Material de Estudo/3-estatistica-basica.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estatística Básica em R\n",
    "\n",
    "Agora que já vimos o básico de importação de dados em **R**, estamos prontos para entender como essa linguagem é utilizada efetivamente para fazer análises estatísticas!\n",
    "\n",
    "Mas não se preocupe, vamos explorando os diversos temas de estatística enquanto aprendemos mais sobre **R**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tipos de Dados\n",
    "\n",
    "Um dos pontos base da Estatística consiste em entender nossas informações e como elas elas são organizadas: os **dados**. Esses dados podem ser divididos em dois tipos diferentes: dados **qualitativos** e dados **quantitativos**.\n",
    "\n",
    "### Dados Qualitativos\n",
    "\n",
    "Esses dados representam **categorias** do nosso espaço, ou seja, eles dividem os indivíduos em diversas classes que não se intersectam. Por exemplo, a cor de uma flor é um tipo de dado qualitativo, já que as categoriza em classes como: *'amarelo'*, *'rosa'*, *'vermelho'*, *'azul'*, etc. Esses dados são bem úteis para representar características que não são facilmente descritas por números, como qualidades (daí vem o nome)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Flor</th><th scope=col>Cor</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>Margarida</td><td>Amarelo  </td></tr>\n",
       "\t<tr><td>Hortênsia</td><td>Azul     </td></tr>\n",
       "\t<tr><td>Violeta  </td><td>Azul     </td></tr>\n",
       "\t<tr><td>Rosa     </td><td>Vermelho </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " Flor & Cor\\\\\n",
       "\\hline\n",
       "\t Margarida & Amarelo  \\\\\n",
       "\t Hortênsia & Azul     \\\\\n",
       "\t Violeta   & Azul     \\\\\n",
       "\t Rosa      & Vermelho \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| Flor | Cor |\n",
       "|---|---|\n",
       "| Margarida | Amarelo   |\n",
       "| Hortênsia | Azul      |\n",
       "| Violeta   | Azul      |\n",
       "| Rosa      | Vermelho  |\n",
       "\n"
      ],
      "text/plain": [
       "  Flor      Cor     \n",
       "1 Margarida Amarelo \n",
       "2 Hortênsia Azul    \n",
       "3 Violeta   Azul    \n",
       "4 Rosa      Vermelho"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Montando um Dataframe de Cores de Flores\n",
    "\n",
    "Flor = c(\"Margarida\", \"Hortênsia\", \"Violeta\", \"Rosa\")\n",
    "Cor = c(\"Amarelo\", \"Azul\", \"Azul\", \"Vermelho\")\n",
    "flores = data.frame(Flor, Cor)\n",
    "flores"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>Flor</th><th scope=col>Cor</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>2</th><td>Hortênsia</td><td>Azul     </td></tr>\n",
       "\t<tr><th scope=row>3</th><td>Violeta  </td><td>Azul     </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       "  & Flor & Cor\\\\\n",
       "\\hline\n",
       "\t2 & Hortênsia & Azul     \\\\\n",
       "\t3 & Violeta   & Azul     \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | Flor | Cor |\n",
       "|---|---|---|\n",
       "| 2 | Hortênsia | Azul      |\n",
       "| 3 | Violeta   | Azul      |\n",
       "\n"
      ],
      "text/plain": [
       "  Flor      Cor \n",
       "2 Hortênsia Azul\n",
       "3 Violeta   Azul"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Mostrando as flores de cor Azul\n",
    "\n",
    "flores[flores$Cor == \"Azul\",]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dados Quantitativos\n",
    "\n",
    "São tipos de dados **numéricos**, ou seja, são mensuráveis e podem ser manipulados matematicamente. Essa categoria engloba os dados representáveis por números, como a altura de uma pessoa ou a área de um terreno. Esses dados são mais versáteis, já que conseguimos fazer comparações, médias e outros tipos de operações aritméticas que não são possíveis com os dados qualitativos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Nome</th><th scope=col>Altura</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>Marcela  </td><td>161      </td></tr>\n",
       "\t<tr><td>Alexandre</td><td>184      </td></tr>\n",
       "\t<tr><td>Rafaela  </td><td>172      </td></tr>\n",
       "\t<tr><td>Mariana  </td><td>174      </td></tr>\n",
       "\t<tr><td>Victor   </td><td>169      </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " Nome & Altura\\\\\n",
       "\\hline\n",
       "\t Marcela   & 161      \\\\\n",
       "\t Alexandre & 184      \\\\\n",
       "\t Rafaela   & 172      \\\\\n",
       "\t Mariana   & 174      \\\\\n",
       "\t Victor    & 169      \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| Nome | Altura |\n",
       "|---|---|\n",
       "| Marcela   | 161       |\n",
       "| Alexandre | 184       |\n",
       "| Rafaela   | 172       |\n",
       "| Mariana   | 174       |\n",
       "| Victor    | 169       |\n",
       "\n"
      ],
      "text/plain": [
       "  Nome      Altura\n",
       "1 Marcela   161   \n",
       "2 Alexandre 184   \n",
       "3 Rafaela   172   \n",
       "4 Mariana   174   \n",
       "5 Victor    169   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Montando um Dataframe de Altura de Pessoas\n",
    "\n",
    "Nome = c(\"Marcela\", \"Alexandre\", \"Rafaela\", \"Mariana\", \"Victor\")\n",
    "Altura = c(161, 184, 172, 174, 169)\n",
    "alturas = data.frame(Nome, Altura)\n",
    "alturas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>Nome</th><th scope=col>Altura</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>2</th><td>Alexandre</td><td>184      </td></tr>\n",
       "\t<tr><th scope=row>3</th><td>Rafaela  </td><td>172      </td></tr>\n",
       "\t<tr><th scope=row>4</th><td>Mariana  </td><td>174      </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       "  & Nome & Altura\\\\\n",
       "\\hline\n",
       "\t2 & Alexandre & 184      \\\\\n",
       "\t3 & Rafaela   & 172      \\\\\n",
       "\t4 & Mariana   & 174      \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | Nome | Altura |\n",
       "|---|---|---|\n",
       "| 2 | Alexandre | 184       |\n",
       "| 3 | Rafaela   | 172       |\n",
       "| 4 | Mariana   | 174       |\n",
       "\n"
      ],
      "text/plain": [
       "  Nome      Altura\n",
       "2 Alexandre 184   \n",
       "3 Rafaela   172   \n",
       "4 Mariana   174   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Selecionando as pessoas com altura maior que 170 cm\n",
    "\n",
    "alturas[alturas$Altura > 170,]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estatística Descritiva\n",
    "\n",
    "O tipo de análise estatística mais básico é a **Estatística Descritiva**, que busca simplesmente resumir e descrever os nossos dados da melhor maneira possível. Esse ramo utiliza de diversas medidas para expressar características diferentes dos nossos dados, como veremos a seguir.\n",
    "\n",
    "### Dataset\n",
    "\n",
    "Antes de qualquer coisa, vamos explicar essas medidas usando o dataset 'mpg', nativo do R.\n",
    "\n",
    "O 'mpg' é uma base de dados que apresenta diversas informações relacionadas à economia de combustível de modelos de automóvel lançados de 1999 a 2008. Algumas das características dos carros que aparecem no dataset são: \n",
    "  - **Marca** ('manufacturer') \n",
    "  - **Modelo** ('model')\n",
    "  - **Ano de Lançamento** ('year')\n",
    "  - **Cilindradas** ('displ')\n",
    "  - **Quantidade de Cilindros** ('cyl')\n",
    "  - **Tipo de Combustível** ('fl') - 'diesel', 'gasolina', 'elétrico'...\n",
    "  - **Classe** ('class') - 'compacto', 'SUV', 'minivan'...\n",
    "  - **Consumo de Combustível na Cidade** ('cty') - em mpg\n",
    "  - **Consumo de Combustível na Estrada** ('hwy') - em mpg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Registered S3 methods overwritten by 'ggplot2':\n",
      "  method         from \n",
      "  [.quosures     rlang\n",
      "  c.quosures     rlang\n",
      "  print.quosures rlang\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>manufacturer</th><th scope=col>model</th><th scope=col>year</th><th scope=col>displ</th><th scope=col>cyl</th><th scope=col>fl</th><th scope=col>class</th><th scope=col>cty</th><th scope=col>hwy</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>audi    </td><td>a4      </td><td>1999    </td><td>1.8     </td><td>4       </td><td>p       </td><td>compact </td><td>7.652592</td><td>12.32918</td></tr>\n",
       "\t<tr><td>audi    </td><td>a4      </td><td>1999    </td><td>1.8     </td><td>4       </td><td>p       </td><td>compact </td><td>8.928024</td><td>12.32918</td></tr>\n",
       "\t<tr><td>audi    </td><td>a4      </td><td>2008    </td><td>2.0     </td><td>4       </td><td>p       </td><td>compact </td><td>8.502880</td><td>13.17946</td></tr>\n",
       "\t<tr><td>audi    </td><td>a4      </td><td>2008    </td><td>2.0     </td><td>4       </td><td>p       </td><td>compact </td><td>8.928024</td><td>12.75432</td></tr>\n",
       "\t<tr><td>audi    </td><td>a4      </td><td>1999    </td><td>2.8     </td><td>6       </td><td>p       </td><td>compact </td><td>6.802304</td><td>11.05374</td></tr>\n",
       "\t<tr><td>audi    </td><td>a4      </td><td>1999    </td><td>2.8     </td><td>6       </td><td>p       </td><td>compact </td><td>7.652592</td><td>11.05374</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllllll}\n",
       " manufacturer & model & year & displ & cyl & fl & class & cty & hwy\\\\\n",
       "\\hline\n",
       "\t audi     & a4       & 1999     & 1.8      & 4        & p        & compact  & 7.652592 & 12.32918\\\\\n",
       "\t audi     & a4       & 1999     & 1.8      & 4        & p        & compact  & 8.928024 & 12.32918\\\\\n",
       "\t audi     & a4       & 2008     & 2.0      & 4        & p        & compact  & 8.502880 & 13.17946\\\\\n",
       "\t audi     & a4       & 2008     & 2.0      & 4        & p        & compact  & 8.928024 & 12.75432\\\\\n",
       "\t audi     & a4       & 1999     & 2.8      & 6        & p        & compact  & 6.802304 & 11.05374\\\\\n",
       "\t audi     & a4       & 1999     & 2.8      & 6        & p        & compact  & 7.652592 & 11.05374\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| manufacturer | model | year | displ | cyl | fl | class | cty | hwy |\n",
       "|---|---|---|---|---|---|---|---|---|\n",
       "| audi     | a4       | 1999     | 1.8      | 4        | p        | compact  | 7.652592 | 12.32918 |\n",
       "| audi     | a4       | 1999     | 1.8      | 4        | p        | compact  | 8.928024 | 12.32918 |\n",
       "| audi     | a4       | 2008     | 2.0      | 4        | p        | compact  | 8.502880 | 13.17946 |\n",
       "| audi     | a4       | 2008     | 2.0      | 4        | p        | compact  | 8.928024 | 12.75432 |\n",
       "| audi     | a4       | 1999     | 2.8      | 6        | p        | compact  | 6.802304 | 11.05374 |\n",
       "| audi     | a4       | 1999     | 2.8      | 6        | p        | compact  | 7.652592 | 11.05374 |\n",
       "\n"
      ],
      "text/plain": [
       "  manufacturer model year displ cyl fl class   cty      hwy     \n",
       "1 audi         a4    1999 1.8   4   p  compact 7.652592 12.32918\n",
       "2 audi         a4    1999 1.8   4   p  compact 8.928024 12.32918\n",
       "3 audi         a4    2008 2.0   4   p  compact 8.502880 13.17946\n",
       "4 audi         a4    2008 2.0   4   p  compact 8.928024 12.75432\n",
       "5 audi         a4    1999 2.8   6   p  compact 6.802304 11.05374\n",
       "6 audi         a4    1999 2.8   6   p  compact 7.652592 11.05374"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Para usar o dataset, precisamos importar a biblioteca ggplot2, que será melhor explicada na parte de Visualização de Dados.\n",
    "library(ggplot2)\n",
    "\n",
    "# Montando o dataset\n",
    "consumo = mpg\n",
    "\n",
    "# Selecionando e reordenando as colunas mais importantes\n",
    "consumo <- consumo[c(\"manufacturer\", \"model\", \"year\", \"displ\", \"cyl\", \"fl\", \"class\", \"cty\", \"hwy\")]\n",
    "\n",
    "# Conversão de milhas por galão de combustível para quilometros por litro\n",
    "consumo$cty <- consumo$cty * 0.425144\n",
    "consumo$hwy <- consumo$hwy * 0.425144\n",
    "\n",
    "# Apresentando o dataset\n",
    "head(consumo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Medidas de Tendência Central\n",
    "\n",
    "As **Medidas de Tendência Central** procuram representrar um valor central em torno do qual os dados se agrupam. Você provavelmente já ouviu falar de alguns exemplos desse tipo de medida, como a **média** e a **mediana**.\n",
    "\n",
    "#### Média\n",
    "\n",
    "A **Média** é a medida numérica que representa a posição central dos nossos dados. Seu valor é obtido ao dividir a soma de todos os nossos dados pela sua quantidade.\n",
    "\n",
    "$$\\large \\bar x ={\\frac {1}{N}}\\sum _{i=1}^{N}x_{i}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "[1] 7.877668"
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   ],
   "source": [
    "# Média do consumo dos automóveis da Toyota\n",
    "\n",
    "mean(consumo$cty[consumo$manufacturer == \"toyota\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.952016  <  8.896532"
     ]
    }
   ],
   "source": [
    "# Média do consumo dos automóveis da Ford e da Volkswagen\n",
    "\n",
    "media_a = mean(consumo$cty[consumo$manufacturer == \"ford\"])\n",
    "media_b = mean(consumo$cty[consumo$manufacturer == \"volkswagen\"])\n",
    "\n",
    "# A eficiência dos carros da Ford é, em média, bem inferior aos da Volkswagen\n",
    "\n",
    "cat(media_a, \" < \", media_b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in mean.default(consumo$manufacturer):\n",
      "\"argument is not numeric or logical: returning NA\""
     ]
    },
    {
     "data": {
      "text/html": [
       "&lt;NA&gt;"
      ],
      "text/latex": [
       "<NA>"
      ],
      "text/markdown": [
       "&lt;NA&gt;"
      ],
      "text/plain": [
       "[1] NA"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Não é possível calcular a média de dados qualitativos\n",
    "\n",
    "mean(consumo$manufacturer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Mediana\n",
    "\n",
    "A **Mediana** é a medida numérica que representa o dado que se encontra na metade da nossa distribuição quanto esta se encontra ordenada. Ou seja, a mediana é o valor que divide os nossos dados de tal forma que metade deles sejam maiores que a mediana e a outra metade seja menor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "[1] 7.865164"
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     "metadata": {},
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    }
   ],
   "source": [
    "# Mediana do consumo dos automóveis da Hyundai\n",
    "\n",
    "median(consumo$cty[consumo$manufacturer == \"hyundai\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in mean.default(sort(x, partial = half + 0L:1L)[half + 0L:1L]):\n",
      "\"argument is not numeric or logical: returning NA\""
     ]
    },
    {
     "data": {
      "text/html": [
       "&lt;NA&gt;"
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       "<NA>"
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       "&lt;NA&gt;"
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      "text/plain": [
       "[1] NA"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Não é possível calcular a mediana de dados qualitativos\n",
    "\n",
    "median(consumo$manufacturer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Percentil\n",
    "\n",
    "O **Percentil** não é uma medida de tendência central, mas se relaciona bastante com a Mediana. Os percentis são medidas que dividem as nossas amostras tal que **p%** delas sejam menores que o percentil. O **50º percentil**, portanto, é a nossa já conhecida mediana, enquanto o **25º percentil** é denominado primeiro quartil."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>0%</th><th scope=col>25%</th><th scope=col>50%</th><th scope=col>75%</th><th scope=col>100%</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>3.826296</td><td>5.952016</td><td>7.227448</td><td>8.077736</td><td>14.88004</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       " 0\\% & 25\\% & 50\\% & 75\\% & 100\\%\\\\\n",
       "\\hline\n",
       "\t 3.826296 & 5.952016 & 7.227448 & 8.077736 & 14.88004\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0% | 25% | 50% | 75% | 100% |\n",
       "|---|---|---|---|---|\n",
       "| 3.826296 | 5.952016 | 7.227448 | 8.077736 | 14.88004 |\n",
       "\n"
      ],
      "text/plain": [
       "     0%       25%      50%      75%      100%    \n",
       "[1,] 3.826296 5.952016 7.227448 8.077736 14.88004"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Obtendo os quartis do consumo de todos os automóveis\n",
    "\n",
    "quartis = quantile(consumo$cty)\n",
    "\n",
    "# Transpondo o resultado para ficar mais visualizável\n",
    "t(quartis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>30%</th><th scope=col>60%</th><th scope=col>90%</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>5.101728</td><td>5.952016</td><td>7.184934</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       " 30\\% & 60\\% & 90\\%\\\\\n",
       "\\hline\n",
       "\t 5.101728 & 5.952016 & 7.184934\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 30% | 60% | 90% |\n",
       "|---|---|---|\n",
       "| 5.101728 | 5.952016 | 7.184934 |\n",
       "\n"
      ],
      "text/plain": [
       "     30%      60%      90%     \n",
       "[1,] 5.101728 5.952016 7.184934"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Obtendo os percentis 30%, 60% e 90% do consumo dos automóveis SUV\n",
    "\n",
    "percentis = quantile(consumo$cty[consumo$class == \"suv\"], c(.30, .60, .90))\n",
    "\n",
    "# Transpondo o resultado para ficar mais visualizável\n",
    "t(percentis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Summary\n",
    "\n",
    "Uma maneira bem simples de mostrar vários dos valores que vimos até agora é utilizando a função ```summary()```. Ela nos apresenta o *mínimo*, o *1º quartil*, a *mediana*, a *média*, o *3º quartil* e o *máximo* da variável em questão, da seguinte forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. \n",
       "  3.826   5.952   7.227   7.167   8.078  14.880 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Resumo do consumo de combustível de todos os automóveis\n",
    "\n",
    "summary(consumo$cty)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Medidas de Variação\n",
    "\n",
    "As **Medidas de Variação** medem o *grau de afastamento* dos dados em torno de um valor central. Elas são importantes para medir a homogeneidade dos dados, indicando se nossas amostras estão próximas entre si ou muito afastadas.\n",
    "\n",
    "#### Amplitude\n",
    "\n",
    "A medida mais simples de variação é a **Amplitude**, que mede simplesmente a diferença entre a maior e a menor amostra. Entretanto, essa medida é bem sensível outliers em suas extremidades.\n",
    "\n",
    "$$ \\large A =  x_{max} - x_{min} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.401152"
      ],
      "text/latex": [
       "3.401152"
      ],
      "text/markdown": [
       "3.401152"
      ],
      "text/plain": [
       "[1] 3.401152"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculando a amplitude do consumo de combustível das pickups\n",
    "\n",
    "amplitude = max(consumo$cty[consumo$class == \"pickup\"]) - min(consumo$cty[consumo$class == \"pickup\"])\n",
    "\n",
    "amplitude"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Amplitude Interquartil\n",
    "\n",
    "Uma outra medida de variação mais precisa é a **Amplitude Interquartil**, definida como a diferença entre o 3º e o 1º quartil. Por desprezar os valores mais extremos, essa medida é menos sujeita a flutuações que a amplitude normal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.275432"
      ],
      "text/latex": [
       "1.275432"
      ],
      "text/markdown": [
       "1.275432"
      ],
      "text/plain": [
       "[1] 1.275432"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculando a amplitude interquartil do consumo de combustível das pickups\n",
    "\n",
    "IQR(consumo$cty[consumo$class == \"pickup\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Box Plot\n",
    "\n",
    "O **Box Plot** é uma representação gráfica das nossas amostras que se aproveita dos conceitos de quartis. Esse gráfico apresenta um quadro entre o 1º e o 3º quartil, com uma linha central na mediana e extremidades nos valores máximo e mínimo dos dados, da seguinte forma:\n",
    "\n",
    "<img src=\"img/boxplot.png\" alt=\"Box Plot\" style=\"width: 300px;\"/>\n",
    "\n",
    "Essa representação é bem relevante para entender a **simetria** das nossas amostras, bem como sua dispersão."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAAMFBMVEUAAABNTU1oaGh8fHyM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\nYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Aprensentando o Box Plot do consumo de combustível dos automóveis da Dodge\n",
    "\n",
    "boxplot(consumo$cty[consumo$manufacturer == \"dodge\"], horizontal=TRUE, \n",
    "        xlab= \"Quilômetros por Litro\", ylab=\"Dodge\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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B8kZyfbOOgiIosya1eWz7N2R+ezdg66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\nkAABggQIECRAgCABAgQJECBIgMB/cDs48mYAlbkAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"Consumo de Combustível por Tipo de Automóvel\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Box Plot comparando o consumo de cada tipo de automóvel\n",
    "\n",
    "# Estabelecendo as margens do gráfico, para facilitar a visualização\n",
    "par(mar=c(7,5,1,1))\n",
    "\n",
    "boxplot(cty~class,data=consumo, main=\"Consumo de Combustível por Tipo de Automóvel\", las=2, \n",
    "   xlab=\"\", ylab=\"Quilômetros por Litro\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As pickups e os SUVs são os carros com menor eficiência de combustível."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Variância\n",
    "\n",
    "A **Variância** é uma medida de variação que mensura a dispersão das amostras com relação à média.\n",
    "\n",
    "Seu valor é obtido a partir do cálculo da média do quadrado das diferenças entre os dados e a média, da seguinte forma:\n",
    "<br>\n",
    "<br>\n",
    "\n",
    "$$\\large s^2 = {\\frac{\\sum_{i=1}^{N}(x_{i} -\\bar{x})^2}{N-1}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.67300412398439"
      ],
      "text/latex": [
       "1.67300412398439"
      ],
      "text/markdown": [
       "1.67300412398439"
      ],
      "text/plain": [
       "[1] 1.673004"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculando a variância do consumo dos carros movidos a gasolina\n",
    "\n",
    "var(consumo$cty[consumo$fl == \"p\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Desvio Padrão\n",
    "\n",
    "O **Desvio Padrão** é uma outra medida de variação obtida simplesmente pela raiz quadrada da variância. Essa raiz é feita para que a unidade de medida seja a mesma dos nossos dados.\n",
    "<br>\n",
    "<br>\n",
    "\n",
    "$$\\large s = \\sqrt{\\frac{\\sum_{i=1}^{N}(x_{i} -\\bar{x})^2}{N-1}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.29344660654562"
      ],
      "text/latex": [
       "1.29344660654562"
      ],
      "text/markdown": [
       "1.29344660654562"
      ],
      "text/plain": [
       "[1] 1.293447"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculando o desvio padrão do consumo dos carros movidos a gasolina\n",
    "\n",
    "sd(consumo$cty[consumo$fl == \"p\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Covariância\n",
    "\n",
    "A **Covariância** é a medida da *relação linear* entre duas variáveis: seu valor representa o quanto elas \"caminham juntas\". Se a covarância é positiva, significa que as duas crescem juntas; se a covariância é negativa, significa que conforme uma cresce, a outra decresce.\n",
    "\n",
    "Seu valor é obtido da seguinte forma:\n",
    "<br>\n",
    "\n",
    "$$\\large s_{x,y}=\\frac{\\sum_{i=1}^{N}(x_{i}-\\bar{x})(y_{i}-\\bar{y})}{N-1}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-1.86667550795642"
      ],
      "text/latex": [
       "-1.86667550795642"
      ],
      "text/markdown": [
       "-1.86667550795642"
      ],
      "text/plain": [
       "[1] -1.866676"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculando a covarância entre as cilindradas e o consumo de combustível\n",
    "\n",
    "cov(consumo$displ, consumo$cty)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A covariância negativa significa que quanto maiores as cilindradas, menor será a eficiência do carro"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Coeficiente de Correlação\n",
    "\n",
    "O **Coeficiente de Correlação** também mede a relação linear entre duas variáveis, mas é obtida a partir da divisão da covariância pelos desvios padrões individuais de cada variável. Dessa forma, encontramos uma medida **normalizada** da relação entre as duas. \n",
    "<br>\n",
    "<br>\n",
    "\n",
    "$$\\large r_{x,y}= \\frac{s_{x,y}}{s_{x} s_{y}}$$\n",
    "\n",
    "Quanto mais próximo de 1, maior a _relação linear positiva_ entre as duas variáveis, e seu gráfico se aproxima de uma linha reta _crescente_. Quanto mais próximo de -1, maior a _relação linear negativa_ entre as duas variáveis, e seu gráfico se aproxima de uma linha reta _decrescente_. Quando seu valor é próximo de 0, não existe uma relação muito grande entre as duas variáveis."
   ]
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   "source": [
    "# Calculando o coeficiente de correlação entre as cilindradas e o consumo de combustível\n",
    "\n",
    "cor(consumo$displ, consumo$cty)"
   ]
  },
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   "metadata": {},
   "source": [
    "O consumo de combustível tem uma relação linear negativa bem forte com as cilindradas."
   ]
  }
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