemente da apar\u00eancia da distribui\u00e7\u00e3o da lista, os limites de Chebychev implicam que as unidades padr\u00e3o estar\u00e3o tipicamente na faixa (-5, 5).<\/p>\n<p style=\"text-align: justify\">Para converter um valor em unidades padr\u00e3o, primeiro encontre o qu\u00e3o distante ele est\u00e1 da m\u00e9dia e depois compare essa varia\u00e7\u00e3o com o desvio padr\u00e3o.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"font-family: serif;font-size: 2.2em;text-align: center\">z = <sup>(valor &#8211; m\u00e9dia)<\/sup>\u2044<sub>SD<\/sub><\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Como veremos, as unidades padr\u00e3o s\u00e3o frequentemente usadas na an\u00e1lise de dados. Portanto, \u00e9 \u00fatil definir uma fun\u00e7\u00e3o que converta um array de n\u00fameros em unidades padr\u00e3o.<\/p>\n<pre><code><span style=\"color: black\">def standard_units(numbers_array):\r\n    \"Converta qualquer array de n\u00fameros em unidades padr\u00e3o.\"\r\n    return (numbers_array - np.mean(numbers_array))\/np.std(numbers_array)    <\/span><\/code><\/pre>\n<p>&nbsp;<\/p>\n<h2 id=\"exemplo\" style=\"text-align: justify\">Exemplo<\/h2>\n<p style=\"text-align: justify\">Como vimos em uma se\u00e7\u00e3o anterior, a tabela <code>united<\/code> cont\u00e9m uma coluna <code>Delay<\/code> que consiste nos tempos de atraso de partida, em minutos, de mais de milhares de voos da United Airlines no ver\u00e3o de 2015. Criaremos uma nova coluna chamada <code>Delay (Standard Units)<\/code> aplicando a fun\u00e7\u00e3o <code>standard_units<\/code> \u00e0 coluna de tempos de atraso. Isso nos permite ver todos os tempos de atraso em minutos, bem como seus valores correspondentes em unidades padr\u00e3o.<\/p>\n<pre><code><span style=\"color: black\">united = Table.read_table(path_data + 'united_summer2015.csv')\r\nunited = united.with_column(\r\n    'Delay (Standard Units)', standard_units(united.column('Delay'))\r\n)\r\nunited<\/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\">Date<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Flight Number<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Destination<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Delay<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Delay (Standard Units)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">73<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">HNL<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">257<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6.08766<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">217<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">EWR<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">28<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">0.287279<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">237<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">STL<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-3<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-0.497924<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">250<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">SAN<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">0<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-0.421937<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">267<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">PHL<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">64<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1.19913<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">273<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">SEA<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-6<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-0.573912<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">278<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">SEA<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-8<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-0.62457<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">292<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">EWR<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">12<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-0.117987<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">300<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">HNL<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">20<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">0.0846461<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/1\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">317<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">IND<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-10<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">-0.675228<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As unidades padr\u00e3o que podemos ver s\u00e3o consistentes com o que esperamos com base nos limites de Chebychev. A maioria \u00e9 de tamanho bastante pequeno; apenas uma est\u00e1 acima de 6.<\/p>\n<p style=\"text-align: justify\">Mas algo bastante alarmante acontece quando classificamos os tempos de atraso do maior para o menor. As unidades padr\u00e3o que podemos ver s\u00e3o extremamente altas!<\/p>\n<pre><code><span style=\"color: black\">united.sort('Delay', descending=True)<\/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\">Date<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Flight Number<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Destination<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Delay<\/th>\n<th style=\"text-align: left;padding: 4px 8px\">Delay (Standard Units)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/21\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1964<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">SEA<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">580<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">14.269<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/22\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">300<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">HNL<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">537<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">13.1798<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/21\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1149<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">IAD<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">508<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">12.4453<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/20\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">353<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">ORD<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">505<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">12.3693<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">8\/23\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1589<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">ORD<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">458<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">11.1788<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">7\/23\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1960<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">LAX<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">438<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">10.6722<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/23\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1606<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">ORD<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">430<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">10.4696<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/4\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1743<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">LAX<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">408<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">9.91236<\/td>\n<\/tr>\n<tr>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">6\/17\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">1122<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">HNL<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">405<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">9.83637<\/td>\n<\/tr>\n<tr style=\"background-color: #f8f8f8\">\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">7\/27\/15<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">572<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">ORD<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">385<\/td>\n<td style=\"padding: 4px 8px;border: 1px solid #ddd\">9.32979<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">O que isso mostra \u00e9 que \u00e9 poss\u00edvel que os dados estejam muitos SDs acima da m\u00e9dia (e que os voos tenham atrasos de quase 10 horas). O maior valor de atraso \u00e9 superior a 14 em unidades padr\u00e3o.<\/p>\n<p style=\"text-align: justify\">No entanto, a propor\u00e7\u00e3o desses valores extremos \u00e9 pequena, e os limites de Chebychev ainda s\u00e3o verdadeiros. Por exemplo, vamos calcular a porcentagem de tempos de atraso que est\u00e3o no intervalo &#8220;media \u00b1 3 SDs&#8221;. Isso \u00e9 o mesmo que a porcentagem de vezes em que as unidades padr\u00e3o est\u00e3o no intervalo (-3, 3). Isso \u00e9 cerca de 98%, conforme calculado abaixo, consistente com o limite de Chebychev de &#8220;pelo menos 89%&#8221;.<\/p>\n<pre><code><span style=\"color: black\">within_3_sd = united.where('Delay (Standard Units)', are.between(-3, 3))\r\nwithin_3_sd.num_rows\/united.num_rows<\/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[14]:<\/td>\n<td style=\"text-align: left\">0.9790235081374322<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">O histograma dos tempos de atraso \u00e9 mostrado abaixo, com o eixo horizontal em unidades padr\u00e3o. Pela tabela acima, a cauda direita continua at\u00e9 z=14,27 unidades padr\u00e3o (580 minutos). A \u00e1rea do histograma fora do intervalo z=-3 a z=3 \u00e9 cerca de 2%, reunidos em pequenos peda\u00e7os que s\u00e3o praticamente invis\u00edveis no histograma.<\/p>\n<pre><code><span style=\"color: black\">united.hist('Delay (Standard Units)', bins=np.arange(-5, 15.5, 0.5))\r\nplots.xticks(np.arange(-6, 17, 3));<\/span><\/code><\/pre>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-741\" src=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-3.png\" alt=\"\" width=\"442\" height=\"284\" srcset=\"https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-3.png 442w, https:\/\/literaciadigital.ufms.br\/files\/2025\/07\/14-2-3-300x193.png 300w\" sizes=\"(max-width: 442px) 100vw, 442px\" \/><\/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\/14-0\/14-1\/\">\u2190 Cap\u00edtulo 14.1 &#8211; Usando os Intervalos<\/a><\/td>\n<td align=\"right\"><a class=\"next-page-link\" href=\"https:\/\/literaciadigital.ufms.br\/data8\/14-0\/14-3\/\">Cap\u00edtulo 14.3 &#8211; DP e Curva Normal \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":722,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-templates\/full-width.php","meta":{"footnotes":""},"coauthors":[14],"class_list":["post-738","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/738","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=738"}],"version-history":[{"count":12,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/738\/revisions"}],"predecessor-version":[{"id":1058,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/738\/revisions\/1058"}],"up":[{"embeddable":true,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/pages\/722"}],"wp:attachment":[{"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/media?parent=738"}],"wp:term":[{"taxonomy":"author","embeddable":true,"href":"https:\/\/literaciadigital.ufms.br\/en\/wp-json\/wp\/v2\/coauthors?post=738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}