[{"data":1,"prerenderedAt":1672},["Reactive",2],{"options:asyncdata:$ogpPUTwkW6:/p/vector-gaussiano-vector-aleatorio-gaussiano:0":3},{"page":4,"book":25,"news":1666,"questionSent":19,"questions":1667,"formData":1668,"attachments":22,"chartData":22,"pending":19,"chartOptions":1669,"afspec":19,"aflink":1671},{"id":5,"book_id":6,"chapter_id":7,"name":8,"slug":9,"html":10,"priority":11,"created_at":12,"updated_at":13,"created_by":14,"updated_by":18,"draft":19,"markdown":20,"revision_count":6,"template":19,"owned_by":21,"editor":20,"trends":22,"raw_html":23,"tags":24},4022,2,0,"Vector gaussiano (vector aleatorio gaussiano)","vector-gaussiano-vector-aleatorio-gaussiano","\u003Cp id=\"bkmrk-un-vector-gaussiano-\">Un \u003Cstrong>vector gaussiano o vector aleatorio gaussiano\u003C/strong> es un \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/vector-aleatorio\">vector aleatorio\u003C/a>\u003C/strong> o variable aleatoria multidimensional en el que cualquier combinación lineal de las variables aleatorias univariante que lo componen sigue una distribución normal unidimensional. Se define con un vector de medias para cada de las variable aleatorias unidimensionales y la \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/matriz-de-varianzas-y-covarianzas-matriz-de-covarianzas\">matriz de varianzas-covarianzas\u003C/a>\u003C/strong> que las relaciona. \u003Cbr>\u003C/p>",328,"2026-02-20T16:00:48.000000Z","2026-02-20T16:10:34.000000Z",{"id":15,"name":16,"slug":17},1,"Admin","admin",{"id":15,"name":16,"slug":17},false,"",{"id":15,"name":16,"slug":17},null,"\u003Cp id=\"bkmrk-un-vector-gaussiano-\">Un \u003Cstrong>vector gaussiano o vector aleatorio gaussiano\u003C/strong> es un \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/vector-aleatorio\">vector aleatorio\u003C/a>\u003C/strong> o variable aleatoria multidimensional en el que cualquier combinación lineal de las variables aleatorias univariante que lo componen sigue una distribución normal unidimensional. Se define con un vector de medias para cada de las variable aleatorias unidimensionales y la \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/matriz-de-varianzas-y-covarianzas-matriz-de-covarianzas\">matriz de varianzas-covarianzas\u003C/a>\u003C/strong> que las relaciona. \u003Cbr>\u003C/p>",[],{"id":6,"name":26,"slug":27,"description":20,"created_at":28,"updated_at":29,"created_by":15,"updated_by":15,"owned_by":15,"default_template_id":22,"pages":30,"index":61,"shelves":1659},"Estadística general","estadistica-general","2023-05-06T08:26:42.000000Z","2023-05-16T06:24:05.000000Z",[31,36,41,46,51,56],{"id":32,"name":33,"slug":34,"html":35},1660,"Plano de regresión","plano-de-regresion","\u003Cp id=\"bkmrk-en-regresi%C3%B3n-m%C3%BAltipl\">En regresión múltiple o modelo de regresión con varias variables independientes), el \u003Cstrong>plano de regresión\u003C/strong> es equivalente a la línea de regresión correspondiente a la regresión simple (una única variable independiente). El plano mostrará una pendiente diferente (igual al coeficiente de regresión correspondiente) para cada variable independiente.\u003C/p>\r\n\u003Cp id=\"bkmrk-puede-interesarte-ta\">\u003Cstrong>Puede interesarte también\u003C/strong>\u003C/p>\r\n\u003Cul id=\"bkmrk-recta-de-regresi%C3%B3n-d\">\r\n\u003Cli class=\"null\">\u003Ca href=\"https://ikusmira.org/p/recta-de-regresion-de-minimos-cuadrados\">\u003Cstrong>Recta de regresión de mínimos cuadrados\u003C/strong>\u003C/a>\u003C/li>\r\n\u003C/ul>",{"id":37,"name":38,"slug":39,"html":40},4003,"Percentiles","percentiles","\u003Cp id=\"bkmrk-un-percentil-es-el-v\">Un \u003Cstrong>percentil\u003C/strong> es el valor de una variable estadística que deja por debajo suyo un porcentaje dado de datos de una muestra. Por ejemplo, el percentil 10, expresado P10, de las calificaciones de un grupo de alumnos, es la calificación en puntos por debajo de la cual se sitúa el 10% de los alumnos del grupo. Una definición alternativa de percentil y al porcentaje que representa incluye al propio valor del percentil, es decir, define el percentil como valor q\u003Cem>ue coincide o es inferior\u003C/em> a un porcentaje de datos datos.\u003Cbr>\u003C/p>\r\n\u003Cp id=\"bkmrk-los-percentiles-son-\">Los percentiles son un tipo de estadístico de orden perteneciente a la familia de los cuantiles; más concretamente los percentiles no son más que 100-cuantiles, por el hecho de que los percentiles P1, P2, ..., P99 dividen en 100 partes que comprenden un 1% cada una de ellas una distribución de datos. Por otra parte, hay una serie de cuantiles que coinciden conceptualmente con ciertos percentiles; por ejemplo, los \u003Ca href=\"https://ikusmira.org/p/cuartiles-primer-cuartil-segundo-cuartil-tercer-cuartil\">cuartiles\u003C/a> o 4-cuantiles coinciden con los percentiles 25, 50 y 75 respectivamente y los \u003Ca href=\"https://ikusmira.org/p/deciles\">deciles\u003C/a> o 10-cuantiles con los percentiles 10, 20, ..., 90. \u003C/p>\r\n\u003Cp id=\"bkmrk-hay-que-recalcar-que\">Hay que recalcar que los percentiles son un valor de variable que se examina y que por tanto vienen expresados en su unidad. Así, si en una muestra de 40 alumnos debe calcularse el percentil 10 de sus notas, este percentil corresponde con una nota, por ejemplo 2.7, de modo que el 10% de alumnos, es decir 4, tiene una nota inferior (o igual). Teniendo en cuenta lo anterior, deben distinguirse el percentil, el rango ordinal y el rango percentil, en el ejemplo dado, el rango ordinal del percentil sería 4, porque hay 4 datos inferiores o iguales a 2.7 y el rango percentil de 2.7 sería 10%. \u003C/p>",{"id":42,"name":43,"slug":44,"html":45},2657,"Sesgo de un estimador (estimador sesgado)","sesgo-de-un-estimador-estimador-sesgado","\u003Cp id=\"bkmrk-en-estad%C3%ADstica%2C-el-s\">En estadística, el \u003Cstrong>sesgo de un estimador\u003C/strong> es el error promedio en el que incurre por utilizar dicho estimador para estimar el valor de un \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/parametro-poblacional\">parámetro estadístico\u003C/a>\u003C/strong> concreto. Más concretamente, para un estimador \\(\\hat{\\theta}\\) de un parámetro \\(\\theta\\), el sesgo de dicho estimador se define de esta forma:\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24e%5B%5Chat%7B%5Ctheta%7D%5D-%5Ct\">$$E[\\hat{\\theta}]-\\theta,$$\u003C/p>\r\n\u003Cp id=\"bkmrk-siendo-%5C%28e%5B%5Chat%7B%5Cthe\">siendo \\(E[\\hat{\\theta}]\\) la \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/esperanza-matematica\">esperanza o valor esperado\u003C/a>\u003C/strong> del estimador. \u003C/p>\r\n\u003Cp id=\"bkmrk-cuando-el-sesgo-es-n\">Cuando el sesgo es nulo, el error promedio es 0, lo cual no quiere decir que no se cometen errores en la estimación, sin que el promedio de dichos errores es nulo; en esos casos, se dice que el estimador es insesgado.\u003C/p>\r\n\u003Cp id=\"bkmrk-cuando-el-sesgo-no-e\">Cuando el sesgo no es nulo, se dice que el \u003Cstrong>estimador es sesgado\u003C/strong>: si el sesgo es positivo, con el estimador se está en promedio sobreestimando el valor del parámetro; si el sesgo es negativo, se esta subestimando en promedio el valor del parámetro. \u003C/p>",{"id":47,"name":48,"slug":49,"html":50},2346,"Media cúbica","media-cubica","\u003Cp id=\"bkmrk-la-media-c%C3%BAbica-es-l\">La \u003Cstrong>media cúbica\u003C/strong> es la raíz cúbica de la media aritmética de los datos al cubo referidos a una variable estadística cuantitativa:\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24%5Coverline%7Bx%7D_%7Bcub%7D\">$$\\overline{x}_{cub}=\\sqrt[3]{\\cfrac{\\sum_i x_i^3}{n}}=\\Bigg(\\cfrac{\\sum_i x_i^3}{n}\\Bigg)^{\\cfrac13}$$\u003C/p>\r\n\u003Cp id=\"bkmrk-es-un-tipo-de-media-\">Es un tipo de media generalizada, al igual que la \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/media-cuadratica\">media cuadrática\u003C/a>\u003C/strong>, y por lo tanto se utiliza como medida de tendencia central, pero su uso es muy restringido.\u003C/p>",{"id":52,"name":53,"slug":54,"html":55},3036,"Número de intervalos de clase","numero-de-intervalos-de-clase","\u003Cp id=\"bkmrk-el-primer-paso-para-\">El primer paso para construir una \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/distribucion-de-frecuencias-agrupada-en-intervalos\">distribución de frecuencias agrupada en intervalos\u003C/a>\u003C/strong> es determinar el \u003Cstrong>número de intervalos de clase\u003C/strong> en los que se van a agrupar los datos. \u003C/p>\r\n\u003Cp id=\"bkmrk-como-regla-general%2C-\">\u003Cstrong>Como regla general, el número de intervalos\u003C/strong> de clase óptimo debe fijarse entre 5 y 15. Un número demasiado pequeño de intervalos, más concretamente inferior a 5, conlleva una gran pérdida de información, mientras que un número demasiado grande de intervalos no resume la información de forma adecuada, que es lo que finalmente se pretende con una distribución de frecuencias agrupada en intervalos. \u003C/p>\r\n\u003Cp id=\"bkmrk-en-todo-caso%2C-existe\">En todo caso, existen también fórmulas que proporcionan el número más adecuado de intervalos para un conjunto de datos, a partir del número de datos a agrupar. La fórmula más utilizada es la \u003Ca href=\"https://ikusmira.org/p/regla-de-sturges-formula-de-sturges\">\u003Cstrong>regla de Sturges\u003C/strong>\u003C/a> (k: número de intervalos; n: tamaño muestral), redondeando el resultado al siguiente número entero:\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24%5Ctext%7Bregla-de-stu\">$$\\text{Regla de Sturges:}\\ k= 1 + 3.322 \\log_{10}(n)= 1 + \\frac{\\ln(n)}{\\ln(2)}$$\u003C/p>\r\n\u003Cp id=\"bkmrk-otra-f%C3%B3rmula-es-la-r\">Existen también otras reglas o fórmulas para la determinación del número de intervalos, redondeando siempre el resultado al siguiente entero:\u003Cbr>\u003C/p>\r\n\u003Cul id=\"bkmrk-regla-de-sturges%3A-%5C%28\">\r\n\u003Cli class=\"null\">regla de la raíz cuadrada de Scott: \\(k=\\sqrt{n}\\);\u003Cbr>\u003C/li>\r\n\u003Cli class=\"null\">regla de Rice: \\( k=2n^{\\frac13}\\), redondeando el resultado al siguiente entero;\u003Cbr>\u003C/li>\r\n\u003Cli class=\"null\"> la regla de Terrell-Scott: \\(k = \\sqrt[3]{2n}\\).\u003C/li>\r\n\u003C/ul>\r\n\u003Cp id=\"bkmrk-por-%C3%BAltino-%28aunque-t\">Por último, proporcionamos la regla de Freedman-Diaconis que nos da la amplitud óptima de intervalo, \u003Cbr>\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24h-%3D-%C2%A02%5Cfrac%7B%5Copera\">$$h =  2\\frac{\\operatorname{IQR}(x)}{\\sqrt[3]{n}}$$\u003C/p>\r\n\u003Cp id=\"bkmrk-el-n%C3%BAmero-de-interva\">El número de intervalos se calcularía según esta fórmula dividiendo el \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/recorrido-rango\">rango o recorrido\u003C/a>\u003C/strong> entre la amplitud obtenida, redondeando siempre por exceso.\u003C/p>",{"id":57,"name":58,"slug":59,"html":60},2926,"Media de medias (media general, media agregada)","media-de-medias-media-general-media-agregada","\u003Cp id=\"bkmrk-la-media-de-medias%2C-\">La \u003Cstrong>media de medias\u003C/strong>, también llamada \u003Cstrong>media general o media agregada\u003C/strong>, es la media de un conjunto de datos calculada a partir de las medias de varias submuestras que engloban aquellos datos. Un ejemplo sería calcular la media de las calificaciones de un conjunto de alumnos en una asignatura a partir de la media de califiaciones para cada aula en la que se imparte en la asignatura. \u003C/p>\r\n\u003Cp id=\"bkmrk-si-denotamos%C2%A0-%5C%28%5Cove\">Si denotamos  \\(\\overline{x}\\) la media general o agregada, \\(\\overline{x}_1,\\overline{x}_2,...,\\overline{x}_k\\) las medias parciales y \\(n_1,n_2,...,n_k\\) los tamaños muestrales de aquellas, cumpliéndose que \\(n_1+n_2+...+n_k=n\\), siendo \\(n\\) el tamaño muestral total, la media general se calcularía a través de esta fórmula, que en última instancia no es más que una aplicación de la \u003Cstrong>\u003Ca href=\"https://ikusmira.org/p/media-aritmetica-ponderada\">media aritmética ponderada\u003C/a>\u003C/strong>:\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24%5Coverline%7Bx%7D%3D%5Ccfra\">$$\\overline{x}=\\cfrac{n_1\\overline{x}_1+n_2\\overline{x}_2+\\cdots+n_k\\overline{x}_k}{n_1+n_2+\\cdots+n_k}$$\u003C/p>\r\n\u003Cp id=\"bkmrk-la-formula-anterior-\">La formula anterior se deduce fácilmente teniendo en cuenta que la suma de un grupo de datos se calcula a partir de su media de esta forma:\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24%5Coverline%7Bx%7D%3D%5Ccfra-1\">$$\\overline{x}=\\cfrac{\\sum_ix_i}{n} \\longrightarrow \\sum_ix_i=n\\overline{x}$$\u003C/p>\r\n\u003Cp id=\"bkmrk-en-la-f%C3%B3rmula-de-la-\">De esta forma, en la fórmula de la media de medias no hacemos más que calcular las sumas parciales para cada grupo de datos según la fórmula anterior y agregar las sumas parciales para cada grupo. \u003C/p>\r\n\u003Cp id=\"bkmrk-ejemplo-de-aplicaci%C3%B3\">\u003Cstrong>Ejemplo de aplicación\u003C/strong>\u003C/p>\r\n\u003Cp id=\"bkmrk-un-profesor-es-respo\">Un profesor es responsable de una asignatura de matemáticas en tres aulas A, B y C. Las medias de las calificaciones en dichas aulas es respectivamente 6.2, 7.5 y 5.4, y los número de alumnos en cada aula 24, 18 y 14. Determinar la media de las calificaciones para las tres aulas:\u003C/p>\r\n\u003Cp id=\"bkmrk-%24%24%5Coverline%7Bx%7D%3D%5Ccfra-2\">$$\\overline{x}=\\cfrac{n_A\\overline{x}_A+n_B\\overline{x}_B+n_C\\overline{x}_C}{n_A+n_B+n_C}=\\cfrac{24 \\times 6.2 + 18 \\times 7.5 + 14 \\times 5.4}{24+18+14}=6.41$$\u003C/p>",{"":62},[63,67,71,75,80,85,90,95,100,105,110,115,120,125,130,135,140,145,150,155,160,165,170,175,180,185,190,195,200,205,210,215,220,225,230,235,239,244,248,253,258,263,268,273,277,281,286,291,296,301,305,310,315,320,325,330,335,340,345,350,355,360,365,370,375,380,385,390,395,400,405,410,415,420,424,429,434,439,444,449,454,459,464,468,473,478,482,487,492,497,502,507,512,517,522,527,531,536,541,546,551,556,561,566,571,576,581,586,591,596,601,606,611,616,621,626,631,636,641,646,651,656,661,666,671,676,681,686,691,696,701,706,711,716,721,726,731,736,741,746,751,756,761,765,769,774,779,783,787,791,796,801,806,811,815,817,819,824,829,834,839,844,848,853,858,863,868,873,878,883,888,893,898,902,907,912,917,922,927,932,937,942,947,951,956,960,965,970,975,980,985,989,994,999,1004,1009,1013,1015,1020,1024,1029,1034,1039,1043,1048,1053,1055,1060,1064,1069,1074,1079,1084,1089,1094,1099,1104,1109,1113,1118,1123,1128,1132,1137,1142,1147,1151,1155,1160,1165,1170,1175,1180,1184,1188,1193,1198,1200,1205,1209,1214,1219,1222,1226,1231,1236,1240,1245,1250,1255,1260,1265,1270,1275,1280,1285,1290,1295,1300,1304,1309,1314,1319,1324,1329,1334,1339,1344,1349,1354,1359,1364,1369,1374,1379,1382,1386,1391,1396,1401,1405,1409,1413,1418,1422,1426,1431,1436,1441,1446,1451,1456,1461,1466,1471,1476,1481,1486,1491,1495,1500,1505,1509,1514,1519,1524,1529,1533,1537,1541,1546,1551,1556,1561,1566,1571,1576,1581,1586,1591,1596,1601,1606,1611,1616,1618,1623,1628,1629,1634,1639,1644,1649,1654],{"id":64,"name":65,"slug":66,"priority":7,"chapter_name":22},3173,"Aleatorización (diseño de experimentos)","aleatorizacion-diseno-de-experimentos",{"id":68,"name":69,"slug":70,"priority":15,"chapter_name":22},631,"Amplitud de clase","amplitud-de-clase",{"id":72,"name":73,"slug":74,"priority":6,"chapter_name":22},387,"Análisis de trayectorias","analisis-de-trayectorias",{"id":76,"name":77,"slug":78,"priority":79,"chapter_name":22},624,"Arranque aleatorio","arranque-aleatorio",3,{"id":81,"name":82,"slug":83,"priority":84,"chapter_name":22},43,"Asociación estadística","asociacion-estadistica",4,{"id":86,"name":87,"slug":88,"priority":89,"chapter_name":22},2019,"Asociación no estadística","asociacion-no-estadistica",5,{"id":91,"name":92,"slug":93,"priority":94,"chapter_name":22},2948,"Banco de datos","banco-de-datos",6,{"id":96,"name":97,"slug":98,"priority":99,"chapter_name":22},1621,"Base del índice (periodo base)","base-del-indice-periodo-base",7,{"id":101,"name":102,"slug":103,"priority":104,"chapter_name":22},35,"Cálculo de la moda estadística para datos agrupados en intervalos","calculo-de-la-moda-estadistica-para-datos-agrupados-en-intervalos",8,{"id":106,"name":107,"slug":108,"priority":109,"chapter_name":22},3063,"Característica cualitativa","caracteristica-cualitativa",9,{"id":111,"name":112,"slug":113,"priority":114,"chapter_name":22},2708,"Casos particulares","casos-particulares",10,{"id":116,"name":117,"slug":118,"priority":119,"chapter_name":22},2730,"Censo estadístico","censo-estadistico",11,{"id":121,"name":122,"slug":123,"priority":124,"chapter_name":22},2551,"Clase mediana","clase-mediana",12,{"id":126,"name":127,"slug":128,"priority":129,"chapter_name":22},1011,"Clase modal","clase-modal",13,{"id":131,"name":132,"slug":133,"priority":134,"chapter_name":22},2133,"Coeficiente de asimetría de Bowley","coeficiente-de-asimetria-de-bowley",14,{"id":136,"name":137,"slug":138,"priority":139,"chapter_name":22},1714,"Coeficiente de asimetría de Fisher","coeficiente-de-asimetria-de-fisher",15,{"id":141,"name":142,"slug":143,"priority":144,"chapter_name":22},1844,"Coeficiente de asimetría de Pearson","coeficiente-de-asimetria-de-pearson",16,{"id":146,"name":147,"slug":148,"priority":149,"chapter_name":22},2057,"Coeficiente de contingencia de Pearson","coeficiente-de-contingencia-de-pearson",17,{"id":151,"name":152,"slug":153,"priority":154,"chapter_name":22},2648,"Coeficiente de correlación biserial puntual","coeficiente-de-correlacion-biserial-puntual",18,{"id":156,"name":157,"slug":158,"priority":159,"chapter_name":22},1938,"Coeficiente de curtosis de Pearson","coeficiente-de-curtosis-de-pearson",19,{"id":161,"name":162,"slug":163,"priority":164,"chapter_name":22},2032,"Coeficiente de determinación ajustado (coeficiente de determinación corregido)","coeficiente-de-determinacion-ajustado-coeficiente-de-determinacion-corregido",20,{"id":166,"name":167,"slug":168,"priority":169,"chapter_name":22},2649,"Coeficiente de 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