Presentación y análisis descriptivo de datos con uso de software R

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.title[
# Presentación y análisis descriptivo de datos con uso de software R
]
.subtitle[
## Variable cuantitativa
]
.author[
### Prof. Roberto Roque <a href="https://rpubs.com/alonso97roque/1335091">   RPub</a>
]
.institute[
### Departamento de Matemática y Estatística, Universidad Catolica San Pablo
]

---

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i+rqVp8GprmU+tzKapF1ViGRkFWV7ltn/9GuNv30lHR0d8cFyXC/bv4/UvXuD//efrCFZdljTuuiCOE0sYjGklXauotfFzaNEwgjBC/QgqEVoOUQ1BQzIa4RIvx9GvEIgII5/VxT46s1Eaa0+R4ldFWNLc71Mb3UNP7fDdn+vX3KQPQrVECrimhaycZCqNU7OypFaUHA+C7+jt5+iXv829z34G5150YX2iT29fP5c848nMnJhjda3CeefspTPf0SYbaxOA6kZbt1EMamPLKQxDOjs7+epX/52sl2GtpOCkJ2uKFA9ztwZpddK00SK4QUq6YeBc22fSbyQKVZhzYLavG48mVboRCGO9k1up0h0qPeLSlfHI1XpUKeR6M1z939/Ntn/5PP1DQ8kWC+Pj4/zlm99IaJUf3vwzwkjIZnMtAgpNZAj1kWDJGDObiERVbV00qjaKJ+MksS5VC7aRE/TDgMN3z5D1Mi2B/F9iWCCdHv0LYpBBM7J7xJglY2+fv51fzjHtN9u2jZvjx39qH62/2a/OJZSGDyct3WSkSZXe9gcbZgMbNYMqcR/1C1/6Yp77rGfV2xdbaymXyywtLzE9OcnEkfuYPHyYE0cm6CxX6VOlz8ngZDIYa/n+u/8Hz3v/X5Lv76s7rY7rEFSq+KV5Pv7VH7IU9NKRy8ffq8TdF2zciC9Mgup+EFIOQ/wwxAYhNgyxYRAXPNcD7XF2MGa7uBODrlS5YF+BrJ3BD/y4/9WDxPjAuLEeA0CPCBOT0xN++2dGhwt9ImZQVCcmZyfvVzJfGB3LoRSssjAzO/krmeA6tnU8r0YLqjI1PTdR2nA7hwuuCgVUSm5oFo+fOP5zXdwjWwueMTKOyKKXc4tHjx5d95mhLaOu45ghhHMWVxYHcIwtjIwdVcuh6bnJxYfyBjI6nN8R+ScvGB0p3JrNeQeOHj1KYXh8PA5ChDOuZvpUtEcNU9PTk35KWL8U4pKfi+t0k1W5CLt2nMbjzjmnXmUoplG7aFWpVn1OFk9y4OBBbrvlFu696ScsHz7OSCgMdHVSPHqY6z/xGZ7xljfiZLN1Sy2Xy/HkC87DMfDBz1/L8kqEl8lhNSKyShRZgjAkDCOiMMIPguTfIX4Y4IchlTDERkGcDbRhnBG0cVZQNB7vpb6PlZgAf94Yls1YFzVvN0aeZa19FXCo7UIwgnmNiLxDsa8Ebrifu7ZBOdeI8xnV6L3A538FZ4tRo+eLmE+j9k3AtzY5QYYccT4Tqb0+1PC9gP9zWCTGccx2EfMvVu3HSqXSJ2uHtvaBrVu3Gtc4vyXwVkTOAs3F55xZRPSG0ZHCu6Znpm59IF+2ZcsWk3WzI1boEdFD09PTYfO2jIxsywn6foHzQZ599OhRA1gj8mEL1lX3d1X09SLmVar6u8AdKWE9BE7h+nNLNmQeobVV+zo3sHmdKo0R8k2DSeMe7o2WNrUi5o6ODvL5DsbHxrj4qU9lbm6eq6+7ltu/fxXL9xxjm5Nl6oc38Z3K3/Ksd/wZmY5cUi0k9PT08rSnXISI4W++9EOKFcgYF2trUoX4EdRIKohlCzWyCm0QE1RCViYhK9TG03cA0fWl3g+a1CMx6miXqgwpusnvrHmUcdR4G9zdDQwS1yqu1roDVVR0QlSX78caaH+9ORttT/HZpuUuE3ejoHkbbPKPY6Bt1tVOA4cZGRlB4/UMiEpXa6J1u4Fjm21Hy2eGh4exqq4jOqhoXja4sWac7H8TeJcqc6r2HYLerogL+mxEXm8w3xjdOvaM6fnJg+3bGR9bLCywdeu5JiMnPbDvcsVcGFn7fGCqaRutaPQ6EWe3RV82PeMdbTr7B0DDSCJEnWVBp0TV3+AYn2q/k/d3AodtSlgbBKNrTfha9Es0YlK6cduDjWmv+VyKLK7rYoypB85rQycgmbZVb6anOI5LoTDKb774xZy/fz8/uv56Dl75fYKlNTI/O8QPPvwP7H/db9NTGEVVEQHPy/LEJ5zHH5YD/ubLN2JtdxKfsvUeVzYK6/82NkJshEMsZRBJHljEJNIHbYTvIydOUKjen9J9c+FovO9JT2bdWL7SfNgymQxdXV2oqjHGkMlk82BcazOl+fnVELDWRgdEeK2qLvf29ppMJsPCwkKrO1YoUC6XiaLILC8vW8CMbh3NWcTDaDg7O12qXTS1z7qui61YN5PN5kW0ND0/bYe2dLrGuDlQ/MArLRZX44stMrdaV19ljBRr39nV1WW6cn5eTME1jpRsWN85U7OMRcR0d9uc6KgnqmG1Ui2dXD3ZcnH29vaafNbm1Y65jssqUe03kbZOrzAyPH4W8BcKR0H/YHm1euPaWnwsRodGrxPMnSryETG8ZXR49O3Ts9MlyJnRwcBTxnIqWCNSmp4jtHYSm/EQNRbECtDT08Py8nJiXY0YkNut2j9Skbtgfp27YYyByPlnJfoajrPYYp0NFXIKnoI/NzfVQvR9fX0m5+XygnEtFX9lNV8plUo2JawW90/WvfRAPZ/1Ewib+mRFSv+WLVyw/3yw2phboeA4Tv3LrMayg2Z3NJvNccbuMxgfG+PG03dw4zeuYPrue/F+dpDr/+4TPPlP/4je0eF6S63u7m6e/uTzmJia4f/zietxV/04RmVDVAOsDbEaoNZHNX4YqqDVRBia9LXSekqh/hxay8JkJ4P9YZ10N4bH/XVrkAd0H1GzdWB4BPQ1IF8Bu1uR1wJdxpgfjQ6NfXJ6bnJGjAwp+jqM863OXOdZKjI4NHT6P8zNHak03b935LyO3xaRq3o6em6xqk9DeLWBEUEWRofG/rEalq9bXFwMsWZ7zsv/pqBfIy8XAC8EPjQ2um1OrX098AQQm83KD0eHx//39OzEjDq2gMprLPJF4CBAd2fvCwR9GciQhvIDI3yltjmO49DR0UHOy/8W8DwRMwJa7OjsuCLXkfvy9Px0PRbX2dH1O6g+H4c+G3I5ynXOBtnZ0S2jrqCvFCRvVT88MNh349RszkJMWNNz0+HI8Ng3HXieiuzFSg9QGh3ecolgXyrIuKIlQa8eGxn7vLhSsqG+XGG/qg4h/EFnvuvry8vLN40MFYaMymuAi5K2Hz8eH+77+MTsarH5V1a1GLH7Vdmvav8ZmCkUCp5aLhaVVysMITJXGClcVvErNy0uLoaF4UIfyG8BzwbpMqIzvd39Xy+VSv+WElbNctKW8DrrTe3mKThNLWWE5nkVrZO/mmy04a4ujlzxbSbyHWAV1/OwUYSTyZDN5+keGqJvbJTurVtxPC/O0NVtG8jn8zztKU9lx2k7uOrKb3Ps29cwPj3D9/6fD/Ibf/Xn9I0OJ7RiGBgY4KXPewanjfYzPDyM52Ub27aRPagPLGwXRRGum+Gz/+ez3Jj9EWulUpusobaCX6y9TJOHbRQ7bozzDrX6bDFmpygHFPIC71Dh+SMj237DEO004rwjsnZOjDlTkP8m4t8EXNeIc8lrjJh3W+xvWOFdIvLniNyOtVMYc45R/Xou0/G2Xbt2fby0WtnpGuctoY2ebMRcKsIhjexOlC+IGGvV3iIiOYO8WbGvHBkZfzrYvUbMu9RGd42PjB+1Vn9fRD4MckDVHhKRlyn6QlT7EDVhZL2cl/8LY8x/U9VrVDkkmJ0ifAiH5/b2nvbqXMa6jqPvjLeV27F2SsS8MY7taQ5V0xL7c00fcDGxb/edO++8s90isTOzk4uDg4P/JW/zbpCNwsJw4fdE5KPADYq9S9QUxMi7VfVVYWBf7Ip5NsIOVdtljLnERvbewsjYqiCfA9muaq9B8YyYt1jVl40Oj/2X6dnJuxqXloOqPtsY8/pI9ZrhgeEiVt5jkD9FuAXVRRF5Euq8IJfJ/+7wcOZbiPknQS5Q1atAjxrkHIQvFIbH3hnB/5qdfeQF7h9+C6t5TL1sRFiysUu44XJrUEuMMHPsOP/+Nx+JK1tcByMGk3HwHI+tw4P0bt1Cz9ZBBk47jZ0XPpGhPWeQ6+5qEqfG/a1O234az3/pS7g6l2PmO9cyXK1w7d9exlPe8kf0jxUS99AwPr6NLYPDXPOju1g+ESDWJnMGbV3OoFZBbV3KoC39rbTG5fX9qfgBxaUT2Chi88Ij4RetJWwkWhVUrFoNEfao1Teheg1GPFXeDfLHBrtPJZbFJuHBy0X4feDiwkDhhqnFKTsyPN6D8HTQW1GMCK+zql8BfSsiq1jbg8inEXlTebV8JQhWbZfAM1X1D9TqDQiXipBXa1+tVm9UI65BXyTGfEisfZqoVDAagthIdYcx5q2KfkdV3yrKRKQ6ZIx+SJHzk3NiRMS8XK39PKrvdDxTjHzNI/IB4Lc6OqIRUbaLyB+qtf+M8G5VXRDsDuBjqjq4Lgak5MXIgKrOobq6WSxvYWHBwpP8wtDxATHm1aDfU7VvtGJnHCSnVt+GyH83ULCqbwPe5xhzQWT1DYI5DPpRYETV/oHA9yI1xqBPQfiCwJtHR7e/maRfmqpFMYHGehrrZNynCPIaVf03Ud4KWsHa7Yh8HeENBu+YCBer1fcq9u8yoRuGblQA+UdEXmzUfhaYe9RnCWszARvlNHKKz8rG2cDN4lgqzBuYK2ylJ5erZ/dcEc7b9zjC79zA/IkiJ+89xuydP2P6J7exZccOHvOi57F9/z6stU0qeGFo61ae/rxLuBaY/MZ3GZ1Vbvj0P/GMt/4Juc7Oenwsm3FYWlrgXX97JVFnjqAaoGEAdQlDLF+IZQwR2CR+rPGAwVa3T6muBjz5yTvIR3NU/WADk6xGWPfvEuoDsLJQqZcuWbXftBp9Z2Zuxh8fH8cG+l3gj4Ehqfk8sY7/LpSDiDxDM/oPwKIR3S3IXkUvE+TZyc/yPuPKjOd5rK2trTrqfsQY85FIOVdgObnQv4hjv0iAQehDMQp5P6r4PZ09Fb8SfsVaOyeqBxDZE1csqEG5AHSnqr6jb3bk4N3MAcePFobGPiaGZ1mwIrKs1n5C4cqV0vLCaO8o90zcUxwbGT+u4InYnGAuVnQQ9LKevp5jBw4cALirMDz2aeBJG9x4DUhOVStqJTzV8d21a57Siq0AX8LKrZKRicpaha6urlUN9HhMdpI3wrJqXPIgaNFAj4VLFb6pxn7TqtpKtY+xMb518kTxeyDPigkoJlMRUyuYNwY1irw0iYR+eKCzf+HwkcOUKN0xNjL+EeCxqIaq+m4V/bwYSt1BN4sszoFZFGQ8nriSyhoa7CKbXVbaOpRCW3uztw56FhphUcARPNfh5f/ltbzkhS+qR4Zcx6U8M8tXfng7pUoZoxCVqxTvm6KytMLMbXew59Ln8LhXvJiO3t5GoB4Y2jrExc+/lOs8j6l/v5zgrp/xvY99imf/6R+TcTOxlCKT4TlPO4/vXn83l99wjMhRKlFIYGsaq2SwBBrXFEqiu5IkjiVNyn8gdOIouda6jcpmttH9u4OiDywmaOvLetvM3EwFYGJiwoyOFEoGCRUxtW76qmAlXDC4/27EvE2VHcACKpciYtVG3zLGeb+qegh/YAP1y2EVR1xQGVDVPoFzVLneGPHV8hPHMZT9svXIfQv0LSLypQ4vf4dfDq4Hrga9cWpuaqowPLYn3i8BYYfGE5AO3M3NtQyYsegBV5wZsRajFNXwWZBn9nT2vaq0Ui6MjW7boarnoxixWAzbBIrW6OGErGpjau9AZLHdepK4lqoiRvIq0abX0NjYmFsul3Ohhquu5P6PiD5TQ/3LzmzXNg1lO3B+MlTESARqGt8TCTsNDFnVH0/P1CQOMxSLmMLw2LVi5EmKHQeTtPaISOIG1sZyh3MUnVGRo/4RnxJxIH1yZuJ/1hMkI2MlVJ5FxNlFb7lgcHcqdj9whz5CJ4s/ItrLtE6Dls2aWG1aVyj1DgaSSA+6GdgywMDAAAMDW+jt7Y0lDZl4DFc8isvBybjYxII5+r1r+MHff5Li5FTSu516B9KBgS1c9Oxn0X/B46lGITPfv56fXnMdYRjWv7O/v583/s4zCU6u4vshK0FAJQyphAHVKKQaxfMGqzaKO4/a5BHZln/7NnEhNwuZt5T25De3nOKDYlVwUbPRRWW1fsNS29QEIlxHjS3eeby/s7OzoSDfUtUuEXlaoVBwReTZqnqrGDkA9IHkUfYB+0XZr8p+0B2qehOw0Ei4qD1+/DgLCwvoWnSrwsuAv1elImJeLmI+JmK+Ojo8fq622JgaZ9Vqgcj6SW3CWPagWEdHVOQTxsiHgRcoDKBcJ8o32+xP277vqlqS+LUWiUakWlFlQWDIYHraCW2QQUZHxjyN+FMi+SfXeGeJ6gfEyGWIeaVCQdHbQb+2YYtJFQT1kn+1x5EsSik5CG7c4oMWKz0Z9JIXxUclPMjBddf9yND4bkG+ZMS8F7g4lr7oFQq3yCNYRf+wdxxdRzY0Wr6cSqMlm7mXbc2w4j7wmtQLtq5Bkn7sTjJivtYiJv59LYt3/4yrP/RRSieXaAosYYxhcMsWnvCiS7FDW8hkPW7+7L8wffxY3YozYnjs2WfygifvprpWwTS1iok7M2iis4rFodSn39h6lF60Pfepm+10LQe2+ZG2EoIso9ojooU4Nd72u4uMxS4Iy7XvlE0uoI02JSK4S4Q7QJ6OZTfofhGumJqZKiFSRJgQeHUEzydynx9EzvOx+nyxzksrAZ9Ndt7WvnHLli3QzaBG0QFL+A418mKr9sWq9h8E9ojqm5K9M4iiKnOJlzDQysTWBXIqhCAXC7zEWvtBizxXxX3l5MzxdyPcDGJVxAjMqdIDZmA725vD67lk/abNwloGvQnYrsr+bdu2tbwfEBhRGQR5tYj0iepegder1c+q8lxRfdnUzMQ7FLk2GUZiWz0MRYQFhRCRoYZ7uYtdu3YZFT09CVrV4meIZJpzWwAlFfqMar55nwojhQsKI2MvMqJvRmS7qn0r6HP9sPJqK/I/SIRq8ghtEvmrsbD01On35gEOtaPf+L826E2aJto0WUQtHUqbxno5mXh8fI2sjOvGE5rdpGuoGKITRb79wb9leWGhQXoSk92OnbvY/6qXsRIE2Ol5bvy3r7O6spK8b8hmc7zxjZcyEvrkJVawx21ltN7vqlZvWC8A140sTamJHu7nAG7uEs4sToSoXgvkEd4g1unatm1b/eIrDI89BdUXKNwlgdwlm6QulQaxNm/Ptm3bjOd5obX2X4F9irxFxExYa6+M/RK9QWDQKvuq1byf63F8x6lW1MjLcextWdc8MxF4GkVMz969ZDO5HrHmc+I4nxOcAWNsaXDW3A5cBnJXLb6FCBJnAG4UkSmFlxZGCt7AwIDZvXs3xsjFqmxPzoQzFMII+ZpxosVMxjI6Ot6jypNrOSCr+gMRfLFcGvQG7lbONf1nnGFEuURVB9uyhHZ6bjpE5dNAKMJfhL7dmxxXA5jOoc6cir5VkHMt+k9qzIBCHtGvu57MeZ2eHRvbkQc9L749WaOeNl2Paqth5pAgtwMvLIyMjwwPD7uAKZX8HhHzAlUOW+GgtF/DgjEiRlVvEBhC9HxF2btjrykUCnkjzocFeYfCXoHDIDd4Hd5qd3e3EbUjcQwSRFKXkA1VoCI0K5HWm7dy/+FjaSIWMdBENI05hLSOkE+IynVdjOMiyQQcjFC+b5Iff/HfKK2ttWx5JpNhz7nnMnbxk1E/YOJ7P+CeO++qiwpV4TFn7uT8C07HK1fxiMhhyYola5SsgCca93s3kDXgGcGT+JFNXq+FOzcXjtIkazjVb6tXoXwP5BIR+XRYtS8ZHR67ZHR47E8E+ZiI5FT5UN9oX13MuU5kqrV1JbeFxCU9fvy4PX78OKh8DzCCvMba6DpxZGZ8fNyAfgmYM0be05Etv6C8Wt3n4P6OEXkPcNAIN8Wuh1gE+koW1UxJ4fuCXCgqb9VA9p8YtvsReRPCHoVvI5Iwp1oj9rCq/ZqIvA6VP+lwO/aVViu/B7wtLh1Qg3KPxOHNN2ho9gbVaL8oHwa9MHGD9hiRO4DviMhbJGde540u7suvlP9URF5buys4rYIsK2U9oKp/KyI7JZYC/NfC8NglheGxlyDmY4L8nqp+zxHnn4EJgVVU3hD59pxKKdhHGL5XRF6gSgjs0SgeMqkwAvISzwlGUL1MYCfohxxxLyyvVi4Waz8MjIN+YnZ2sqiITSKftd/Lajxq6QvADCLvjobtJUul5QvEmner6jmqfE7gFlXdA/qb1bK/u1oKngXyN6iOi2qfKju39m11H2mE9asLuj+g19hgap80EVmTNSUmyfDJpms2bkxUBsGpWVlOEteSxMoCsrkcU9f/mFu3j/HE51+Ka5z6Srq6u3nCy17Ml//ju+jyKrde/m3OOOccerq7AejMd/CkfSN84ytfIDAOEdW6Zbj54Ne2IHikLEz1srU/2kQ4aprCLpvCTs1NVQpD294oou+P5QLmUoQQ1RxwNM4SRZ9fWloysaGhIdIaMxEk1PiiDVEson7satW8Ju5Q1VtAnqnwr9PTk6vJW3eMDY+/HeEDRuQLYmTVWu0BrlflL6ZmJ+ZGR8b2aNw8zD92bBHAHx3OfFyVJwryZzjyx7EoUl1V/aKK+Wdj7QUIFYVwenZqtTA0/h417ETkfSKyCniKfh50GRWL6JUg/wb8uXHk9arkBbnSou8U5QMi5sOgLwyFtzsqn0PkI4IsK+qp6qdE5DmsjyMxuTJpx/JjH1D0pCBvRuRDCJUkrlQBvhypfc/U9MRyYev4dTjycUT/q2AuFdRFuQOr70bk3UbMB63qUUG/raq/B/I3oK6K/STIgCBvBbk0iRn4qvougS8mxljzb+ZrXJZjp2Yn7yiMjL8d+ICIfEnEVFQ1J8LfG8PnFenDcqmIfMCIWVTVnMKXgfch8kGBD3qe92JgkUdQTOsRxKBNI+nbhlCsG4TaZDWB1FsUN3pcrc+CiQiOm4mJSgTHdeIRXU7cx71ujYlgbUTOcbnzy19n/LHnML59e9yCJrHaxnfs4LQnn8+RK7/PsR/8iGOvOMJjzjobx3FQhd/57ZfynGc/DavrdqJF2d4cJ2h2fKMowhjDpz/9KW7M3rCBcLS2vvsVjlp37vghGSy8MjBmt4rdK4qrwgJibynMFFZv5mYLMDg4eIfjZM/zItPchcGaTOY634/OBL/o+2qz2Y7HepHWSMlOzk6WhodHX+2o05XNZ1p0O9tmJ751dGTkOqPOuVbJo7KYq5g7zls65n8JCKPMjSLhE631V2HBAkzPsrh7YOAVa15+R3K3N6geNnNTx6bADoyffl3WDx9jJSgCTM1NzAwObn+ha6I9KlpAOYZjD1t1eyJrfOazy33jpdeu+vo3KtoHMuXOHj8w2jnEVKf7vUiky1o5PDg34S8MbXuGI5xlYTCKOCiBTInHB8TV1ZmpmXUX7eTs5Op58LcTw9s+5aD7iTVbq2rkDgmrE2Z+HsCuzU8sPwfe9oPh8c9Y7JCILmRscMduEXuX8a4Ua0fCwD8wvzhfKQwWHmMdyasfTs2enC2dBf+/4lDhUyqcJSKhYA794ezE3GW1eJnqK1GXublsaXzY/7sI+ZTJ6CJgt89MfPPI8GnXOATnguQUJi6amTx4OZjtUCwWChdpKOci5DEcNEFl4oyFBXvP8NiVKlLp6MouMvfICsC7Dz8tbWZXbay5ag/QN9lYLa5fre9VzfJavxJpWFgi8bOJB6PWyE4aYbJ4lP3KKjdfcSW9r3olvT29LWs864XP5ch/fBuZOsGB227j9J276OrqwhhDX/8A1/34Z0zPLyUxLGnEglQ5ZcGMQLkaUCwuUPUrtCXAHjQuBpzBHtvR3X1w6Mc/PgDAvn3mYBQZf2WWnauxuZaJIlse7F7os9auLjS5haXlkNHRhd5Kxa4s9lPqXV4chJbP5AcHSiWo7MhkONoUZtg+MMBZo6MVXVu7ofvgHCt7zmY1WGVs6aR5BSt05KJwKd+56K2NW1gwAFkgyp5uTad/9IxD5cP3AOXdHcZxRmH6ZkNpa6gjqwvdYWhXZ2dNFvB0i422+nf13nnnXeETngB3wsmecjHs8ezI/L1mJjNgu/KFW4YPzjC3rccUT3PoXFzk8eNnHOvL5Zi8S+hjwnDaiN9h7a3dN9/MveeeSzCfo9q7vNhnrV3dJHxiu7o4c6hvdY8xV3HbbfZOIHfuuabzXiVknqXk7zp37eIieu8YvfckK3sKlIJVhhbmGN7RvzDW3T3HrT7CsDk8ZOdEBJa6LSdnjfbtg0G/ePpdmesm+7pg27JZmOkwl3CQEmC29heDQEx+rkiPHS0tjVQqA+WynQGzfWCAswoDlS3W3uBMTTE3Pm5yXTnz0nvvpRvMyuBgaehw5oaVVaV8bkjHvRHs6DKF7u6DJ48csZfPrD7isoUPe2mO6P05gbROb94gIC+JSFSk3bpqCgC1xb9EwKkNkTBN5OWYJOAuTSU1cYehXL6DiWt/xLEnPoGzz31cnFGUOAs59pgz6dw1RvWeY9z7gx9x8ukX05WISV3HYWp6hr/6wP8m09VJZJNsoG4QSt+Av8prARc/+XSy/ix+EGzQD+t+SC/Bn+8+N280uoBA4WSJYPfZRICuBuwAdKQXpbexxhAMDrr77PraI4AqOJJj65YyajOxPm332Q0LtpoYgNWAncnrNWxd8WNLcNcYHbbIiAO6azu7knDYWAiaW0F3nx0X8UqSTFAo784ynmRw6awgu89GdA18wZJhZNfZiUzPj/9k99noyRKMQq+AUkV3jbNTAFtEd+XYKj5DdKFDXbBcRZarbNsCMng2nCwhCuXdZzNWitD8GhI6KA7Du85e1zWkdoOzZUsZi+w6m90CthRhRkDlbLbUkz/x9phdeTptERywQwPkyxbKS8hovM5dvsTXSP2YBIgv6O6QbVpEyqC7M3RpvD1ajbt76O5eDCW6SyCSZdfus1GF4RU/zkP3DDG04qOSoyc5bt2lCIYj+kdgoAQ6AiKdbClbs33k9EOXH/rpsUc3YYm0TIpoURu19HGXTZ3GikYxKSWt1I1o4gYqxhBPpwnCetC6OeNYrVSpoogqxkY4xD+2GINGlozjkHEcbBSP5lK1SHGZO3/wQ07buYve3t56TKmrr48znnMxP7zzI5z84W0cvvcQo4VCPP7LdbjogsdSXVZMD1STPld117apQ2r7fdsgiFsjz/vr1hBseqj/GqhImBd4bm0dYdPFY1Xr4gkRTb5PiGQjzVfb600/nkry/qZ6sdYusVZab0y22TUWjd1okU27dDS3i7HNXra0i5VqX1/f0IZX3tT0X6UpQ6uN9WvTEPJY2aGNDzbVsDaLm5vVH1Y20Bpqs0C39TiptIYHRFszxu0SRW3qtlv7He06GWPr8azvY/M9r/0YxrtjQO1fw7G/ToPurYRUtzlkownOTZOcRQhUmejMstbT1bThiaTBxNZWb38/27aNr/u+1Sjk1pWTrEV+XadVyy5aaxnfcRo7cXHLPmrjdsY2jMgaYeqndzEzO0NvTQWfbFnh7HMQrxONqhy75x6e8MQn0ZHvQK1SGNlKtq8bCSKqahPLYYMeOdruuCrYRnvlU3fw809JWH/QnVvYUi6/ywJVtcaGIVnXIQNWRYisxYjgGkMAVAPfeK6Litjw12D6hUaRsaobmOnrTff4OAhqjHWMwaoSPcTHwCTWuQ1DYzcgWdd1rT6IdQVhaDZK3hhjcB3H/rzbnwFMybF//Qj8TR82wjKbBLKkLTje/qGGODRuoZEd6OcN7/y/GR8b2zCD1pnvZHRkpMUlFBG2jI7yO//PO1leXllnrKi1nLHjdA7905dZuOfeuGg5ioiiCMcYykcnOX70KGfsPqM+4RlVtu48nVzew10qMXvvUdZKa/WJO/19PTzutK3cPTWFiCb1xUkVTvK9LTf/FnNT6r3hN48ECtAJrGx6zD9xyy384XnnWdcYJldWbH8Ymrzn2azv89cLCwbHccnl7BcvvNC/7tAhU6pUbNV1MV1dhPahDV+0WLsP1/LJk7ZqrdlwDEDbctYY2+O6LGUyTJbLnDM0xHKl8pBuTz6TIaxWqRSLNlQ1zdtgRejq77dhcrO6v3V2uC4nFxasG0WmfV+cXM5u6e//hba/d+G+R6Ta/VfecXRjsmoKq4u09NHKZ3Ps3bOH7du2n8KAW/8D9HR3c/7+8zb9m8rKKoc0jl1ZG7c4tmGIjSKkVOHIgZ/hP+3pZLPZ+vZ09HTTubWXk0tlFo4d5+TJkwxuGQQgl8syXtjCHZNTGCN18tGmnbVNRNW8XJtpqPb+dFgr93vEP37zzbavr890ZDsLIvIsyktPFZE9hVznEIhHqOFbr/3hoiqHQX8oa/6V0Xzx0MyJmf/Ugw7OPvtsc3Kt+iKFHjnFuQdiFfUFWYZqEUpzRpi6/ND1/gPRjjxYFIYKI2p4ljTp3TQWOy/LsdnvTM4eW30g6xkbGfMUfSbKUNvJb6iEd00dOHAjv4Z42AjLnoK49JT3v4bjVytI1l+Su1IbK2+Tnuy1h7ERJ45PUKlUYsKqHbyMR8/oDuTwDKXFRYrFYsv6+no7cPwl+rOgzYNfN9z/xl77EuGXA2J1ufPzxNvrGBkqdBkjvwPyRwJ7FfJaTz3WWtvoTuACkN/E8GYjzudGto5+fGZ+euY/6XltyuWyQeQ9BnbX4p+ySV4nKY0JiYWbq1b1QGF4/F8UvXJ6dvKhbLFiVNgrmI8iuG0/4yGr4S1A6YGQpLXSZYy8VYWn0BZAUdW/B1LCegiDWPfz2npxaI2woqhVTLlRLLhmWbWMENu0tWmj2yeq2MgSRRE2cQmjMEIiy9L0LKVSib7ehrzBOIaO8VGILMHKKssrK3XLTlV56lN241ePJEXSjZKc5nFgthavqvfPUsIwZHh0iKmZIsY8gEN3qivEmHcB/xUl12SvmVa7tn6qu6rsBHmncZzHj4yMvHZmZmb1P+OJnZwjRhVPaZTWNHveti28lVhjIyA7QZ8lIt8qjIy/eWpmYuIhdjTc2sCKpvuP92CmI8UttHFJGt+3BPMVw68p3F81UWmb6SDNBFPvJSOIY6BUYd/eMxkdHiHy/XUB/Iai3W2cmE3uYdwFQTc6uxEEk8kQhWFiWcVjuaIoBGtZKy6xVlprqW8UY8gPboG1ANtVoVRaSwgpllH8xm88h3Mf90TCKGqtiayV3KiiWGzU3PBPCYKQmflZrrj8BJ5j4itrw46j3Zu6hYUtBRfHvB7480TMZdppOlmL1Xp/ijqRuai8yOC+d2jr+F/MzU9U/pMSFqBG4ok/jbihbHw2Nt5TA3goLwLGR0fHnzs9PVF8iNxD03S61467FeIM8YO6lARTIyfZzJlJCeuXFcuSpgC0tATca5ZWLpenN4R7vndVnYxqswdrRdD3HT7MrmddzM5du1tO3urKCiePHEODoMmuSDqAquJ4GXq3jXH0+hvjGYOJhWXDCGst1Wq1bik1W23Zzk7ICDYMqVarca8oazHGcOU1t/CvX70GJWhqGdMSW28Kakn9Nd8PqEiZjuocfhj9XHMJcWUI4Q/ayUobscFVAV8VT9CuNovTCGoR+W1X+MLG7sUekpbqbXfz7SQF/6e4cHYSdxdujkMe2/zifiguRNnwntk62UfWfbdF2C+qv7916/Dfzs/PbpCW7TTQQaO3YQ1dFvqB4w/o3q2IacsIn3LfVThl68uNsdlvBudxnr2P+1hYtx+PYsKSB/Bq7Pq1OIIxKRG3PJ784c3MXfsTHONishmcrIvrZXA8Fyfjcri6Rml0kF27z2hZb3mxyB1f/AqVlRXUxiJOa6NYcxVGRH5Ada1EFFmiqDb0NCKyiXtobVs755joomoVyXuoQhhF9bbIqFBcLfPtH96JSAkb2paBGS0RFW297QfVkIsuOI1S1ScyziaOr3CKshyTnJ1ntR5nrU0/OyjoOxVZJO5b9XbQ8xPaNNJwVIYwXLwBYdWLGQuFgmdC06WqOSugJixF0dbV+fl5n82LHs0pwpz1vxkeHjaZTMaIiH/8+PGHfkK1SLO1YzXpytAkujEoVkRe6Rrn/wAzbdtpx7f05yIv6pKw4EncILvkZJ3lidgo3WD/zboYWvMNcGhoyMzNzTE+Pg4lvBP9Wb98770NAm22Bh+UQdDYlvHx03K+b3tcE7kSSSU04TKz+AGBeUhuDr8uhNVwyNoLcrTtBthEViYpm0mU7FYtUc5BHXBcRRwlkniuH2rBMbELRmuWECAMqlRLJTSM6i6YDUOiMCQMIsIwSDKEYWxh2djCCqIIdQwZr7XnkFolrFTAMVixGOMkWcbYwir7EXRmsKFD4DrrhaO10FmNoBNXEhNrpJxNkwsPyOLaqUpXoxmbJrpN8UE/4IfVfyuXy3R3d4OaWx0x/4KyU9rqNlX1sdu2bXOPHz9ezxqODo/tF9YugbGLiGS3NdqH4gmCqFSMk10ojIwfAL1W4MrJmcawhJGt4yMild8WxnJx59VaMHrsdozzLVXbZ1R/G5Hngu6wIQZ0oTAydj1WP1FcLR79RcdQiciURd8s6JTE95YuYLfAX7SZfTXsBrMTmBkbGstZ0fOB5xsxF1rVcbFOH0Y9RazglGzITGE4d0Cx34exby4WF6aq1WpLzytpHumUBJ6sBUcyTyoMj7/WBlyApz39a5Xl/uGxOxA+oWpvmJ6d9mMfUmOT8AHmn0aHCjuEtUsw4y+2YbTbFQZQcdWRikNmYXpk9qa8dn+9w3ZdOTM/ufxIJa2HvTTnVGQm7YMn6r2uamU4EivBG42U4l7kxEI6U28vs56slJhgsBEaRYl1FdXjVZEf1GUM8cMm7mBEYEM0kyGXzbXER2wU4q+soo4hFPA8r+5OGmNYKvtE0uT5Nd/92pKi2lazpMnQCt1Q9V/74xybqd0VXGlSytdjVKoe8KqM412XyXsTQehXfN8/1tnR/V9QutqFFCIsF4tFC9jRoUKfGPl9VN4KDIJ6cZMvSWJiCkhXUgR8FiKXAm8pjIy/K4zCL87Nz4RitGDEeZuqDtU2TASD6ufRaE6U94nIUzTuKGGbwuQXipHn9HYNvE1ErllbW/t5LygL+KLcOjU7ebT2Yn9///c6vA7A/A2qubhXuqCoQbVLYHx0ZLxHVT8gYl6E6qC29smq/S5dxPt2roi8COUPtvRufV+g0X9EUdU2ui+vi6cahJcYkb+oH5vG2/sEeY5iPjy6dfRvp+enK8Y+sPvWUP+Q53qZl4O8A9gTd+oQWy/AV/KgA8BeI/IideSG0ZHxd1T96h2Li/Pho5awxEpr7KbWX0mkXhcoTeXNzf2s4iLluOav1uLYOA7GjbWPtUZ8xnHi79GNpvEkQW5by8g1MoJRnaSipgxhGFtYVinbCK+vm858Z50IVS1REFI6sUgQRUR5j+6urthqS+QRKyur9GcgUlMvXdF2sbvWAuDJRS9CxVpcQ22O58/pfmsRTChNGbIaaQk8S8R8Q41elSH7XS/nXZ+J3AP3Ldxn97Of5eFltzpbtccb8Rc7NjZmNJI/Bt4Ncbu5RqBerdRdlrhHeaKc8GKLRT7iGrcEfCVJuxtQt6k8xSSWzWXAvlofMInvSU2szj4R+UBvV/8r1tbWJh6MFdDedDo5P+p/f/LkSZMb7jhghJJCTpOruTanVOIM4rsR+f16AiUxcgS1qhgBizQEoUk7l/0i8iFXnYWhodHrFxeKtoXe6v3GGAfenxC1aY4UJH30B0XknRhTAv6Xmvrr60MMwK5du0xlqWLUkd8U4TJV7Wo6lk1iWm3URKJdAs8yIp/LZrK/C9zySLO0HhbCKhaLLJeW/Uq5VGkErrV+sLQ2raYpkN4eUK91Vogb8NXaG7s4btyQz3Hjh7Fm81O2JiOIIrQlVhVnA8Oa9iqKiJJ6wshGrKoytnsXXV2djYyjQmV1hfLaKuVqBR0bZMvAAGEQxC5mGOKtHGKMSSI3RK1B1cQNkhWslUTeECcBtekRRSGZbAHP5JMBsOtM0eSqq2x2MlmF240wobqhixNrr0R2C/w+UArc6I7C8Ph1M8x+VZDbJ5koNt/9NWQfwpsTC601DSBSAqYkni5cQNVLCKH2Y/SAvn10ePSqpjxDeyrh4oRVbDIvybZ+j5i4KR/7UX0NjP6/8evTD4i+pS0frdaaXbt2mXvvDejomGdsbIy1ldI+oKst+WxFJET1CcDLE4aIu57G67QisirCFJBXKIiqS41wY0LZbuDtd95554sLw+NtN5C6GCHXZAHaZBta21qr5hB5T2GkcCUw05IfF2ncDQVz77332sJw4WmIfMTGFqJp+z7bNG+9tj81ojxLhI8URsZfNjUzMcOjrR9WkMlY349C13FKzd0U1CqRtagj9S4IDTcwjlkZMXULy6lZVslzzbpykodxnEbpzLr0dpy9U2tjy6oWVA9rqvYIG8Uj5qPEZVRrCaylkjHsOvNMHMeNp0Ynq1xdOEF5rUSpVKZndJS+3l6CJP4lCG94/et5yYteTBAGRInVFTU/bKL5qllliayhXKlQ9Stc8e0r0MBvv/Z84iJCGxswmxVA6zFV/Q/gTzZyJjW+AGqncA7V84HzReR1qnprYXj8H23EV2YWTlaS28sFojJQd93rF4dMgb5dLddbUdeIvERE3qOq+YYVpgYYF8z4A7LFhVVVck0XcSN7qWoQeSqc+f+Fn/28soKciFxcWq3sHR0GYaCntFreLcgbk0Z27YbtcqzNom+D8MYhq/pmlIMikkN4LchfanMr73gs2VnbtmzLR6pm84lR8V1I4gaF+aTRYvPnDNCFyu8KfEg3DLfEv25hZLwL9M2oDrS2UVaAInAVosdQ9gMXkAy9qFtdKvtAfwf4n486C+vwxARrlVLouu5qPIVBTNLlFsIImzMN66rJHTR1K8tJrCsXx9QsLBfj1gZKmJbeVu2ZF016p6tqS4wqCqMWErFhw7KyUZztK9mQymAf55x9dtN6LIiwODnJ6okTlFxh97YxOjo6CMMQtcpqqcLlV93O8tJyS88FYX13QaEReBfgxNIqyyuT+DaKb22tJBxSr3refJDq9Ox0qTAy/n7gmSh7aye8bGBxtMgeVAdBninCk8To+Mhg79/NLDg+2IWky6VpWHlqQG82Kl/x5rzKYQ7bwsjYfwBvAXKS3LnVWsQYDyWPqt/omtA85RsEVkE/bq1+W0T2I/LWZJBpc0cPCzpeKBx0p6amH3D5kLa0hNERRD4aT9sREFxiq8hq/ejUydwITCn6XZAZmgdGCEbQb+e7Or5z6NAhADs6Mv51gb9sv0EAfdaxedqaNtaHpsRhkhlF36XKYUT2CfJu0J4WlzYm7ItV9dPUzu36WaUNz0XZi8iFoKbNHbbA/zLI+ydmJiqFobECIpeBviB2feu/qyciL9u9e/f/TPbt0UNY13/zm/aVL3xu2JHLLYuITcxbxBh83yfokPqJW+9tZeLBDrHrZxquXybuFGrchhvY3OoYIxuqAJSErBKrJkrcwCjJEkZRSGhrMay4ntCqshBU2XX+49l5+unYZIqzWksUBMzfe5jl6TnWero4/Yw9OMbgBwFGhImpBf7XF65k4t4jrcNjLfW+YLUYljT3NhElDC2vfMXZrK6uYsIobn9j68m+EkIFIcQ99cU6NTMxNTYy9jwVeV88M5CBRHNhkotzvcIrDqwYVfLGyHvVyCoc//j0LF8G/q2ZIAYHBz3PzfZZdLwyVB0oyNg5qvKGJB5TZ4om8W5L0woxpmUeI/BJHH379JRPYUvHd3Bsr4j8eZ08pBaslh5wc5CMu7rfyFVtG1rmBrj19xovG6nLTqi7TVb1SyrR/5iZ6QnhYH3/RwZPyxk36iutlPcUhscHUd2P6htbYmW12CQYdTXOzrQQiCLxTXwVq29zs+afOzo6rO/7V/nl0AN5j6p1RaRZcrBDYUC0trb2m4/4IGfVLMKWjjwiR621n+nd2ucPzQ5xaO7QTNdwz0cReZEk2et6Ub7qvrXl8jjwoOKFvxZB96gahR2eV6ydBIriZjJxYLsjC0trid5KYrISp+4Kxt1BnfpzPWblNILt9UZ8baneZpdQrW3UCNYtqpo7aBvWl8auWUVDTnbnecFFF9UD7qqKCqzMzTF/7xFWKiWc0UF2nHYaURQRBAGu43B08gRGqzi92cTKqxklmygTEgIzAlFo8VyTSCdscheuqz0qoD6ofSCn0OTM5LHRkfE3gZ4ryisQngYyDvRJnOWyaJNostHyydQGoY6OjH9lemZirr9/0HR42RHgAkSeAHoWyk6gR4zkVOkTGm5Mi2vV3Oq63kxKaY3b6A/X1taAjJ06MWRHh2d+atqOWXJNuY7/QMpPhFMU3D+Qv7cKtyN8PJ/vDof6S67rjW9XdL/AEyA6B2UckR6J41ddrW5sW1a4+UC0+52qiwo3zc6usGNHB+Vy2Rrca0SkCDLYtsoegcFTkEio6C5RvHZ3WGFORVbvvPPO+t/m6T5qFB9waxOCEmsrJ8I4m6h6f60Ja6m05Gdcd1ZVfRE8TdyKUqVC4GXiIRLSpLky0iCppqB6LV4Vu4Sx5WWMU+8a2t4hueYSqmri+oV1SUOYuIS2ZlklMS61lshaTmrIlsc+lgue8IR6zM3aCMd1WTh6HwvHjrNULjN41l6Gh4bx/YAwCIkiy5GJhbhURyFsbti3WehCGxZJZ8bBceKGg6ENUWOa9ym2sBBL8VSU1WXiaydPcckpFQrmBlW9oVKqDKK6V5Fni8iTUN2H1Ob6SYsmLraEZbtg9w4MDCxmM9nfFpE3q3IudQsluRC0pvOq5+0bqg1Nin9sk2ylJUYMxMMuFpeWkpmQLACFVa1bKNJy83mwxXJNLl4z/9lN41yxVXgTqm+enlldKGwl73r8noi8CZUddUlH4iJbxSaGpG1kDxtub8tA4OYbQyMeuCyiy76/k4MHbwEwhZFCEaWy7j4n6sZ1j41jIm17KyI9tbKd1huHlpqqH2zyI5fAVDTOErY2DNTmKohHEWEduedY6HnehEAFkS6ASC2LS0uEWQ8rhoxJtFa1YHs9uN46nsu4Lk4mJi8xySAJMS1hzo3Ml7ogNIjdwFqAPWrWXmn8WNKQ+zoyvP33fpfhoaEkKB4XXpeKS9z3k9uYuP1u5vJZfvvpT6cjl8X3q0SRZW21ysT0AstrpZhstO2M01N7Mf15DyOWSrlE0NqXygJFrFQwsilZ7dixw1SrUR/WemiAEFJaU0QNoibM+O41UXd0zfLyMh0dXSOivEZE3oJqYQNTpEstQ7lM/o8R3h8Hw9v1R7WsllSACVS3N1sa7RUCNJdeNZFH60BREGkRw9zvobs/W6tt2UeaCUtqU5aLqnpYVP8Jhy9OTk2uFgYLLob3IvKnVtsIvbGtBmFZYAp0b+N1cyorr0XnEgdFbmmRAqmsu4mANmcSZQN9o9Q+07Te+mdco+vSHEZNQ+ogzYE/eXQOobB3HTlgz95z1pzjuquIDNZ8JPF9qq5BXQexGltXTUTl1KY1t8StmtzERAnf0gZ2wyyhxuUxYUPJXtNLRWEjbqVqKduImQw87pLn8IT9++sZxiiKEBFOHDnK4et/xInqGrlz9vCYPXsIwhDf93Ecw5GJeSampgiDiEzWxL2u6mMoalN92hvNSP2O3NudQ9US+j6OGKLGLTIkHrtUQdeNUK+jUqm4os57BZ6GJM0ttV6ruVjNBm/KLXgHl8pLtqenZ65Uqv5t1s3kgPe035El7k7ag/BHxJm79ru2TzzX73uq+l2BVRH5J1Xdvt4ltK09jk9hcNYuGN3oBqQ/D121zMKcUdW/QJlrDoGr6Kq1zBmRBX/FX5wvzcdvObIXkZeguj7mJ1JCuR34BqrXW8EI8m1aK1bXhdo3QI/CQC6XmxsfH2dlZcUqDIqQr3kjTSZpiGoxnruoG7u7qrPU4sWt+96lRr3mc0eNDoDkNrpuBV18VFpYN/3gpvDJ5z+pmPW8hXJJtiNiavqq1TAk6vAwa35Db9XmDhonHnhanydoTFy2UxsgUdfFaNvvI3XdVGxR2Ua8KnEJY1V5/OxHEbMSkTnrMbzmNa8mm8slbmO80srSEhM/vYMjN95AsSvPuU+5kJ7ubnzfp1r1UbV841tXcNXVV+H7it/UE2FdLKdxF4tPHyeOePaeNUIQBKysrCYuRf3zFWABkQom2DRDNjMz4xeGx0OEszY4oUOBNy3nl99CGXv8+HE7PDycBxluLZyqZ68qxLKHPdS7ObTgU1h939Tc5MTu3btNabV8jsZ6i5qYtJFUtJJkPLXlZ5JNwnr1Ibva1jD7QYtpWxT/VuLjeE2z0n0ji7GJYnYnMaP2bfZR/Z+IfnhqZmphz549ZmV57Wmn5tYNqcsAQ6o8befOnXf5vm86OjrwK+HzNAmcN7t+ghQ1qbbeMCSq6qrIIYOUlIZgNNn3cZCRsbGxuSiKbBRFIHIuiKuqzcfZghRVOEZbHeOjgrAIgjCyUsx63iFEzo+b8QniOEzMzfOYvq3I2ol6kL0hCI2JqmZdGcdBnEYAvm5d0RBgbnyz1sSSSrKDzXKGKAmy24g5iVjeczr/15+9hTN2nxH330qsK4Mwe+Agt3796yxYi7N3Dxc/9WkEvk/V94miiM6uLv7wdb/Jiy95CkEQ1PVWNZfSJllGmxRPN4YCxBu+tLJC1nO54aYfs1I8Ca4DkW3Er1RnMVSoin9qrZF+F+QPAS8RXdYsDQ/4k2ym48LCyPhdKCHCPuCsFhO1ofJeRJlAxG34tsnNIZ5m8a9O1sz0nnGGq2VrVHmBoEMt7lKL42dbJIwPzD5qm+L4c/iF67W3wgO8CA0qfXGpToui3IpIUVX/FUNx79697nJx1RPlZc1W1anc2Lb9z4uR9xcXlowKB1CeJMKf1a7RtrL7QwJFaY0j1rdLRVxBbleYAvZIa6xxCHg3Ie80ahZEnH0o72y4nQ3JiwhXTU1PPqLaNzyctYThifn5kus4B2PXRlwRIZvPs3RigXDHTnT6RGJBmZZx8nXyqsWxkjIdSSQMsbXcJBVn4yyhjcKG+1cPuMfW1VoUMmMsJ8dHefOfvYV9555bF49GSTHz8sQUh679Iffe8BPmh7fwjKc/jb7eXqrVKn4Q0NGRY2ZujbvvmWRpZbU1lV6bTViXem8woVqEe+5bY/eYx+LiSYIgaPZoLMIiIotYfIwJT3HBWZQbgeuBZ27onqiei8i+hDFtI9CsDbkFWFSvE9GD8RCwuGyzUYemIPLOMLAjeb9UKcNFgvxOLca1zvkR2zKGbcPROG0uIRvNmZRf7EQUHlTXWiuGRSBUbQzpTYS3g8C7NZIvLJ9ccYFnAL+9Ycxso6G67fuv2ofI3xhhVWP9lbd+2wVV/Qboom56IARBj6rI19AG6TV9+6WK7EMoxg0LGWyzCW2cneRzPMLwsBY/H5ucrBjXuVNEimJkwBhjrMDi8jKrGYeBXK5JptAcZHdwXTeZ1FzLCjYGqdbH/SVyhHYSqHf3DGs6rKROUC2hjVi2IUckwNt7Jm/6kz/moic+KTEGYssIEfzVNY7ddhs3/vuXOdHh0nfu2Zz/+McTRRG+7xOpBRxuvPVu/vHTX2S+HGw4GeeUjktg2b1nmKGe7czNzBJY2xw5tcAcyCKojxOdsjC1uHpyrq+7/32CPElp1KdJczRYtTV4rC2ZNIvIBKqXRURHHcxRkB01MmoEe/ViUZ4CYhMB5oZ0JICoadF2KHpq/mka0dYm5nzQaM95iDzgtRi1HBZhWYSetvUYgZcIvCB5pSWe3ZLdk9pxXu8StsW4PNU4a9taS6iIiFWYEeGzNnbt2XAokKqdnBsOR4dmPyYizwE9R8E0Behd0HFivVx73WJN2P01Ub3ykeQO/jzZ4V8o8L64sOB3dXYddYxz1HEc4zgOgkEUZlaW0cG+eNCp03AHHdfB3UhvJaZJ9piIOa22NMprJS9tCEWjuGVMMQq4N6pwoCfHY1/7Kt73wQ/w9Isuwiatkq2NS5ajcpmZu+/m2x/8H0wsrzI1OsRvveIVdHV2Uq1UKFereBmPucUSP775Lmb9CM1msDk3fnRksB1u8si0/Tt+aD5D1JXhjNO3YKOAiePH24RblIBDYJcx+GTDU55EpVLJTs1Ofk/RNyByLDnprNaiaRtbGLV1WoMcw+orp2bLN83O9lYUPipSKwnCtujbYqur1uZ1IibWFu/SAq6iOdV6Sxnbbju1z7hSFdv4e603HXogNwDXdWnPMMrPF7m3flS+A+GfRSQUbYwcbOmbhZqkxP+QwGrzZxILy5NQXGl3dVWtQKjwH4g094830pCsWRGxiCyAvmVyZnJGFCO6LtNoGxt2i5memzysIq9A5IBAmNxgmvt51Sxhi9QlKQCfVLVvnZydrMCjeFT99Oyx8KzH7is6xtwqRs43EuuovFyWuycm2b//fKITh3Cz2Th2VasRdJP6QWPq5Tr1EHtTBwZN4k3tLqECkbUsnSxSjAKW/DKLYlnpyrHtvAt4+aXP46kXXkgu1xHXNmoclEcEDXzmDx7iqg9+hOPT88wPDfDMFz6fwvAwVd/Hr1aT0I/D0Yk5rrn5DkIgCKMH90MY6MvmKAx2cfLkHCcWFlC3pS9yEeEIkZQwkc8aD6j1hx9FX84a96gKb0bkmQJ9rTVqTWQVyxIWUL3SCpdNz04mOfZFkLFPqcrWpFvBYOuFJxZhGdVrQD4A+rsi8pI63SZ3bjHyMqv6CSMsqorbbIKJ4CPa4uYKagUz1+xBJ7nUZZuxwOnAETbJlAKyKqILmrTnlHoAX4qNQRz3jxMnTtjCcOFDivSJkZerak+LhRpnYhdQ/SaiHxKcD4Ge22rZCUr0IuAOESkq4tYG6Sak8yXgcwjvR6VAkwBXIBSRQ6r6fsV+LeEoK0aKAnPtbcJVqRPN9Mzxg4XhwisQeTPIC0QZBHLN5l9C6SWEYyifw9q/n56bWuYRiIeVsJZni35wZriayXg/imz0+47j4jouXjZHZXWNpXyWLi9DPolduc1lN8aJNVemabhEbRSWtWhk0STmVCeyJitg1Ybc1qEsovSMjLPjcefwrN/4Dc456yx6enrqrmJNYGqMQcOQxSP3cfVHP86dN97I8aE+TnvqhTz1widTDQKqfhW/6rNlYIDiSsC3rr2Z5bUK0um11+ptEm5tOAU2tJxxxgD9PR3cfNO9CdG23DnngAmMqWDx8YMHdMwXFmZC4MaRkZE3GHXOBZ4k8DgVGZH497cKFVWdQO1tInKDWntwen66Jag/PTO5XBguvEctl2PMSyVu9pcDVlX1kKpeLcp1y6WlUndn75Ra+zkRsSoNuQaKL3BQ0TeAuM2Gjo354642j/AGVF+WxOTqbqSiYbbqleDwpjG8iYkJWxgeexOQqyUT6o6oxVeRBzURaGp2amp0YPQtmjHfQHmGoDsQ8YBlVA8Al4dhcOvciblKYWT8Laj2NdK/icDTsIgwo6qvaIyYTpjF6uHp+cmZwkjhdhW5FJUnGrRHYRn0J1i+HFT8w/PL8/FQa2OWLfouUelppsVEPDPVtu0HRraOvMUY5zKFi0V4HMigCJ5CJf68Xoty45RER5mdecR2HJWH+fvM2O7dfTjO9uWV5a96mcz2jJvBy+UQx2XnrtO5JN9Pz9xJvGwWN+Mm+qvEJay5gtLcRC+ql9yEYcg9pRUe94bf4eUve3ldiqCqlMolZmdnAejr7aOnuwvEJMr32KqKkmyh67iEpRInjhzha29/F4d+dpBjW3vZevFTeNVLX0q+o4NKtUKlWkWtUlzx+erXr+Z7195OiDYl9TYIZmzyMwSZLG9+9UV4js9nPvW/WVhdbgTdY3fwK8BnEL0LN1xgKXqwgyHMqVL3nXSaAQY4znF7irhFso7twDHb9rp9pMU7fkkhlA32c7tpql6xv/h31Hret7Zjfii2fZBBU6bMGmv/KX+/h3sIhV0+ebLSPTi4aOB6182MuxnPuJkMWS/L/PwClSfuJrdQpMNttJFpEFWSxVKpd01Qq42Ge1GYjJlvjNCqjdXKZbOctn173X2MIovVBllZGxfIOsZQWVzk8I9v4tp/+AcOHzvCxHA/Pefv4xUveiHZrBe3f6lWsWrp6uxiZa3CtkIHL3v+vkaBNI1BqG3xHmq5uFp0per7iJdhx7atXH/9D5hfPIE6pjkcuwj8DHSZiArl6OfpBGkfxPv2Qa7DPkQX6yMdm+2n/SV8x0O9fvsA1v+I//0e9qk5pcpK2FHt9K3Vb7sZ9xLXy/RlMhkyXgZjDEdLq5w9PIAuler9reri0Fq6pVaEbDVpd6xNLY1t0gQvyfCREFvT39SKm2tkJUnPLfyAtcVFfvzPX+LGf/o/nIiUI115tl6wn5f/5kvp7MjHEgY/wA98BrcM0t29hWrYwdOevhWbyCoaGezmEbEbqbZjZf99xyfJZgIc8bnlppsxroMfhrXUkgWOgh4GWcUYH/vQn1hryX8PMN9Lil/m8Tj8S9vSR/pUnEccYUVrfriWL1eiKLrFGOdWz/OelslkjOdl6MjlODY9zWl7H0Nu9Rg5160XQ0uT6pn6SPlYI1VvEdOkXK834UvEmbaZtGrpexGMCrbqszw7xz0/vIGbvvAFjh84yFJvL5PdWfa98Hk86ylPIetlKZXL+IGP7wcMbtlCPt/HT+44yocv+zRR4NXzL6dqytBoiBy/qdYyevpWXv68x3HjDTcxPTOTBKrrf7kKekvcKM+sIiX/UWDJpEjxyCAswK4tLlbyXV3F0Pe/nentfVLW83JexsN1HVxjOLZcpLuwlWBuiVwuG1/YSfpDk57CmtQBaq3NcdMMQdvU3rgWhK+ND0NBw5CoGuCvrHByepqJn/6UOy+/nOm772ZVXKb7+6huG+M3nv1MnvD4xwNCOSGrIAjp6e6it2eAg4dn+OhlX2Di5Bp++ABDSk3znYzEruqLHrsNG4V85zvfIbSWoNW6mgL5KUoRMausEqanbYqUsB5WMysKMaa0dvLkDYNDWw94WW+fl3Hx3AxZz+Pk0jLLZ5yBd2KJLPE4kXrDg9o8Qa21hbFNva3CutUVhkF94rICUaXKysQkUbXK0sICC/cd59gNNzB3550sn1imknMoZvPc5wg7nvxEXvobz2B8ZJQgDKj6PmEYUvV9erq72bp1mIVilcs+9mUOn1hkxbi1nNfGaO/ZQSyy0WrIc558JmfuGuVb37yc2ZlZIkfarCtuj30OXSYKH3G6mBQpfv0JC8KgurpqneyxleLS57Zs2XKOl3HdjOvGVpbrcs/kcbrP2E7x4DH6cx1xDVpd0GmJwjhmFdZ6syd1gbXhEkEQEIZRXR6wND3Dt/7yXRTni5QnJomoYjv6KffkmO/McrKzg97HnMlvPvUpnLVnD6CUymWCMCAIQ4IwZLC/n+HhMdbKlv/+7r/np7NT2IyLtM0Z3HA54aq6oFCVx5wxzvMuegxzMzNcddX3MTkP3/ebA6GHgatBFxFdRqqpO5giJaxfBYJSELp9ncuzx4/fOL5t7PbOjo59rusY13FwjIFIOVpaZcfwAN7sSTo9L4lbNbos1IZH1BTstf7sYdLqpTGJWSlXKyxOLrDkKKXBXlYFVlxDtb+Xwhm7uXD/49l52nY6OjoIgoAgjAkvTAZIDPT3s2VwlKmZZT78d1/kzpkpTNaNZz1t0Gxps2ULZFQZ6OnlJc84h2zG8NnPfIal1VUqrcm/InFzpKNAEROVHqhYNEWKlLB+CVaWH66VOtzOqenjE/+ypb9/Z8Z1e4xxjJG4bczJkyfJDw0jnVnctSpuMtDA1nqytwXba8XMQeK+hWFYL7GZLZ7ktoEsqwq5zi30jo5w3mPOZNeOHWwd2ILjGIIgpFyuxBZaGNa7LTiOw9Hjc/zNP/47k3f+lAOHK+TyJtFbtVeEret928pYrkOuq5fffMlj2X/W6Xz+85/n+NQUQavwOgQOAD8CFlCzTInUukqREtav8LtttBpUqnm/ePzo0euGRka+efqOHb8VD8sxSbmG4fjsDIyMoMfn2FIKcBIlej1mFbVNvtG4GLlcKhGEYb3xXr67h9f9+dsxInTkcrhubB3F7l5AuRIRRvE8wTAKybgZhrdupburlzCKWCuVePULnoL/nCcm32Prwy1UGwLVenOOZMpEnA1M2t9YZXp2lsLIEGeesZurr7qaf//3r4DnEjVKiiwwA1wBHAJdQFklilLrKkVKWL/i77cBlDo7clP33nPP1wcHB8/q6+vdV2sVU+uWcO/xCRgawk6doHeljGO1aa5g2DK52WLx/YBypRIPNY0sYRSiaunv6SGMIsIoolQuEyaEFyZupKriui5Dg0MM9G8hl+vi0NEZ7jo0kcgiQCUDZNYp11vsqg26pgjgV32ynSOce9ZjuOrqq7js7y/D8TJUAr850F4EbkC4HZUFomiZapBaVylSPBIIi1IpLOMXNYzuuvmHN3zh4uc8a4cxpk81HrJaI5M7JybYPTREaCO6FpZxldY2x4nuyori+1VKpXI9/hQmpBYmjftiKyoJ0qviOA6d+Tw93d309vTT0dFNcbnMv379Gr70L99iqhoHwkNtKmprDk81DUJp8QwT60oE+sjy0ksfz8ueez43334rH/noR4kEyn4LWVWI6+m+D8wguoirJapp7CpFikcCYQGEFq8URP5McenkVTfd/JNt5+3f9xpV7YsiW48l+UHILffey7aREc6gh/zUAl5kGzGsWldPiahUq5TKa/hBbEG5rovjCEgc0M9ms/Gzl8XzsmSzHThulmpgmZw9wVcvv5wrrrmJk8UVvP4u3IyTHKwNmv2fop+kI/Gkne7ubt748qdy4f49/OhHP+Kv/uqvyHZ2slatNJNVLW71deAAqjPgFCmFlfQ0TZHikUNYUCqFgcsq2fzU9LH7Lj/c0z20bcdpLwijKB+GEX4QUkm6et516BCl0VH2bN9K58Q8XjWCZGxXZC3WKNWqz1qpRBhGGCP09W8BMoRJF4aaEjWMLMXFCjMn5jl6z1FuueNe7rnrEPMoTsaBnk7Kod2QpuJiG7sJVYFDLFZ97O7tvPTZ+znv7B1cccUVfOxjHyPbmafkV9vJagL4FsgdqM6hUsSvpLqrFCkecYQVNxDzA42KxkYH77rjjn+NrO0aHStcEkahiWUGIdWqT+AH/OzoUWZ7+3jc6QUGJxdwZuZxrNYHj/qBH8ewwggv43L9TXdy1z0zZByDRVkrVTlRXGNlYoallSITSz6OhkjGQ/MZAqtUw7A+mVdbB8xvOF6w1swxYwQbWHIdnTz/4sfzwmc8jm2FrXzyk//IV772VXxrKfvV5vHzFmUG4TsgP0LsFIYFVqullKxSpHhkEhYEhATVStXKnM1kbrn1ppttuVLJDQ4PX1gNgly16uP7Pn7Vp1opU1pZZW5hnj2nncaZj91Nx/FZZH6RSCxVP6BcLhOGEeUKrCzPcPzgtVTW1hKKcVHNYTo76dvSSa92QjmIxamyyUjmNu8vaVkb01kc3KKSsQSrPo+94Aye8dQns+/sM/jZwYO87rVvY3ZhAd+GVIOglaxgAuGbwHcRPYYyQ6BF4hl5KVKkWOfdPJLgOC6e2+NmvCFXedLI+PjL+gYHL/aDoKta9QmqcWuXwI9HavX09jE8PMTpg1sZDy1LM/Ow7zE8/txzCa3FdRy2je8APMqVamNgBbVmCk3zAbXR+mUddPOjF8seKiwtF9kxtoXxwiiVSoXPff4LXHHFFaysrVIO/GbpAgkhTQHfBK4FjqJMEAWL+FHqCqZI8Z+CsGLS8shm845jChLZfV39fS/bMjR8SRgEeb9axa9WCZOSmSgIyHfk2DI8wvj4OKdt3coTLngC+Xye4tISGddhpZzh+7dNQVgh0maikvrMvOY+543Wvq2HxwBOUkFdDS1RqPR059k10svo1m4eu3c7I1v7uebaa/nMZz7DfcePE1hLJfDb+a5C3JPkO8B3gSlUpgjtIkG1ksS0UqRI8Yh1CZsRRT6lElFn55QY9ZdPnCiWVpYXu/sHLjFiCmEYmLBWP4iyvLJK8eQBlubnmRkeZsX3eeL+x1MYGSHf0cGBwwscOHCcYnGeUhgSYeNmfcQdE+pjCur6qYbVFWn8iEtwDD3GpdfJUjhjhPPO3sbZuwucsbOAEeH666/nbZ/5LJNTU2RyWZYr5XbSszQKmq9A9XaMTGDNDKEWE7JKLasUKf5TWVh1OnVdPK8L1T5B9xojT83kOp7jOM65kbVeGDY6hpK0nfGMQ29/H6Pj45y2fTs7d5xGPt/P/FLIWiVioVhiuVRhea1KsVTG90M0jOJ+8AlR1aZPS8ahK59lS3cnQ32djI0MMLqlh8H+POMjW3AdYW5mmltvv53vfue7HD9+HJNxKVcr+JFtP7o+cV/2W4ErQA/FwXaZIdRlfCpQTckqRYr/tIRVG0XkujkymQGMFrC6z7juC8VxzldlSJsJi0TEaWMLqiObpbOnh127dnHW3jPJ5TrIZDvo6uzG4oAaIjWEkSWM4r8xxpDLeuS8DPkOj76ePJ0dWTpyHp7rsLRUZH5hnttvv4Pbbv0Jd999ABWpzwEvVyrNQfUaVoFDwHXAj4FjKHMYXSCsLlNN6wRTpPh1IKyG65rBw+voQnUAZCfoEzHmYmAfql3JXLgGcTUP4fRDHGOIoojRQoGhrVvJ5zvYWigw0NdDf18fWc+js7MLI4LruJQrZVZX15ifX2BlqUixWOTOO+/mxOIJXMchUksml0OBil+tD1xtc/9KwAzCrVi+i+EYKguIzoAsY8slymk2MEWKXzfCiq2tjoyLsXnI9AEjKDuBi0DOB/aAdm0cl5N4KiWKYwyOMXHtYKTkvAyZTCYe3GoacwBrLWpKlTId+TwZ18UPQ4LEdYxLh+ptBZsREgfVFxL37ycgB5NRSgtgipigSMn6SdP51LJKkeLXkLAa1la3kwOvi9D2IDICuhvkPJQLEHYAfZwqoZDoqhzjYASMmHhCb9JovTZHz6JEqvXOpYnpttla/YSoZoCDMVnJnQgLqC6CWcCJlilHFcIwdQFTpHiUEFZsbWVwyWY9Arow9CD0gdkOuhd4HMpZCENAF+Cxfi7fLwqbWFOlxJo6BhxE9afxwAiKCEWgiNVlxClRKoWkkoUUKR51hNUgLsdxyZFDTR5jerDSgzCIZTvC6cAO4smUg0APkAdcBIO2kJhps8Js09GxTQRVSUhqkVj4eQy4N7GsFoEiyDLCMthV3LDEWuQTkrp/KVI8ygmrQTYduNhsDsfkgDyqXQhdKF0YRrAMIYwCBWAgefQAucQCc5uIq0ZOzQS1nFhSc8AcyjRGF1FZRVnF6CoqJUSWCW0Fx6lQqo/kSokqRYqUsDYgLjB04qJ5jyjK4ZJDJQfkUPUQycfL5BHtAskDHoqH1GbfqEU1RCSJS2kFpARUUPXj17WCmhLYCiIVlApu5OMEPkXClKRSpEgJ68GRl8HgOi5WXLIZD9TD4qG4GHURPFRd1LjJoTA1fzA5PBZJyEdsCBISEROWwccPfVRDHEKC0CYTmVOiSpEiJaxf2PKKnzO4uFmDxaBqEHXJJMs+4LhxMysNwWKJxCItj5By2Ta5eilBpUiREtbDRmIPBik5pUiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFClSpEiRIkWKFCkekfj/AyUrcVhOmTwrAAAAAElFTkSuQmCC);
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# Presentación de datos de una Variable Cuantitativa

Frente a un conjunto de datos numéricos, el primer paso a dar, es expresarlo y clasificarlo en alguna manera simple que permita ver rápidamente todas las características posibles para obtener conclusiones útiles.

Dicha clasificación se realiza en un cuadro o tabla, a las que en adelante llamaremos Tablas de Distribuciones de Frecuencias.

---

# Distribución de Frecuencias de variables cuantitativas discretas

Si `$n$` valores de una variable discreta `$X$` observados en una muestra de una población tiene `$k(k\leq n)$` valores distintos, `$x_1,x_2,...,x_k$`, que se repiten respectivamente `$f_1,f_2,...,f_k$` veces, entonces, la organización o agrupación de estos `$n$` datos origina la distribución de frecuencias del siguiente cuadro:

---

|Valores de la variable `$X$`|Frecuencia absoluta `$f_i$`|Frecuencia relativa `$h_i$`|Frecuencia porcentual `$p_i$`|Frecuencia absoluta acumulada `$F_i$`|Frecuencia relativa acumulada `$H_i$`|Frecuencia porcentual acumulada `$P_i$`|
|:--:|:--:|:--:|:--:|:--:|:--:|:--:|
|$$x_1$$|$$f_1$$|$$h_1$$|$$p_1$$|$$F_1$$|$$H_1$$|$$P_1$$|
|$$x_2$$|$$f_2$$|$$h_2$$|$$p_2$$|$$F_2$$|$$H_2$$|$$P_2$$|
|$$\vdots$$|$$\vdots$$|$$\vdots$$|$$\vdots$$|$$\vdots$$|$$\vdots$$|$$\vdots$$|
|$$x_k$$|$$f_k$$|$$h_k$$|$$p_k$$|$$F_k=n$$|$$H_k=1$$|$$P_2=100$$|
|Total|$$n$$|1|100|

---

# Frecuencias

- **Frecuencia absoluta (`$f_i$`):** es el número de datos que resulta del conteo en la categoría respectiva `$C_i$`, donde `$i=1,2,...,k$`. La suma de todas las frecuencias absolutas es igual a `$n$`, el total de datos observados.

- **Frecuencia relativa (`$h_i$`):** se define en cada categoría `$C_i$` por `$h_i=\frac{f_i}{n}$`. La suma de todas las frecuencias relativas es igual a uno.

- **Frecuencia porcentual (`$p_i$`):** se define en cada categoría `$C_i$` por `$p_i=h_i\times 100\%$`. La suma de todas las frecuencias porcentajes es igual a `$100\%$`.

---

# Frecuencias acumuladas

- **Frecuencia Acumulada Absoluta, `$F_i$` hasta el valor `$x_i$`** es la suma de las frecuencias absolutas `$f_i$` de todos los valores menores o iguales a `$x_i$`, esto es,
`$$F_i=f_1+f_2+...+f_i.$$`
		
- **Frecuencia Acumulada Relativa, `$H_i$` hasta el valor `$x_i$`** es la suma de las frecuencias relativas `$h_i$` de todos los valores menores o iguales a `$x_i$`, esto es,
`$$H_i=h_1+h_2+...+h_i\quad\text{o}\quad H_i=\frac{F_i}{n}$$`
		
- **Frecuencia Acumulada Porcentual, `$P_i$` hasta el valor `$x_i$`** es la suma de las frecuencias porcentuales `$p_i$` de todos los valores menores o iguales a `$x_i$`, esto es,
`$$P_i=p_1+p_2+...+p_i\quad\text{o}\quad P_i=H_i\times100\%.$$`

---

# Data

> Considere la siguiente data:

``` r
library("readxl")
dato5 <- read_excel("D:/Backup/Documentos/Cursos/aModulo1/dato5.xlsx",sheet = "Hoja1")
dato5
```

```
## # A tibble: 50 × 7
## Numero Sexo Hermanos Edad Peso Ciudad Puntaje
## <dbl> <chr> <dbl> <dbl> <dbl> <chr> <dbl>
## 1 1 F 1 18 60 Lima 69
## 2 2 F 2 19 58 Arequipa 78
## 3 3 M 0 18 59 Cuzco 68
## 4 4 M 1 20 58 Tacna 69
## 5 5 F 2 21 61 Moquegua 70
## 6 6 M 2 22 62 Lima 68
## 7 7 M 1 20 63 Cuzco 70
## 8 8 F 3 21 58 Cuzco 74
## 9 9 M 1 18 59 Arequipa 75
## 10 10 F 0 19 60 Tacna 75
## # ℹ 40 more rows
```

---

### Diagrama de barras
> Haga la tabla de distribución de frecuencias del número de hermanos

.panelset[
.panel[.panel-name[Table]

```
##   frec frec_rel frec_porc Acum Rel_acum Porc_acum
## 0    4     0.08         8    4     0.08         8
## 1   18     0.36        36   22     0.44        44
## 2   21     0.42        42   43     0.86        86
## 3    7     0.14        14   50     1.00       100
```

]

.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-3-1.png)

]

.panel[.panel-name[R Code]

``` r
frec<-table(dato5$Hermanos)
tb=cbind(frec,
 frec_rel=prop.table(frec),
 frec_porc=prop.table(frec)*100,
 Acum=cumsum(frec),
 Rel_acum=cumsum(prop.table(frec)),
 Porc_acum=cumsum(prop.table(frec)*100)
 )
print(tb)
g3=barplot(frec,
 main = "Diagrama de barras",
 xlab = "Número de hermanos",
 ylab = "Número de personas",
 col=rainbow(4),
 ylim = c(0,max(frec)*1.2))
text(x=g3,y=frec,labels=frec,pos=3)
```
]

]

---

# Nube de puntos

.panelset[

.panel[.panel-name[R Code]

``` r
x<-1:10
y<-c(1,3,2,6,3,5,8,6,7,5)
plot(x,y,pch=19,col="red")
```
]

.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-6-1.png)

]

---

### Diagrama de bastones
> Grafique el diagrama de bastones del numero de hermanos

.panelset[

.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-7-1.png)

]

.panel[.panel-name[R Code]

``` r
frec<-table(dato5$Hermanos)
plot(frec,
 type = "h", 
 lwd = 3,
 ylim=c(0,max(frec)*1.2),
 col="blue")
text(x=names(frec),y=frec,labels=frec,pos=3)
```
]

]

---

### Poligono de frecuencias
> Grafique el poligono de frecuencias del numero de hermanos

.panelset[

.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-9-1.png)

]

.panel[.panel-name[R Code]

``` r
frec<-table(dato5$Hermanos)
valores=as.numeric(names(frec))
plot(c(min(valores)-1,valores,max(valores)+1),
 c(0,as.numeric(frec),0), 
 type = "l", 
 lwd = 3, 
 col = "blue",
 main="Poligono de frecuencias",
 xlab = "Número de hermanos",
 ylab = "Nro de personas",
 ylim=c(0,max(frec)*1.2))
text(x=valores,y=as.numeric(frec),labels=frec,pos=3)
```
]

]

---

# Distribución de frecuencias para variable cuantitativa continua

Finalmente estamos listos para proceder con la tabulación para determinar las frecuencias de
cada uno de los intervalos de clase.

<table border="1" cellspacing="0" cellpadding="5" style="border-collapse: collapse; text-align: center; font-size: 85%; margin: auto;">
 <caption style="font-weight: bold; margin-bottom: 8px;">
 Distribución de frecuencias de una variable continua
 </caption>
 <tr>
 <th rowspan="2">Intervalos</th>
 <th rowspan="2">Marca de clase xi</th>
 <th colspan="3">Frecuencias</th>
 <th colspan="3">Frecuencias acumuladas</th>
 </tr>
 <tr>
 <th>fi</th>
 <th>hi</th>
 <th>pi</th>
 <th>Fi</th>
 <th>Hi</th>
 <th>Pi</th>
 </tr>
 <tr>
 <td>I1</td>
 <td>x1</td>
 <td>f1</td>
 <td>h1</td>
 <td>p1</td>
 <td>F1</td>
 <td>H1</td>
 <td>P1</td>
 </tr>
 <tr>
 <td>I2</td>
 <td>x2</td>
 <td>f2</td>
 <td>h2</td>
 <td>p2</td>
 <td>F2</td>
 <td>H2</td>
 <td>P2</td>
 </tr>
 <tr>
 <td>⋮</td>
 <td>⋮</td>
 <td>⋮</td>
 <td>⋮</td>
 <td>⋮</td>
 <td>⋮</td>
 <td>⋮</td>
 <td>⋮</td>
 </tr>
 <tr>
 <td>Ik</td>
 <td>xk</td>
 <td>fk</td>
 <td>hk</td>
 <td>pk</td>
 <td>Fk = n</td>
 <td>Hk = 1</td>
 <td>Pk = 100</td>
 </tr>
 <tr>
 <td>Total</td>
 <td></td>
 <td>n</td>
 <td>1</td>
 <td>100%</td>
 <td></td>
 <td></td>
 <td></td>
 </tr>
</table>

---

### Construcción de la distribución de frecuencias

Sean `$n$` valores de alguna variable cuantitativa `$X$` continua (o discreta con más de 20 valores distintos). Debemos calcular:

- Rango `$R$`: Es la diferencia entre los límites reales de la muestra.
`$$R=x_{\max}-x_{\min}$$`
	
	
- Número de intervalos `$k$`: La formula de Sturges nos sirve para decidir el número de intervalos:
`$$k=1+3.3\log_{10}(n),\;n>10$$`
	El resultado puede aproximarse por exceso, la regla no es absoluta, va a depender de la decisión del investigador. Se recomienda como mínimo 5 y como máximo 20 intervalos.

- Amplitud `$A$`: es el tamaño de cada intervalo es `$A=\frac{R}{k}$`.
Si el valor obtenido en `$A$` no coincide con el número de decimales de los datos, entonces `$A$` se aproxima por exceso. Por ejemplo: si los datos tienen 2 decimales y si `$\frac{R}{k}=7.2516$`, se elige `$A=7.26$`.

---

- Intervalos de clase: A partir del extremo inferior `$x_{\min}$` y de la amplitud `$A$`, definimos los intervalos de clase:

<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async
 src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>

<div style="font-size:90%; margin: 20px;">
 \[
 \begin{aligned}
 I_1 & = [x_{\min},\, x_{\min} + A), \\[6pt]
 I_2 & = [x_{\min} + A,\, x_{\min} + 2A), \\[6pt]
 & \;\;\vdots \\[6pt]
 I_k & = [x_{\min} + (k-1)A,\, x_{\min} + kA).
 \end{aligned}
 \]
</div>

- Marca de clase del intervalo `$I_i=[L_i,U_i]$`: Cada intervalo de clase tiene un punto marca de clase, denotado por `$x_i$`, que representa a cada intervalo, y es dado por
`$$x_i=\frac{L_i+U_i}{2}.$$`

---

#### Función para construir tabla de frecuencias

``` r
tabHist<-function(x,k,A){
 R=max(x)-min(x)
 if(A*k>R){
 print("Si cumple que A*k>R")
 t<-seq(from=min(x),by=A,length.out=k+1)
 Intervalos<-cut(variable,breaks = t,right = FALSE)
 ##Marcas de clase
 M.clase<-seq((t[1]+t[2])/2,by=A,length.out=length(t)-1)
 frec<-table(Intervalos)
 respuesta=cbind(M.clase,
 frec.abs=frec,
 frec.rel=round(prop.table(frec),3),
 frec.porc=round(prop.table(frec),3)*100,
 abs.acum=cumsum(frec),
 porc.acum=cumsum(prop.table(frec)*100))
 return(list(tabla = respuesta,
 particion = t,
 frec_intervalica =table(Intervalos)))
 } else{
 return(print("No cumple A*k>R, cambie los valores de A y k"))
 }
}
```

---

# Verificando Sturges

``` r
variable<-dato5$Peso
n=length(variable);n
```

```
## [1] 50
```

``` r
R=max(variable)-min(variable);R
```

```
## [1] 5
```

``` r
#
k=1+3.3*log10(n);k
```

```
## [1] 6.606601
```

``` r
## Modifique el numero de intervalos:
k=7
```

---

``` r
## Amplitud
A=R/k;A
```

```
## [1] 0.7142857
```

``` r
## Modifique la amplitud:
A=1
## Cumplir la desigualdad
A*k>R
```

```
## [1] TRUE
```

``` r
##Si es posible puede reducir el valor de k 
k=6;A=1;A*k>R
```

```
## [1] TRUE
```

``` r
g=tabHist(variable,k=6,A=1)
```

```
## [1] "Si cumple que A*k>R"
```

---

### Tabla

``` r
g$tabla
```

```
##         M.clase frec.abs frec.rel frec.porc abs.acum porc.acum
## [58,59)    58.5        8     0.16        16        8        16
## [59,60)    59.5       11     0.22        22       19        38
## [60,61)    60.5        6     0.12        12       25        50
## [61,62)    61.5       10     0.20        20       35        70
## [62,63)    62.5        9     0.18        18       44        88
## [63,64)    63.5        6     0.12        12       50       100
```

---

### Histograma

.panelset[
.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-15-1.png)

]

.panel[.panel-name[R Code]

``` r
hist(variable,
     breaks = g$particion,
     right = F,
     labels =as.character(g$frec_intervalica),
     ylim = c(0,max(g$frec_intervalica*1.1)),
     col="cyan",
     xaxt="n")
axis(side = 1, at = g$particion)
```

]

---

### Obteniendo tabla a partir de `hist()`
> Grafique el histograma de los puntajes. Luego, a partir del histograma construya la tabla de frecuencias.

.panelset[
.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-17-1.png)

]

.panel[.panel-name[Table]

```
##   Intervalo Frecuencia FrecRel
## 1   68 - 70         17    0.34
## 2   70 - 72         14    0.28
## 3   72 - 74          7    0.14
## 4   74 - 76          9    0.18
## 5   76 - 78          3    0.06
```

]

.panel[.panel-name[R Code]

``` r
variable<-dato5$Puntaje
h <- hist(variable,plot = F,right = F)
ymax=max(h$counts)*1.1
## Arreglando ejey
hist(variable,right = F,labels = T,ylim = c(0,ymax))
## Tabla
tabla <- data.frame(
 Intervalo = paste(h$breaks[-length(h$breaks)], h$breaks[-1], sep = " - "),
 Frecuencia = h$counts,
 FrecRel = h$counts / sum(h$counts)
)
tabla
```
]

]

---

#Comparando histogramas

.panelset[

.panel[.panel-name[Plot]

```
## [1] "Si cumple que A*k>R"
```

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-20-1.png)

]

.panel[.panel-name[R Code]

``` r
variable<-dato5$Puntaje
g=tabHist(variable,A=2,k=6)
par(mfrow=c(2,2))
hist(variable,labels = T,
 main = "Right-open (left closed)")
hist(variable,labels = T,right = F,
 main = "Right-closed (left open)")
hist(variable,labels = T,right = F,ylim = c(0,20),
 main = "Eje y (ampliado)")
hist(variable,labels = T,right = F,ylim = c(0,20),breaks = g$particion,
 main="Nuestra partición")
```
]

]

---

# Poligono de frecuencias

> Grafique el poligono de frecuencias de los puntajes

.panelset[

.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-22-1.png)

]

.panel[.panel-name[Table]

``` r
tabla
```

```
##   Intervalo Marca Frecuencia FrecRel
## 1   68 - 70    69         17    0.34
## 2   70 - 72    71         14    0.28
## 3   72 - 74    73          7    0.14
## 4   74 - 76    75          9    0.18
## 5   76 - 78    77          3    0.06
```

]

.panel[.panel-name[R Code]

``` r
variable<-dato5$Puntaje
h=hist(variable,right = F,plot = F)
ymax=max(h$counts)*1.1
A=diff(h$breaks)[1]
tabla <- data.frame(
 Intervalo = paste(h$breaks[-length(h$breaks)], h$breaks[-1], sep = " - "),
 Marca=h$breaks[-length(h$breaks)]+A/2,
 Frecuencia = h$counts,
 FrecRel = h$counts / sum(h$counts)
)
xi=c(tabla$Marca[1]-A,tabla$Marca,tabla$Marca[length(h$breaks)-1]+A)
fi=c(0,tabla$Frecuencia,0)
hist(variable,right = F,labels = T,ylim = c(0,ymax),
 xlim =c(min(xi),max(xi)))
lines(xi,fi,lwd=3,col="blue")
```
]

]

---

### Ejemplo
> Grafique la Ojiva de la variable puntaje.

.panelset[

.panel[.panel-name[Plot]

![](data:image/png;base64,#D5-VCuantitativa_files/figure-html/unnamed-chunk-25-1.png)

]

.panel[.panel-name[Table]

```
##   Intervalo Marca Frecuencia Acumalada FrecRel
## 1   68 - 70    69         17        17    0.34
## 2   70 - 72    71         14        31    0.28
## 3   72 - 74    73          7        38    0.14
## 4   74 - 76    75          9        47    0.18
## 5   76 - 78    77          3        50    0.06
```

]

.panel[.panel-name[R Code]

``` r
variable<-dato5$Puntaje
h=hist(variable,right = F,plot = F)
A=diff(h$breaks)[1]
ti=h$breaks
Fi=c(0,cumsum(h$counts))
plot(ti,Fi, type = "b",
 main="Ojiva",xlab="Puntajes",ylab="Frec. Acumuladas",
 lwd=2,col="blue",
 ylim=c(0,max(Fi)*1.1))
text(x=ti[-1],y=Fi[-1], labels=cumsum(h$counts),pos = 3) 
tabla <- data.frame(
 Intervalo = paste(h$breaks[-length(h$breaks)], h$breaks[-1], sep = " - "),
 Marca=h$breaks[-length(h$breaks)]+A/2,
 Frecuencia = h$counts,
 Acumalada=cumsum(h$counts),
 FrecRel = h$counts / sum(h$counts))
```
]

]