Un curso de estadstica para principiantes

(Alumnos y profesores...)

Fernando San Segundo

15 de enero de 2013

2

Indice general

I Estadstica descriptiva 5

1. Introduccion a la estadstica descriptiva 71.2. Poblacion y muestra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3. Estadstica descriptiva. Tipos de Variables. . . . . . . . . . . . . . . . . . . . . . . 71.4. Tablas y visualizacion grafica de datos. . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Valores centrales y dispersion 132.1. La media aritmetica (y otros conceptos de media). . . . . . . . . . . . . . . . . . . 132.2. Otros valores centrales. Mediana, percentiles, moda. . . . . . . . . . . . . . . . . . 152.3. Medidas de dispersion. Primera parte . . . . . . . . . . . . . . . . . . . . . . . . . 172.4. Varianza y desviacion tpica. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

II Probabilidad y variables aleatorias 21

3. Probabilidad 233.1. El lenguaje de la Probabilidad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2. Probabilidad mas alla de la Regla de Laplace. . . . . . . . . . . . . . . . . . . . . . 273.3. Probabilidad condicionada. Sucesos Independientes. . . . . . . . . . . . . . . . . . 293.4. Probabilidades totales y Teorema de Bayes. . . . . . . . . . . . . . . . . . . . . . . 313.5. Combinatoria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6. Otras formulas combinatorias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4. Variables aleatorias 394.1. Variables aleatorias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2. Media y varianza de variables aleatorias . . . . . . . . . . . . . . . . . . . . . . . . 424.3. Operaciones con variables aleatorias. Esperanza y varianza. . . . . . . . . . . . . . 44

5. Teorema central del lmite 475.1. Experimentos de Bernouilli y distribucion binomial. . . . . . . . . . . . . . . . . . 475.2. Distribuciones Binomiales con n muy grande . . . . . . . . . . . . . . . . . . . . . 505.3. Las distribuciones continuas entran en escena... . . . . . . . . . . . . . . . . . . . . 525.4. Mas sobre distribuciones continuas . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5. Distribucion normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.6. El teorema central del lmite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

III Inferencia Estadstica 65

6. Inferencia e intervalos de confianza 676.1. Distribucion muestral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2. Intervalos de confianza para la media en poblaciones normales . . . . . . . . . . . . 736.3. Desviacion tpica muestral (y el misterio del n 1). . . . . . . . . . . . . . . . . . . 786.4. Distribucion t de Student. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7. Contraste de hipotesis 857.1. El lenguaje del contraste de hipotesis . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2. Un contraste de hipotesis, paso a paso. Region de rechazo y p-valor. . . . . . . . . 877.3. Contrastes de una y dos colas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.4. Contraste de hipotesis para con muestras pequenas y varianza desconocida. . . . 91

3

8. Distribuciones relacionadas con la binomial 938.1. Proporciones y su distribucion muestral . . . . . . . . . . . . . . . . . . . . . . . . 938.2. Inferencia estadstica sobre proporciones . . . . . . . . . . . . . . . . . . . . . . . . 958.3. Distribucion de Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.4. Inferencia estadstica para la distribucion de Poisson . . . . . . . . . . . . . . . . . 100

9. Inferencia sobre la varianza 1019.1. Inferencia sobre la varianza y 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

10.Inferencia sobre dos poblaciones 10710.1. Diferencia de proporciones en dos poblaciones . . . . . . . . . . . . . . . . . . . . . 10710.2. Diferencia de medias en dos poblaciones . . . . . . . . . . . . . . . . . . . . . . . . 10910.3. Inferencias para comparar varianzas en poblaciones normales. Distribucion F de

Fisher-Snedecor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

IV Regresion y correlacion 117

11.Regresion lineal simple. 11911.1. Variables correlacionadas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11911.2. Error cuadratico medio, recta de regresion, correlacion. . . . . . . . . . . . . . . . . 12211.3. Analisis de la varianza. Coeficiente r de correlacion lineal de Pearson. . . . . . . . 12411.4. Regresion lineal y muestreo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

12.Otros modelos de regresion. (*) 131

13.Tablas de contingencia y test 2.(*) 133

V ANOVA 135

14.Introduccion al ANOVA 13714.1. ANOVA unifactorial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13714.2. Identidad ANOVA. Residuos e hipotesis necesarias. . . . . . . . . . . . . . . . . . . 13914.3. Tablas ANOVA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

15.ANOVA con dos factores. (*) 145

VI Metodos no parametricos. 147

16.Metodos no parametricos. (*) 149

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Parte I

Estadstica descriptiva

5

Captulo 1

Introduccion a la estadsticadescriptiva

1.2. Poblacion y muestra.

La estadstica se divide habitualmente en varios temas relacionados. Esa division resultaevidente mirando el programa de esta asignatura, o la division por captulos de cualquiera delos manuales que aparecen en la bibliografa. Algunos de los temas habituales en cualquiercurso de introduccion a la Estadstica son

Estadstica =

Estadstica Descriptiva

Distribuciones de Probabilidad

Inferencia Estadstica

Diseno de Experimentos

Regresion

...

Cuando tenemos un conjunto de datos y queremos describirlos, representarlos o visualizarlos,utilizamos tecnicas que corresponden a la Estadstica Descriptiva. As que la Estadstica Des-criptiva se encarga del trabajo con los datos que de hecho tenemos, y con los que podemoshacer operaciones. Es el tema con el que vamos a empezar el curso.

En muchas ocasiones esos datos corresponden a una muestra, es decir, a un subconjunto(mas o menos pequeno), de una poblacion (mas o menos grande), que nos gustara estudiar.El problema es que estudiar toda la poblacion puede ser demasiado difcil o directamenteimposible. En ese caso surge la pregunta hasta que punto los datos de la muestra sonrepresentativos de la poblacion? Es decir, podemos usar los datos de la muestra para inferirlos datos de la poblacion completa? La Inferencia Estadstica se encarga de dar sentido yformalizar estas preguntas.

La razon por la que la inferencia, que es la esencia de la Estadstica, funciona es porquemuchos casos (ya discutiremos cuales), cualquier muestra bien elegida (ya veremos lo quesignifica esto) es bastante representativa de la poblacion. Dicho de otra manera, si pensamosen el conjunto de todas las posibles muestras bien elegidas que podramos tomar, la inmensamayora de ellas seran representativas de la poblacion. As que si tomamos una al azar, casicon seguridad habremos tomado una muestra representativa. Puesto que hemos mencionadoel azar, parece evidente que la manera de hacer que estas frases un poco imprecisas seconviertan en afirmaciones rigurosas y medibles es utilizar el lenguaje de la Probabilidad. Poresa razon necesitamos hablar en ese lenguaje para poder hacer Estadstica rigurosa.

Las tecnicas de muestreo y las de diseno de experimentos tambien forman parte de la Es-tadstica, para garantizar el buen funcionamiento de las tecnicas que hemos descrito.

1.3. Estadstica descriptiva. Tipos de Variables.

Contenido:

Variables cualitativas y cuantitativas.

7

Variables cuantitativas discretas y continuas.

Notacion para las variables. Frecuencia.

1.3.1. Variables cualitativas y cuantitativas

A veces se dice que las variables cuantitativas son las variables numericas, y las cualitativaslas no numericas. La diferencia es, en realidad, un poco mas sutil. Una variable es cualitativanominal cuando solo se utiliza para establecer categoras, y no para hacer operaciones conella. Es decir, para poner nombres, crear clases o especies dentro de los individuos que esta-mos estudiando. Por ejemplo, cuando clasificamos a los seres vivos en especies, no estamosmidiendo nada. Podemos representar esas especies mediante numeros, naturalmente, pero eneste caso la utilidad de ese numero se acaba en la propia representacion, y en la clasificacionque los numeros permiten. Pero no utilizamos las propiedades de los numeros (las operacionesaritmeticas, suma, resta, etc.). Volviendo al ejemplo de las especies, no tiene sentido sumarespecies de seres vivos.

Una variable cuantitativa, por el contrario, tiene un valor numerico, y las operaciones ma-tematicas que se pueden hacer con ese numero son importantes para nosotros. Por ejemplo,podemos medir la presion arterial de un animal y utilizar formulas de la mecanica de fluidospara estudiar el flujo sanguneo.

En la frontera, entre las variables cuantitativas y las cualitativas, se incluyen las cualitativasordenadas. En este caso existe una ordenacion dentro de los valores de la variable. Por ejemplo,la gravedad del pronostico de un enfermo ingresado en un hospital. Pero no tiene sentido hacerotras operaciones con esos valores: no podemos sumar grave con leve. Como ya hemos dicho,se pueden codificar mediante numeros de manera que el orden se corresponda con el de loscodigos numericos1:

Pronostico CodigoLeve 1

Moderado 2Grave 3

En este caso es especialmente importante no usar esos numeros para operaciones estadsticasque pueden no tener significado (por ejemplo, calcular la media).

1.3.2. Variables cuantitativas discretas y continuas.

A su vez, las variables cuantitativas se dividen en discretas y continuas. Puesto que se trata denumeros, y queremos hacer operaciones con ellos, la clasificacion depende de las operacionesmatematicas que vamos a realizar.

Cuando utilizamos los numeros enteros (Z) o un subconjunto de ellos como modelo, la variablees discreta. Y entonces con esos numeros podemos sumar, restar, multiplicar pero no siempredividir.

Por el contrario, si usamos los numeros reales (R), entonces la variable aleatoria es continua.La diferencia entre un tipo de datos y el otro se corresponde en general con la diferenciaentre digital y analogico, o con la diferencia entre ecuaciones en diferencias y ecuacionesdiferenciales.

Es importante entender que la diferencia entre discreto y continuo es, en general, una dife-rencia que establecemos nosotros al crear un modelo con el que estudiar un fenomeno, y quela eleccion correcta determina la utilidad del modelo.

1.3.3. Notacion para las variables. Tablas de frecuencia. Datos agrupa-dos

En cualquier caso, vamos a tener siempre una serie de valores (observaciones, medidas) deuna variable, que representaremos con smbolos como

x1, x2, . . . , xn

1Donde encaja el pronostico reservado en esta escala?

8

El numero n se utiliza habitualmente en Estadstica para referirse al numero total de valoresde los que se dispone). Por ejemplo, en el fichero( y version ) tenemos los pesos (en libras) de los n = 92 alumnos de una clase deUniversidad de los Estados Unidos2 (los datos llegan a la fila 93, pero la primera fila es elnombre de las variables).

Si utilizamos p1, p2, . . . , p92 para referirnos a estos datos de peso, entonces p1 es el dato enla segunda fila, p2 el dato en la tercera, y p35 el dato de la fila 36. En estos casos puedeser una buena idea introducir una columna adicional con el ndice i que corresponde a pi (ies el numero de la observacion). Porque, como ya veremos, puede ser comodo conservar losnombres de las variables en la primera fila de la hoja de calculo.

Un mismo valor de la variable puede aparecer repetido varias veces en la serie de observacio-nes. En el fichero de los alumnos de nuestra clase, la variable genero solo toma dos valores,Hombre o Mujer. Pero cada uno de esos valores aparece repetido bastantes veces. En concretoen la clase hay 35 alumnas y 57 alumnos. Este numero de repeticiones de un valor es lo quellamamos la frecuencia de ese valor.

El numero de repeticiones de un valor, del que hemos hablado en el anterior parrafo, se llamafrecuencia absoluta, para distinguirlo de la frecuencia relativa, que se obtiene dividiendo lafrecuencia absoluta por n (el total de observaciones).

Tambien se pueden utilizar porcentajes en lugar de frecuencias relativas (la frecuencia relativaes el tanto por uno, y el porcentaje el tanto por ciento).

Cuando tratamos con variables cualitativas o con variables discretas, muchas veces, en lugardel valor de cada observacion la informacion que tenemos es la de las frecuencias de cada unode los posibles valores distintos de esas variables. Esto es lo que se conoce como una tablade frecuencias. Vamos a ver como se obtiene (usando la funcion FRECUENCIA()) la tabla defrecuencias de la variable genero en el ejemplo de la clase que acabamos de ver (el resultadoesta en estos ficheros y ).

Que sucede en este ejemplo con la variable peso? Podemos calcular una tabla de frecuencias?S, en principio, podemos. Pero hay demasiados valores distintos, y la informacion presentadaas no es util. De hecho, como el peso es una variable continua, si nos dieran los pesos delos alumnos con, por ejemplo, dos cifras decimales, es muy posible que no haya dos valoresiguales de la variable peso. Por otra parte, si los pesos de dos alumnos se diferencian en unasdecimas, seguramente preferiremos representarlos todos por un valor comun. En el caso devariables continuas, lo habitual es dividir el rango de posibles valores de esa variable continuaen intervalos. La tabla de frecuencia por intervalos mide, para estas variables, cuantos de losvalores observados caen dentro de cada uno de los intervalos.

En el ejemplo que estamos utilizando, podemos dividir arbitrariamente los valores del pesoen intervalos de 10 libras, desde 85 hasta 215, y obtenemos esta tabla de frecuencias:

Intervalo Significado Frecuencia(75,85] Peso

Los intervalos, insistimos, se han elegido de manera arbitraria en este ejemplo. Piensas queeso afecta de alguna manera a la informacion de la tabla de frecuencias?

En cualquier caso, conviene recordar que una tabla de frecuencias es una forma de resumir lainformacion, y que al pasar del conjunto de datos inicial a las tablas de frecuencias de Pesoy Genero se pierde informacion.

Cuando los valores de una variable continua se presentan en forma de tabla de frecuenciaspor intervalos hablaremos de datos agrupados.

1.4. Tablas y visualizacion grafica de datos.

Contenido:

1. Diagramas de lneas, sectores y barras

2. Histogramas

1.4.1. Diagramas de lneas, sectores y barras

Los diagramas de lneas se utilizan para mostrar tendencias temporales. Como el que semuestra en este fichero (version )

Los diagramas de sectores y barras se utilizan cuando queremos mostrar frecuencias (o por-centajes, recuentos, etcetera). Se pueden utilizar para ilustrar las frecuencias de variablestanto cualitativas como cuantitativas.

Los diagramas de sectores circulares son utiles para mostrar proporciones cuando los valoresson bastante distintos entre s. Pero en muchas ocasiones pueden resultar confusos o pocoprecisos. Hay un ejemplo en este fichero (version ).

Los diagramas de barras tienen, en general mas precision que los de sectores. Ademas sepueden utilizar para mostrar varios conjuntos de datos simultaneamente. Ver un ejemplo eneste fichero (version ).

10

Mes

Ingresos

Jul

2

Ag

2,1

Sep

2,2

Oct

2,1

Nov

2,3

Dic

2,4

???

Pgina

??? (???)

21/09/2011, 12:16:19

Pgina /

Ingresos

Hoja1.B1:Hoja1.B1

Jul

Hoja1.A2:Hoja1.A7

2

Hoja1.B2:Hoja1.B7

Ag

2.1

Sep

2.2

Oct

2.1

Nov

2.3

Dic

2.4

Hoja1MesIngresosJul2Ag2.1Sep2.2Oct2.1Nov2.3Dic2.4&C&A&CPgina &PHoja1Hoja2&C&A&CPgina &PHoja3&C&A&CPgina &P

genre

unitsSold

Sports

27500

Strategy

11500

Action

6000

Shooter

5000

Other

1500

???

Pgina

??? (???)

27/07/2011, 13:35:55

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Columna B

Sports

27500

Strategy

11500

Action

6000

Shooter

5000

Other

1500

Columna B

Sports

27500

Strategy

11500

Action

6000

Shooter

5000

Other

1500

Hoja1genreunitsSoldSports27500Strategy11500Action6000Shooter5000Other1500&C&A&CPgina &PHoja1Hoja2Hoja3&C&A&CPgina &P&C&A&CPgina &P

Region

Sales2006

Sales2007

A

1000

800

B

5000

4500

C

7500

9000

D

8000

12000

E

9500

9300

???

Pgina

??? (???)

20/09/2011, 18:55:40

Pgina /

Sales2006

Hoja1.B1:Hoja1.B1

A

Hoja1.A2:Hoja1.A6

1000

Hoja1.B2:Hoja1.B6

B

5000

C

7500

D

8000

E

9500

Hoja1RegionSales2006Sales2007A1000800B50004500C75009000D800012000E95009300&C&A&CPgina &PHoja1Hoja2&C&A&CPgina &PHoja3&C&A&CPgina &P

1.4.2. Histogramas

Un histograma es un tipo especial de diagrama de barras que se utiliza para variables cuali-tativas agrupadas en intervalos. Las dos propiedades que definen el histograma son:

1. Las bases de cada una de las barras se corresponden con los intervalos en los que hemosdividido el rango de valores de la variable continua.

2. El area de cada barra es proporcional a la frecuencia correspondiente a ese intervalo.

Dos observaciones adicionales: en primer lugar, puesto que los intervalos deben cubrir todo elrecorrido de la variable, en un histograma no hay espacio entre las barras. Y, como practicarecomendable, para que la visualizacion sea efectiva, no es conveniente utilizar un histogramacon mas de 10 o 12 intervalos como maximo.

En el caso de variables cuantitativas discretas, normalmente los intervalos se extienden a valoresintermedios (que la variable no puede alcanzar) para que no quede espacio entre las barrasdel histograma.

Los pasos para obtener el histograma son estos:

1. Si no nos lo dan hecho, debemos empezar determinar los intervalos. Para ello podemoslocalizar el valor maximo y el mnimo de los valores, restarlos y obtenemos el rango.

2. Dividimos el rango entre el numero de intervalos deseados, para obtener la longitud decada uno de los intervalos. Construimos los intervalos y la tabla de frecuencias corres-pondiente.

11

3. Calculamos la altura de cada barra, teniendo en cuenta que area=base altura, y queel area (no la altura!) es proporcional a la frecuencia. Por lo tanto podemos usar:

altura =frecuencia

base=

frecuencia del intervalo

anchura del intervalo.

para calcular la altura de cada una de las barras.

Pronto aprenderemos a obtener muchos de los resultados de la clase de hoy usando R. Comoaperitivo, aqu tenemos un de instrucciones para el ejemplo que hemos usado hoy.

12

setwd("C:/Users/Fernando/Documents/Dropbox/Clases/EstadisticaGradoBiologia/Esquemas/")datos

Captulo 2

Valores centrales y dispersion

2.1. La media aritmetica (y otros conceptos de media).

Contenido:

Definicion de la media aritmetica.

La media aritmetica a partir de una tabla de frecuencias.

2.1.1. Definicion de la media aritmetica.

La idea de media aritmetica apenas necesita presentacion. Dados n valores de una variablecuantitativa, sean x1, x2, . . . , xn, su media aritmetica es:

x =x1 + + xn

n=

ni=1

xi

n.

Algunos comentarios sobre la notacion. El smbolo x refleja la notacion establecida en Es-tadstica: la media de una variable se representa con una barra sobre el nombre de esa

variable. Y el smboloni=1

, que espero que ya conozcais, es un sumatorio, y representa en

forma abreviada, la frase suma todos estos valores xi donde i es un numero que va desde 1hasta n.

Insistimos en esto: la media aritmetica solo tiene sentido para variables cuantitati-vas (discretas o continuas). Aunque una variable cualitativa se represente numericamente,la media aritmetica de esos numeros seguramente sea una cantidad sin ningun significadoestadstico.

Las Hojas de calculo incluyen normalmente una funcion PROMEDIO() que calcula la mediaaritmetica de un rango de celdas de la hoja. Ademas suelen incluir otras funciones promedio,que suponen pequenas modificaciones sobre esta funcion, y que es bueno conocer.

Para practicar esto, en el se han recogido unos datos (ficticios) sobre lostamanos de los grupos de Grulla Comun que se han visto abandonando el dormidero de laLaguna de Gallocanta por la manana1. Cual es el tamano medio del grupo?

2.1.2. La media aritmetica a partir de una tabla de frecuencias.

Supongamos que queremos calcular la media a partir de la tabla de frecuencias de una variablecuantitativa, como esta:

1Si vives en Espana y no sabes de que va esto de las grullas de Gallocanta, creeme que te estas perdiendo algo

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Grulla comn (Grus Grus), Laguna de Gallocanta

Observacin n:

Tamao de la bandada

1

109

2

175

3

4

4

174

5

188

6

200

7

34

8

125

9

159

10

162

11

56

12

116

13

74

14

193

15

132

16

88

17

19

18

188

19

10

20

13

21

128

22

107

23

27

24

116

25

129

26

49

27

131

Tamao medio del grupo=

??

28

125

Nmero total de individuos

3031

???

Pgina

??? (???)

22/09/2011, 12:45:50

Pgina /

Intervalo Frecuencia

x1 f1

x2 f2

......

xk fk

Aqu los valores distintos de la variable2 son x1, . . . , xk y sus frecuencias absolutas respectivasson f1, f2, . . . , fk. Esta claro entonces que:

f1 + f2 + + fk = (nro. de observ. de xk) + + (nro. de observ. del valor xk) =

= (suma del numero de observaciones de todos los valores distintos) = n

Recordemos que para calcular la media tenemos que sumar el valor de todas (las n observa-ciones). Y como el valor xi se ha observado fi veces, su contribucion a la suma es

xi fi = xi + xi + + xi (sumamos fi veces)

Teniendo en cuenta entonces la contribucion cada uno de los k valores distintos, vemos quepara calcular la media debemos hacer:

x =x1 f1 + x2 f2 + + xk fk

f1 + f2 + + fk=

ki=1

xi fi

ki=1

fi

Ejemplo: en una instalacion deportiva el precio de la entrada para adultos es de 10e y de 4epara menores. Hoy han visitado esa instalacion 230 adultos y 45 menores. Cual es el ingresomedio por visitante que recibe esa instalacion?Tenemos dos posibles valores de la variable x =precio de la entrada, que son x1 = 10 yx2 = 4. Ademas sabemos las frecuencias correspondientes: f1 = 230 y f2 = 45. Por lo tanto:

x =x1 f1 + x2 f2

f1 + f2=

10 230 + 4 45230 + 45

= 9.02

El ingreso medio es de 9.02e por visitante.2

Si lo que queremos es calcular la media aritmetica a partir de la tabla de frecuencias agrupadaspor intervalos de una variable cuantitativa, las cosas son solo un poco mas complicadas. Eneste caso vamos a tener una tabla de frecuencias por intervalos3 como esta:

Intervalo Frecuencia

[a1, b1) f1

[a2, b2) f2

......

[ak, bk) fk

Comparando esta tabla con el caso anterior esta claro que lo que nos falta son los valoresx1, . . . , xk y, en su lugar, tenemos los intervalos [a1, b1), . . . , [ak, bk). Lo que hacemos en estoscasos es fabricar unos valores xi a partir de los intervalos. Se toma como valor xi el puntomedio del intervalo [ai, bi); es decir:

xi =ai + bi

2, para i = 1, . . . , n.

2Acuerdate de que tenemos n observaciones de la variable, pero puede haber valores repetidos. Aqu estamosusando el numero de valores distintos, sin repeticiones, y ese numero es k.

3los intervalos a veces se llaman tambien clases.

14

Estos valores xi se denominan marcas de clase (o marcas de intervalo). Una vez calculadaslas marcas de clase, podemos usar la misma formula que en el caso anterior.

Para practicar, aqu tienes un en el que aparecen datos sobre la altura (en pulga-das) de los estudiantes de un instituto americano. El fichero contiene la tabla de frecuenciasagrupadas por intervalo de la variable altura, y a partir de ella debemos calcular las marcasde clase y la media. Aqu tienes .

2.2. Otros valores centrales. Mediana, percentiles, moda.

Contenido:

Mediana.

Moda

2.2.1. Mediana.

Como en el caso de la media aritmetica, vamos a suponer que tenemos n observaciones deuna variable cuantitativa:

x1, x2, . . . , xn.

Antes de seguir: la variable ha de ser cuantitativa, y estamos suponiendo que los datos noestan agrupados en una tabla de frecuencia. Mas abajo veremos el caso de datos agrupados.

Como los xi son numeros, vamos a suponer que los hemos ordenado de menor a mayor:

x1 x2 xn1 xn.

Entonces, la mediana (ingles: median) de ese conjunto de datos es el valor central de esa serieordenada. Es decir:

Caso impar: si tenemos una cantidad impar de datos, solo hay un valor central, y esees la mediana. Por ejemplo, para siete datos:

x1 x2 x3 mitad izda.

x4 x5 x6 x7 mitad dcha.

mediana

Caso par: Por contra, si el numero de datos es par, entonces tomamos el valor maximode la mitad izquierda, y el valor mnimo de la mitad derecha y hacemos la media. Porejemplo, para seis datos:

x1 x2 x3 mitad izda.

x3 + x42

x4 x5 x6 mitad dcha.

mediana

En el caso de un numero impar de datos la mediana siempre coincide con uno de los datosoriginales. Pero en el caso de un numero par de datos la mediana pueden darse los dos casos.Por ejemplo, si tenemos estos seis datos ordenados:

2 5 6 7 11 15,

Entonces la mediana es 6.5

2 5 6 6.5 7 11 15,

que no apareca en el conjunto original (fjate en particular en que, como pasaba con lamedia aritmetica, aunque todos los datos originales sean enteros, la mediana puede no serlo).Mientras que si tenemos estos seis datos, con los dos datos centrales iguales:

2 5 6 6 11 15,

Entonces la mediana es 6, que ya estaba (repetido) entre los datos originales.

2 5 6 6 8 11 15,

15

ndice i

Ext. Inf. Intervalo

Ext. Sup. Intervalo

Marca de clase xi

Frecuencia fi

Productos xi*fi

1

60

62

5

0

2

63

65

18

0

3

66

68

42

0

4

69

71

27

0

5

72

74

8

0

Suma xi*fi=

Suma de frecuencias=

Media=

Ejemplo tomado de Estadstica 3a. Ed, (Serie Schaum), M.Spiegel, pg.73

???

Pgina

??? (???)

22/09/2011, 13:17:48

Pgina /

ndice i

Ext. Inf. Intervalo

Ext. Sup. Intervalo

Marca de clase xi

Frecuencia fi

Productos xi*fi

1

60

62

61

5

305

2

63

65

64

18

1152

3

66

68

67

42

2814

4

69

71

70

27

1890

5

72

74

73

8

584

Suma xi*fi=

6745

Suma de frecuencias=

100

Media=

67,45

Ejemplo tomado de Estadstica 3a. Ed, (Serie Schaum), M.Spiegel, pg.73

???

Pgina

??? (???)

22/09/2011, 13:16:37

Pgina /

Que ventajas aporta la mediana frente a la media aritmetica? Fundamentalmente, la medianase comporta mejor cuando el conjunto de datos contiene datos atpicos (ingles: outliers). Esdecir, datos cuyo valor se aleja mucho de la media. Todava no podemos precisar esto porquepara hacerlo necesitamos la nocion de medidas de dispersion que vamos a ver en la proximaseccion. Pero la idea intuitiva es que si tenemos un conjunto de datos, e introducimos un datoadicional que se aleja mucho de la media aritmetica inicial, entonces en el nuevo conjuntode datos podemos tener una media aritmetica bastante distinta de la inicial. En cambio lamediana sufre modificaciones mucho menores frente a esos datos atpicos.

Para observar este comportamiento podeis abrir el (se abre en el nave-gador, y requiere Java), y experimentar con el.

En el caso de que queramos calcular la mediana a partir de la tabla de frecuencias para los dis-tintos valores de una variable cuantitativa discreta, construimos la tabla de frecuencias relativasacumuladas (f.r.a.):

Intervalo Frecuencia F.r.a.

x1 f1 g1

x2 f2 g2

......

...

xk fk gk = 1

Que que es eso de las frecuencias relativas acumuladas? En definitiva, se trata de los tantospor uno acumulados: es decir, que para cada uno de los valores x1, . . . , xn vamos sumandolas frecuencias de ese valor y de todos los que le preceden. En formulas:

g1 =f1n, g2 =

f1 + f2n

, g3 =f1 + f2 + f3

n, etc.

En el puedes ver como se hace se calculo para ua tabla de frecuencias (devalores aleatorios). Cada vez que se abre el fichero genera un conjunto de datos distinto,as que puedes recargarlo varias veces para ver ejemplos variados.

Y si lo que necesitamos es calcular la mediana a partir de la tabla de frecuencias de una variablecuantitativa, pero agrupada en intervalos? En este caso las cosas se complican un poco. Sihemos entendido la idea de histograma, esta forma de verlo nos puede ayudar: la mediana esel valor de la variable (por lo tanto es el punto del eje horizontal) que divide el histogramaen dos mitades con el mismo area. Existen formulas para calcular la mediana en estos casos(usando interpolacion). Pero aqu no nos vamos a entretener.

2.2.2. Moda.

La media aritmetica y la mediana se utilizan para variables cuantitativas. La moda en cambiopuede utilizarse ademas con variables de tipo cualitativo (y es, de los que vamos a ver, elunico tipo de valor promedio que puede usarse con variables cualitativas). La moda de unaserie de valores agrupados en una tabla de frecuencias es el valor con la frecuencia mas alta.

Puesto que puede haber dos o mas valores que tengan la misma frecuencia, hay conjuntos dedatos que tienen mas de una moda.

El calculo de la moda (o modas) es inmediato a partir de las tablas de frecuencias. Aqu tienescon datos de viajeros del Titanic (tomado de los ejemplos que acompanan

al libro Estadstica Basica con R y RCommander de la Univ. de Cadiz). Para practicarpodeis calcular la moda de todas las variables que aparecen en el (aunque algunas modasson evidentes por simple inspeccion de los datos: la mayora de los viajeros eran adultos, porejemplo).

16

Media, mediana y datos atpicos

(Espera un poco a que termine de cargar el applet java...)Tenemos un conjunto de datos (los puntos verdes), e introducimos un dato adicional: el punto B de color naranja, que puedes ver marcando la casilla correspondiente. Tambin puedes ver las medias aritmticas (con y sin B) y las medianas (con y sin B) seleccionando las casillas adecuadas. Prueba a arrastrar B con el ratn y observa como cambian los valores. Cul es un mejor "valor central" frente a los atpicos, la media o la mediana?Puedes hacer zoom con la rueda del ratn mientras pulsas la tecla maysculas. Pulsa sobre el icono de reciclaje para obtener otro conjunto de datos.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Fernando San Segundo, Creacin realizada con GeoGebra

ndice

Valor

Frecuencia

Frecuencia acumulada

Frecuencia relativa acumulada

1

7

7

7

0,156

2

9

3

10

0,222

3

10

1

11

0,244

4

11

4

15

0,333

5

14

6

21

0,467

6

17

5

26

0,578

mediana

7

19

2

28

0,622

8

21

6

34

0,756

9

23

5

39

0,867

10

25

6

45

1,000

n=

45

???

Pgina

??? (???)

23/09/2011, 17:12:13

Pgina /

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"2nd" "Male" "Child" "Yes""1500" "2nd" "Male" "Child" "Yes""1501" "2nd" "Male" "Child" "Yes""1502" "2nd" "Male" "Child" "Yes""1503" "2nd" "Male" "Child" "Yes""1504" "2nd" "Male" "Child" "Yes""1505" "2nd" "Male" "Child" "Yes""1506" "2nd" "Male" "Child" "Yes""1507" "3rd" "Male" "Child" "Yes""1508" "3rd" "Male" "Child" "Yes""1509" "3rd" "Male" "Child" "Yes""1510" "3rd" "Male" "Child" "Yes""1511" "3rd" "Male" "Child" "Yes""1512" "3rd" "Male" "Child" "Yes""1513" "3rd" "Male" "Child" "Yes""1514" "3rd" "Male" "Child" "Yes""1515" "3rd" "Male" "Child" "Yes""1516" "3rd" "Male" "Child" "Yes""1517" "3rd" "Male" "Child" "Yes""1518" "3rd" "Male" "Child" "Yes""1519" "3rd" "Male" "Child" "Yes""1520" "1st" "Female" "Child" "Yes""1521" "2nd" "Female" "Child" "Yes""1522" "2nd" "Female" "Child" "Yes""1523" "2nd" "Female" "Child" "Yes""1524" "2nd" "Female" "Child" "Yes""1525" "2nd" "Female" "Child" "Yes""1526" "2nd" "Female" "Child" "Yes""1527" "2nd" "Female" "Child" "Yes""1528" "2nd" "Female" "Child" "Yes""1529" "2nd" "Female" "Child" "Yes""1530" "2nd" "Female" "Child" "Yes""1531" "2nd" "Female" "Child" "Yes""1532" "2nd" "Female" "Child" "Yes""1533" "2nd" "Female" "Child" "Yes""1534" "3rd" "Female" "Child" "Yes""1535" "3rd" "Female" "Child" "Yes""1536" "3rd" "Female" "Child" "Yes""1537" "3rd" "Female" "Child" "Yes""1538" "3rd" "Female" "Child" "Yes""1539" "3rd" "Female" "Child" "Yes""1540" "3rd" "Female" "Child" "Yes""1541" "3rd" "Female" "Child" "Yes""1542" "3rd" "Female" "Child" "Yes""1543" "3rd" "Female" "Child" "Yes""1544" "3rd" "Female" "Child" "Yes""1545" "3rd" "Female" "Child" "Yes""1546" "3rd" "Female" "Child" "Yes""1547" "3rd" "Female" "Child" "Yes""1548" "1st" "Male" "Adult" "Yes""1549" "1st" "Male" "Adult" "Yes""1550" "1st" "Male" "Adult" "Yes""1551" "1st" "Male" "Adult" "Yes""1552" "1st" "Male" "Adult" "Yes""1553" "1st" "Male" "Adult" "Yes""1554" "1st" "Male" "Adult" "Yes""1555" "1st" "Male" "Adult" "Yes""1556" "1st" "Male" "Adult" "Yes""1557" "1st" "Male" "Adult" "Yes""1558" "1st" "Male" "Adult" "Yes""1559" "1st" "Male" "Adult" "Yes""1560" "1st" "Male" "Adult" "Yes""1561" "1st" "Male" "Adult" "Yes""1562" "1st" "Male" "Adult" "Yes""1563" "1st" "Male" "Adult" "Yes""1564" "1st" "Male" "Adult" "Yes""1565" "1st" "Male" "Adult" "Yes""1566" "1st" "Male" "Adult" "Yes""1567" "1st" "Male" "Adult" "Yes""1568" "1st" "Male" "Adult" "Yes""1569" "1st" "Male" "Adult" "Yes""1570" "1st" "Male" "Adult" "Yes""1571" "1st" "Male" "Adult" "Yes""1572" "1st" "Male" "Adult" "Yes""1573" "1st" "Male" "Adult" "Yes""1574" "1st" "Male" "Adult" "Yes""1575" "1st" "Male" "Adult" "Yes""1576" "1st" "Male" "Adult" "Yes""1577" "1st" "Male" "Adult" "Yes""1578" "1st" "Male" "Adult" "Yes""1579" "1st" "Male" "Adult" "Yes""1580" "1st" "Male" "Adult" "Yes""1581" "1st" "Male" "Adult" "Yes""1582" "1st" "Male" "Adult" "Yes""1583" "1st" "Male" "Adult" "Yes""1584" "1st" "Male" "Adult" "Yes""1585" "1st" "Male" "Adult" "Yes""1586" "1st" "Male" "Adult" "Yes""1587" "1st" "Male" "Adult" "Yes""1588" "1st" "Male" "Adult" "Yes""1589" "1st" "Male" "Adult" "Yes""1590" "1st" "Male" "Adult" "Yes""1591" "1st" "Male" "Adult" "Yes""1592" "1st" "Male" "Adult" "Yes""1593" "1st" "Male" "Adult" "Yes""1594" "1st" "Male" "Adult" "Yes""1595" "1st" "Male" "Adult" "Yes""1596" "1st" "Male" "Adult" "Yes""1597" "1st" "Male" "Adult" "Yes""1598" "1st" "Male" "Adult" "Yes""1599" "1st" "Male" "Adult" "Yes""1600" "1st" "Male" "Adult" "Yes""1601" "1st" "Male" "Adult" "Yes""1602" "1st" "Male" "Adult" "Yes""1603" "1st" "Male" "Adult" "Yes""1604" "1st" "Male" "Adult" "Yes""1605" "2nd" "Male" "Adult" "Yes""1606" "2nd" "Male" "Adult" "Yes""1607" "2nd" "Male" "Adult" "Yes""1608" "2nd" "Male" "Adult" "Yes""1609" "2nd" "Male" "Adult" "Yes""1610" "2nd" "Male" "Adult" "Yes""1611" "2nd" "Male" "Adult" "Yes""1612" "2nd" "Male" "Adult" "Yes""1613" "2nd" "Male" "Adult" "Yes""1614" "2nd" "Male" "Adult" "Yes""1615" "2nd" "Male" "Adult" "Yes""1616" "2nd" "Male" "Adult" "Yes""1617" "2nd" "Male" "Adult" "Yes""1618" "2nd" "Male" "Adult" "Yes""1619" "3rd" "Male" "Adult" "Yes""1620" "3rd" "Male" "Adult" "Yes""1621" "3rd" "Male" "Adult" "Yes""1622" "3rd" "Male" "Adult" "Yes""1623" "3rd" "Male" "Adult" "Yes""1624" "3rd" "Male" "Adult" "Yes""1625" "3rd" "Male" "Adult" "Yes""1626" "3rd" "Male" "Adult" "Yes""1627" "3rd" "Male" "Adult" "Yes""1628" "3rd" "Male" "Adult" "Yes""1629" "3rd" "Male" "Adult" "Yes""1630" "3rd" "Male" "Adult" "Yes""1631" "3rd" "Male" "Adult" "Yes""1632" "3rd" "Male" "Adult" "Yes""1633" "3rd" "Male" "Adult" "Yes""1634" "3rd" "Male" "Adult" "Yes""1635" "3rd" "Male" "Adult" "Yes""1636" "3rd" "Male" "Adult" "Yes""1637" "3rd" "Male" "Adult" "Yes""1638" "3rd" "Male" "Adult" "Yes""1639" "3rd" "Male" "Adult" "Yes""1640" "3rd" "Male" "Adult" "Yes""1641" "3rd" "Male" "Adult" "Yes""1642" "3rd" "Male" "Adult" "Yes""1643" "3rd" "Male" "Adult" "Yes""1644" "3rd" "Male" "Adult" "Yes""1645" "3rd" "Male" "Adult" "Yes""1646" "3rd" "Male" "Adult" "Yes""1647" "3rd" "Male" "Adult" "Yes""1648" "3rd" "Male" "Adult" "Yes""1649" "3rd" "Male" "Adult" "Yes""1650" "3rd" "Male" "Adult" "Yes""1651" "3rd" "Male" "Adult" "Yes""1652" "3rd" "Male" "Adult" "Yes""1653" "3rd" "Male" "Adult" "Yes""1654" "3rd" "Male" "Adult" "Yes""1655" "3rd" "Male" "Adult" "Yes""1656" "3rd" "Male" "Adult" "Yes""1657" "3rd" "Male" "Adult" "Yes""1658" "3rd" "Male" "Adult" "Yes""1659" "3rd" "Male" "Adult" "Yes""1660" "3rd" "Male" "Adult" "Yes""1661" "3rd" "Male" "Adult" "Yes""1662" "3rd" "Male" "Adult" "Yes""1663" "3rd" "Male" "Adult" "Yes""1664" "3rd" "Male" "Adult" "Yes""1665" "3rd" "Male" "Adult" "Yes""1666" "3rd" "Male" "Adult" "Yes""1667" "3rd" "Male" "Adult" "Yes""1668" "3rd" "Male" "Adult" "Yes""1669" "3rd" "Male" "Adult" "Yes""1670" "3rd" "Male" "Adult" "Yes""1671" "3rd" "Male" "Adult" "Yes""1672" "3rd" "Male" "Adult" "Yes""1673" "3rd" "Male" "Adult" "Yes""1674" "3rd" "Male" "Adult" "Yes""1675" "3rd" "Male" "Adult" "Yes""1676" "3rd" "Male" "Adult" "Yes""1677" "3rd" "Male" "Adult" "Yes""1678" "3rd" "Male" "Adult" "Yes""1679" "3rd" "Male" "Adult" "Yes""1680" "3rd" "Male" "Adult" "Yes""1681" "3rd" "Male" "Adult" "Yes""1682" "3rd" "Male" "Adult" "Yes""1683" "3rd" "Male" "Adult" "Yes""1684" "3rd" "Male" "Adult" "Yes""1685" "3rd" "Male" "Adult" "Yes""1686" "3rd" "Male" "Adult" "Yes""1687" "3rd" "Male" "Adult" "Yes""1688" "3rd" "Male" "Adult" "Yes""1689" "3rd" "Male" "Adult" "Yes""1690" "3rd" "Male" "Adult" "Yes""1691" "3rd" "Male" "Adult" "Yes""1692" "3rd" "Male" "Adult" "Yes""1693" "3rd" "Male" "Adult" "Yes""1694" "Crew" "Male" "Adult" "Yes""1695" "Crew" "Male" "Adult" "Yes""1696" "Crew" "Male" "Adult" "Yes""1697" "Crew" "Male" "Adult" "Yes""1698" "Crew" "Male" "Adult" "Yes""1699" "Crew" "Male" "Adult" "Yes""1700" "Crew" "Male" "Adult" "Yes""1701" "Crew" "Male" "Adult" "Yes""1702" "Crew" "Male" "Adult" "Yes""1703" "Crew" "Male" "Adult" "Yes""1704" "Crew" "Male" "Adult" "Yes""1705" "Crew" "Male" "Adult" "Yes""1706" "Crew" "Male" "Adult" "Yes""1707" "Crew" "Male" "Adult" "Yes""1708" "Crew" "Male" "Adult" "Yes""1709" "Crew" "Male" "Adult" "Yes""1710" "Crew" "Male" "Adult" "Yes""1711" "Crew" "Male" "Adult" "Yes""1712" "Crew" "Male" "Adult" "Yes""1713" "Crew" "Male" "Adult" "Yes""1714" "Crew" "Male" "Adult" "Yes""1715" "Crew" "Male" "Adult" "Yes""1716" "Crew" "Male" "Adult" "Yes""1717" "Crew" "Male" "Adult" "Yes""1718" "Crew" "Male" "Adult" "Yes""1719" "Crew" "Male" "Adult" "Yes""1720" "Crew" "Male" "Adult" "Yes""1721" "Crew" "Male" "Adult" "Yes""1722" "Crew" "Male" "Adult" "Yes""1723" "Crew" "Male" "Adult" "Yes""1724" "Crew" "Male" "Adult" "Yes""1725" "Crew" "Male" "Adult" "Yes""1726" "Crew" "Male" "Adult" "Yes""1727" "Crew" "Male" "Adult" "Yes""1728" "Crew" "Male" "Adult" "Yes""1729" "Crew" "Male" "Adult" "Yes""1730" "Crew" "Male" "Adult" "Yes""1731" "Crew" "Male" "Adult" "Yes""1732" "Crew" "Male" "Adult" "Yes""1733" "Crew" "Male" "Adult" "Yes""1734" "Crew" "Male" "Adult" "Yes""1735" "Crew" "Male" "Adult" "Yes""1736" "Crew" "Male" "Adult" "Yes""1737" "Crew" "Male" "Adult" "Yes""1738" "Crew" "Male" "Adult" "Yes""1739" "Crew" "Male" "Adult" "Yes""1740" "Crew" "Male" "Adult" "Yes""1741" "Crew" "Male" "Adult" "Yes""1742" "Crew" "Male" "Adult" "Yes""1743" "Crew" "Male" "Adult" "Yes""1744" "Crew" "Male" "Adult" "Yes""1745" "Crew" "Male" "Adult" "Yes""1746" "Crew" "Male" "Adult" "Yes""1747" "Crew" "Male" "Adult" "Yes""1748" "Crew" "Male" "Adult" "Yes""1749" "Crew" "Male" "Adult" "Yes""1750" "Crew" "Mal