5.1 Estadística descriptiva variables cuantitativas

Posgrados CUCS CUCS/UDG

Instituto de Investigación en Ciencias Biomédicas
Universidad de Guadalajara

2023-09-15

Introducción

flowchart TD
  A[Estadísitica descriptiva] --> B(Variables cuantitativas)
  B --> C[Medidas de tendencia central]
  B --> D[Medidas de dispersión]
  B --> E[Medidas de posición]
  B --> F[Gráficos]
  C --> G[Media \n Mediana \n Moda]
  D --> H[Rango \n Varianza \n Desviación estándar \n IQR...]
  E --> I[Percentiles \n cuartiles \n deciles...]
  F --> J[Histogramas \n boxplot \n densidad \n violín \n dispersión]

Introducción

• Medidas de tendencia central
  • Media
  • Mediana
  • Moda
• Medidas de dispersión
  • Varianza y desviación estándar
  • Coeficiente de variación
  • Error estándar
  • Rango intercuartil

Introducción

• Medidas de posición
  • Cuaratiles
  • Deciles
  • Percentiles

Edad de pacientes

ID	EDAD	ID	EDAD	ID	EDAD
Paciente1	61	Paciente21	53	Paciente41	58
Paciente2	46	Paciente22	50	Paciente42	46
Paciente3	66	Paciente23	54	Paciente43	52
Paciente4	42	Paciente24	64	Paciente44	47
Paciente5	89	Paciente25	73	Paciente45	54
Paciente6	63	Paciente26	61	Paciente46	59
Paciente7	49	Paciente27	53	Paciente47	61

¿Cuál es la mejor manera de describir estos resultados?

Descripción de datos cuantitativos

• En la descripción de variable cuantitativas se debe hacer:
  • Tabulación de los datos. Dividir la variable, obtener intervalos de clase y realizar una tabla de frecuencias
  • Hacer una representación gráfica (histograma, poligono de frecuencias etc.)
  • Cálculo de parámetros de centralización (medidas de tendencia central)
  • Cálculo de parámetros de dispersión (coeficiente de variación, rango etc.)
  • Cálculo de parámetros de posición (percentiles, cuantiles, etc.)

Medidas de tendencia central

Cálculo de medidas de tendencia central

• Es necesario conocer otros parámetros que informen de la tendencia central de los datos
  • Media arimetica
  • Media goemétrica
  • Media armónica
  • Media cuadratica
  • Mediana
  • Moda
• Las unidades de medida de todos los parámetros de tendecnia central son las mismas que la de los datos sobre los que se calculan

Media aritmética

Se refiere al valor que tendría cada elemento de la serie de datos si todos tuvieran el mismo valor

• Es el parámetro de centralización más utilizado, su valor es el centro aritmético de los datos
• Se suele emplear el simbolo \(\mu\) para la media poblacional
• Se emplea \(\bar{x}\) muestral
• Su formula es:
  • \(\bar{x}=\displaystyle\sum_{i=1}^n \frac{x_i}{n}\)

Media aritmética

• Su formula es:
  • \(\bar{x}=\displaystyle\sum_{i=1}^n \frac{x_i}{n}\)
• Donde: \({x_i}\) es el i-ésimo dato; si la suma es desde \(i\) es igual a 1 hasta \(n\)

La media aritmética de los datos: 2,4,6,8 y 9 es:

\(\bar{x}=\displaystyle\sum_{i=1}^n \frac{x_i}{n}=\frac{(2+4+6+8+9)}{5}=5.8\)

Media aritmética en \(R\)

En \(R\) para calcular la media se utiliza la función:

mean()

Dado el conjunto de datos estime la media en la consola de R:

18	22	24	26	27	29	30	32	37	40	43	47
19	23	24	26	27	29	31	33	37	40	43	48
20	23	25	26	28	29	24	26	27	29	30	40

Media aritmética en \(R\)

En \(R\) para calcular la media se utiliza la función: 1. Crear un objeto con los datos

x<-c(18,22,24,26,27,29,30,32,37,40,43,47,19,23,24,
     26,27,29,31,33,37,40,43,48,20,23,25,26,28,29)

2. Estimar la media del objeto

mean(x)

[1] 30.2

Media aritmética en \(R\) con NA

En R para calcular la media se utiliza la función: 1. Crear un objeto con los datos

x<-c(18,22,24,26,27,29,30,32,37,40,43,47,19,23,24,
     26,27,29,31,33,37,40,43,48,20,23,25,26,28,29, NA)

2. Estimar la media del objeto

mean(x)

[1] NA

Media aritmética en \(R\) con NA

En R para calcular la media se utiliza la función: 1. Crear un objeto con los datos

x<-c(18,22,24,26,27,29,30,32,37,40,43,47,19,23,24,
     26,27,29,31,33,37,40,43,48,20,23,25,26,28,29, NA)

2. Estimar la media del objeto

mean(x, na.rm = T)

[1] 30.2

Media aritmética ponderada

• En algunas ocasiones no todos los datos de una serie tienen la misma importancia por lo que se hace una Ponderacion
• Si se tiene un conjunto de datos \({x_1}, {x_2},{x_3},...,{x_n}\) y cada uno de ellos tiene los pesos: \({k_1}, {k_2},{k_3},...,{k_n}\) la media aritmética ponderada se puede calcular:

\(\bar{x_p}= \frac {\displaystyle\sum_{i=1}^n {k_i}{x_i}}{\displaystyle\sum_{i=1}^n {k_i}} = \frac{{{k_1}{k_1}}+{{k_2}{k_2}}...{{k_n}{k_in}}}{{k_1}+{k_2}...{k_n}}\)

Media aritmética ponderada. Ejemplo

• En la evaluación de un servicio sanitario, han sido calculados tres índices. La evaluación total del servicio se obtiene calculando la media ponderada de los índices yq que no tienen el mismo valor. Los pesos asignados son 3 al primer índice, 5 al segundo y 9 al tercero

• Los datos que se obtuvieron son los siguientes:

Primer índice	7
Segundo índice	8
Tercer índice	7

Media aritmética ponderada. Ejemplo

Primer índice	7
Segundo índice	8
Tercer índice	7

• \(\bar{x_p}= \frac{(3x7+5x8+9x7)}{3+5+9}=7.29\)

Media geométrica

• La media geométrica de un conjunto de datos de \(n\) datos se calcula obteniendo la ríaz enésima del producto de todos los datos:
  • \({\bar{x_G}}=\sqrt[n]{x_1}{x_2}...{x_n}\)
• En la expresión anterior \(n\) debe ser igual a la suma de todas la frecuencias

Media geométrica. Ejemplo

Calcular la media geométrica de los datos siguientes: 4, 5, 6, 8, 9, 12

• \({\bar{x_G}}=\sqrt[6] (4 \cdot\ 5\cdot\ 6\cdot\ 8\cdot\ 9\cdot\ 12) =6.85\)

La media geométrica sólo es preferible a la aritmética en los casos que se presentan progresión geométricas. por ejemplo (cromatografía líquidos, citometría)

Otras medias

• Media armónica
  • Es al inversa de la media aritmética de los inversos de una sere de datos. Se calcula mediante la siguiente expresión:
  • \({\bar{X_a}}=\frac{n}{\sum_{i=1}^n \frac{1}{x_i}}\)
  • útil para el caso de parámetros como velocidades
• Media cuadrática
  • Es la raíz cuadrada de media aritmética
  • útil para promediar series de números al cuadrado

Mediana

• Es el valor central de un conjunto de datos de \(n\) datos ordenados de menor a mayor
• Divde al conjunto de datos ordenados en dos partes iguales
• Cuando se trabaja con una \(n\) impar la formula es la siguiente:
  • \(M= \frac{X(n+1)}{2}\)
• Si \(n\) es par, al mediana es la media aritmética de los dos valores centrales:
  • \(M=\frac{{X_{\frac{n}{2}}}+{X_{\frac{n}{2}+1}}}{2}\)

Mediana. Ejemplo

• Calcular la mediana de los conjuntos de datos siguientes: 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 20
  • Es un número impar por lo tanto la mediana es: 10
• Calcular la mediana de: 3, 6, 8, 12, 17, 38, 32, 34
  • Número par de datos, se toma el promedio. \(M=14.5\)

Mediana. Cálculo en R

• En R se utiliza la función

median()

median(c(2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 20))

[1] 10

median(c(3, 6, 8, 12, 17, 38, 32, 34))

[1] 14.5

Moda

• La moda de un conjunto de datos es el valor que más veces se repite.
• La moda absoluta es el valor que más veces se repite
• La moda relativa es le valor que sin ser el que más veces se repite, se repite más veces que el resto de los datos.

Moda. Ejemplo

• En el siguiente conjunto de datos: 2, 2, 2, 3, 7, 8, 9, 11, 11, 11, 11, 34, 56, 78.
• Identifique:
  • Moda absoluta
  • Moda relativa

Moda. Ejemplo

• En el siguiente conjunto de datos: 2, 2, 2, 3, 7, 8, 9, 11, 11, 11, 11, 34, 56, 78.
• Identifique:
  • Moda absoluta= 11
  • Moda relativa= 2

Propiedades de la media

1. Unicidad. Para un conjunto determinado de datos, sólo existe una media aritmética.

2. Simplicidad. La media aritmética es fácil de comprender y calcular.

3. Todos los valores en la serie de datos se utilizan para su cálculo. Por ello, los valores extremos pueden sesgar el resultado.

4. Se puede estimar una media de varios grupos.

Propiedades de la mediana

1. Única.

2. Simple.

3. Los valores extremos no le afectan como a la media.

4. Divide al grupo de valores en dos partes iguales, cada una con el 50% de las observaciones.

Propiedades de la mediana

Sus desventajas en relación con el promedio son:

1. Desprecia información, porque sólo considera los valores de 1 o 2 observaciones.

2. Cuando dos o más grupos se unen en uno solo, no es posible calcularla a partir de la mediana de cada grupo.

Medidas de dispersión

Medidas de dispersión

• Ofrecen información sobre el grado de variabilidad de una variable
• Indican si una variable tiene datos más dispersos (variación) que otra

Medidas de dispersión

• Junto con las medidas de tendencia central y dispersión son las medidas que se utilizan para presentar una variable cuantitativa.
• Las importantes son:
  • rango
  • Varianza
  • Desviación estándar
  • Coeficiente de variación

Rango

• Es la diferencia entre el valor máximo y el valor mínimo de los datos observados.
• Aporta información sobre el recorrido de una variable
• Puede ser engañosa con datos extremos
• Nunca se debe evaluar solo, se necesitan de otras medidas de dispersión

Rango. Ejemplo

En la medidas de presión arterial sistólica en milímetros de mercurio en un grupo de pacientes se obtiene los siguientes resultados: 120, 135, 160, 100, 155, 115, 165, 125, 130. - Calcular el rango:

• \(Máximo=165\)
• \(Mínimo=100\)
• \(Rango= 165-100=65\)

Estimación del rango en R

• La siguiente tabla contiene las edades de un grupo de pacientes:

18	22	24	26	27	29	30	32	37	40	43	47
19	23	24	26	27	29	31	33	37	40	43	48
20	23	25	26	28	29	24	26	27	29	30	40

• En R el rango se estima con la función:

range()

Estimación del rango en R

1. Crear un objeto llamado x

x<-c(18,22,24,26,27,29,30,32,37,40,
     43,47,19,23,24,26,27,29,31,33,
     37,40,43,48,20,23,25,26,28,29)

2. Calcular el rango

range(x)

[1] 18 48

Desviación media

• \({D_m}=\displaystyle\sum_{i=1}^n \frac{|{X_i}-\bar{X}|}{n}\)

• Características:

  • Da buena información de la desviación con respecto a la media
  • Se eliminan valores negativos
  • El valor absoluto dificulta su trabajo

Desviación media. Ejemplo

• Las tallas en centímetros de un grupo de personas se detallan a continuación: 180, 165, 160, 175

\({D_m}=\displaystyle\sum_{i=1}^n \frac{|{X_i}-\bar{X}|}{n}\)

• \(\bar{x}=170\)
• \({D_m}= 7.5\)

Desviación media. Ejemplo

• Las tallas en centímetros de un grupo de personas se detallan a continuación: 180, 165, 160, 175

\({D_m}=\frac{|180-170|+|165-170|+|160-170|+|175-170|}{4}\)

• \(\bar{x}=170\)
• \({D_m}= 7.5\)

Desviación media en R. Ejemplo

• Las tallas en centímetros de un grupo de personas se detallan a continuación: 180, 165, 160, 175
• \({D_m}=\displaystyle\sum_{i=1}^n \frac{|{X_i}-\bar{X}|}{n}\)

y<-c(180,165,160,175)#un objeto con los datos
z<-mean(y)# un objeto de la media de los datos
sum(abs(y-z))/length(y)

[1] 7.5

Varianza

• Es el promedio de las diferencias cuadráticas de los datos respecto a la media
• Sus unidades son las de los datos al cuadrado
• Se representa por:\(\sigma^2\)
• Para una población su formula es:
  • \(\sigma^2 = \displaystyle\sum_{i=1}^{n}\frac{(x_i - \mu)^2} {N}\)

Varianza

• Se representa por:\(S^2\)
• Para una muestra su formula es:
  • \(\sigma^2 = \displaystyle\sum_{i=1}^{n}\frac{(x_i - \bar{x})^2} {n}\)
• Cuando se utiliza como estimador población se utiliza \(n-1\):
  • \(\sigma^2_{n-1} = \displaystyle\sum_{i=1}^{n}\frac{(x_i - \bar{x})^2} {n-1}\)

Varianza en R

1. Crear un objeto llamado x

x<-c(18,22,24,26,27,29,30,32,37,40,
     43,47,19,23,24,26,27,29,31,33,
     37,40,43,48,20,23,25,26,28,29)

2. Calcular la varianza

var(x)

[1] 67.54483

Desviación estándar

• Es la raíz cuadrada de la varianza
  • \(\sigma=\sqrt{\displaystyle\sum_{i=1}^{n}\frac{(x_i - \mu)^2} {N}}\)
• Más fácil de manejar que la varianza
• Mide que tanto se desvían los datos con respecto a su media
• Mientras mayor sea la desviación estándar, mayor será la dispersión de los datos

Desviación estándar R

1. Crear un objeto llamado x

x<-c(18,22,24,26,27,29,30,32,37,40,
     43,47,19,23,24,26,27,29,31,33,
     37,40,43,48,20,23,25,26,28,29)

2. Calcular la varianza

sd(x)

[1] 8.218566

Coeficiente de variación

• Es una medida de dispersión sin unidades y es el cociente de la desviación estándar respecto a la media, multiplicado por 100 \(CV=\frac{S}{\bar{X}} 100\)
• Permite la comparación de datos en distintas unidades y en distintas poblaciones o muestras

Coeficiente de variación en R

1. Crear un objeto llamado x

x<-c(18,22,24,26,27,29,30,32,37,40,
     43,47,19,23,24,26,27,29,31,33,
     37,40,43,48,20,23,25,26,28,29)

2. Calcular el coeficiente de variación

(sd(x)/mean(x))*100

[1] 27.21379

Coeficiente de variación en R creando una función

Es necesario crear una función

CV<-function(x){
  (sd(x)/mean(x))*100
}
x<-c(18,22,24,26,27,29,30,32,37,40,
     43,47,19,23,24,26,27,29,31,33,
     37,40,43,48,20,23,25,26,28,29)
CV(x)

[1] 27.21379

Propiedades de la varianza y la desviación típica

1. Son mayores o iguales a cero

2. A mayor dispersión de los datos mayor varianza y mayor desviación estándar

Error estándar

El error estándar, es una medida que se utiliza para conocer que tanto varían un estadístico de una muestra a otra. El error estándar tiende a disminuir cuando aumenta el tamaño de la(s) muestra(s). Se puede establecer por lo tanto que cuanto menor sea el error estándar, más representativa será la muestra de la población en su conjunto

Error Estándar

• La variabilidad de las medias muestrales se puede medir por su desviación estándar. Esta medida se conoce como el error estándar y tiende a disminuir cuando aumenta el tamaño de la(s) muestra(s).

\(SE=\frac{\sigma}{\sqrt{N}}\)

\(SE=\frac{S}{\sqrt{n}}\)

• El error estándar no disminuye en relación directamente proporcional con el tamaño de la muestra. Ya que tomamos la raíz cuadrada de N

Medidas de posición

Medidas de posición

Medidas de posición

\(n-\)tiles

• Los \(n-\)tiles dividen al conjunto de datos en un número determinado de grupos con el mismo número de datos cada uno de ellos.
• El término \(n-\)tiles se refiere a \(n\) grupos:
  • Los Terciles dividen a un conjunto de datos en 3
  • Los cuartiles lo dividen en 4 partes iguales
  • Los quintiles en 5
• Los más importantes son los cuartiles, los deciles y percentiles

Cuartiles

• Dividen al conjunto de datos en cuatro partes iguales, en cada una de ella hay 25% de los datos

25%	\(Q_1\)	25%	\(Q_2\)	25%	\(Q_3\)	25%

Deciles

• Se consideran como medidas de dispersión o de posición, las cuales dividen un conjunto de datos en 10 partes iguales en cuanto al número de datos.

10%		10%		10%		10%		10%		10%		10%		10%		10%
	\(D_1\)		\(D_2\)		\(D_3\)		\(D_4\)		\(D_5\)		\(D_6\)		\(D_7\)		\(D_8\)		\(D9\)
10%		20%		30%		40%		50%		60%		70%		80%		90%

Deciles

• Se consideran como medidas de dispersión o de posición, las cuales dividen un conjunto de datos en 10 partes iguales en cuanto al número de datos

Deciles

Cuartil	Decil	Percentil
		10	10%
		20	20%
	Q1		25%
		30	30%
		40	40%
Mediana	Q2	50	50%

¿Cómo calcularlos?

• Hay muchas formas de calcularlos

  • \(P(n +1)\) es la que utilizan normalmente los programas estadísticos
  • Donde:
    • \(P=\) al percentil que se desea calcular divido entre 100
    • \(n=\) es el número de datos

¿Cómo calcularlos?

• Dado el siguiente conjunto de datos calcule el percentil 25
  • 2, 3, 5, 6, 7, 8, 9 y 10
• \(P(n +1)\) Sustituyendo \(0.25(8+1)=2.25\)

¿Cómo calcularlos?

• El primer cuartil está situado entre el segundo y tercer dato

• Dado que la distancia entre el segundo (3) y el tercer dato (5) es de dos, 0.25 corresponde a 0.5

• El primer cuartil es 3.5

¿Cómo calcularlos?

• Dado el siguiente conjunto de datos calcule el percentil 25
  • 22, 43, 65, 76, 87, 98, 109 y 210
• Determine:
  • Percentil 80
  • Percentil 50
  • Percentil 14

¿Cómo calcularlos?

• Percentil 80: \(0.80(8+1)=7.2\)
  • Corresponde a un valor entre el valor 7 y 8 a una distancia de 0.2
  • \(210-109=101\)
  • por lo tanto \(0.2*101=20.2\)
  • por lo tanto el percentil 80 corresponde: \(109+20.2=129.2\)
• Percentil 50: \(0.50(8+1)=4.5\)
• Percentil 14: \(0.14(8+1)=1.26\)

¿Cómo calcularlo en R?

• Crear un objeto

x=c(22, 43, 65, 76, 87, 98, 109, 210)

• Utilice la función:

quantile()

• Calcule

quantile(x, 0.80)

  80% 
104.6 

¿Cómo calcularlo en R?

• R tiene varios algoritmos para calcular los percentiles
• Utilizando el argumento type se pueden cambias

quantile(x, 0.80, type=6)

  80% 
129.2 

# El tipo 6 fue el que utilizamos en la clase 
# Para estimar el percentil manualmente

https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/quantile

Discrepancias en el cálculo de los percentiles

• No existe unanimidad para el cálculo de los percentiles
• Es común que el cálculo de los percentiles no coincida
• Otra manera de calcularlos es como sigue:
  • \(P(n-1)+1\)
  • Dividir la \(n\) entre el percentil que se desea buscar
  • Para el ejemplo anterior:
    • \(25/8=3.12\) se busca el dato de la posición \(3 \sim 4\). El dato de la posición 4 corresponde a \(76\)

Calcular varios percentiles en R

• Utilizando el objeto x

quantile(x, c(0.10,0.30,0.40,0.50,0.75, 0.99), type=6)

Calcular varios percentiles en R

• Utilizando el objeto x

quantile(x, c(0.10,0.30,0.40,0.50,0.75, 0.99), type=6)

   10%    30%    40%    50%    75%    99% 
 22.00  58.40  71.60  81.50 106.25 210.00 

Calcular cuartiles en R

• Utilizar la función:

quantile(x)

• Con el objeto x

quantile(x)

    0%    25%    50%    75%   100% 
 22.00  59.50  81.50 100.75 210.00 

De ahora en adelante no cambiaremos el argumento type

Recorrido intercuartil

• El rango intercuartílico IQR (o rango intercuartil) es una estimación estadística de la dispersión de una distribución de datos.
• Consiste en la diferencia entre el tercer y el primer cuartil. Mediante esta medida se eliminan los valores extremadamente alejados.
• El rango intercuartílico es altamente recomendable cuando la medida de tendencia central utilizada es la mediana (ya que este estadístico es insensible a posibles irregularidades en los extremos).

Recorrido intercuartil

Recorrido intercuartil

• Se calcula de la siguiente manera:

\(IQR=Q_3-Q_1\)

• Por ejemplo, para el conjunto de datos x
  • Obtenemos el \(Q_3=\) 106.25
  • Obtenemos el \(Q_1=\) 48.5
  • Realizamos la resta \(Q_3-Q_1=\) 57.75

Recorrido intercuartil en R

• Utilice la función:

IQR(x)

• Con los datos del objeto x

IQR(x)

[1] 41.25

Algunos gráficos

Representación gráfica

Representación gráfica

La representación gráfica de variables cuantitativas puede ser muy variada. Los gráficos que más se utilizan son:

• Histograma de frecuencias
• Histograma de frecuencias acumulado
• Polígono de frecuencias
• Polígono de frecuencias acumuladas
• Gráfico de caja y bigotes

Histograma de frecuencias

Dado el siguiente conjunto de datos realice un histograma en la consola de \(R\)

18	22	24	26	27	29	30	32	37	40	43	47
19	23	24	26	27	29	31	33	37	40	43	48
20	23	25	26	28	29	24	26	27	29	30	40

Histograma de frecuencias

Pasos 1. Cree un objeto

set.seed(1234)
x<- sample(15:45, 200, replace = T) #Creación de un objeto (paso 1)

2. Utilice la función:

  hist()

Histograma de frecuencias

#Creación de un objeto (paso 1)
hist(x, main="Mi primer histograma", col = "gray",
     ylab = "Frecuencias", 
     xlab = "Intervalos de clase")

Histograma de frecuencias

¿Qué es un histograma?

• Se usan comúnmente para visualizar variables numéricas.
• Un histograma es similar a un gráfico de barras después de que los valores de la variable se agrupan en un número finito de intervalos (bins).
• Para cada intervalo, la altura de la barra corresponde a la frecuencia (recuento) de observación en ese intervalo.

Histograma

Histograma con densidad

Densidad

• La densidad es la frecuencia relativa para un intervalo unitario. Se obtiene dividiendo la frecuencia relativa por el ancho del intervalo:

\[f _{c} = p_{c} /w_{c}\]

Densidad

\[f _{c} = p_{c} /w_{c}\]

• En donde:

  • \(p_{c}=n_{c}/n\) Es decir, la frecuencia relativa con \(n_{c}\) como la frecuencia de intervalo \(c\).
  • \(n\): es el tamaño de la muestra
  • El ancho del intervalo \(c\) se denota como \(w_{c}\)

Histograma de densidad en R

# Es necesario utilizar el objeto x, creado anteriormente
hist(x, freq=F, main="Mi primer histograma", col = "gray",
     ylab = "Frecuencias", 
     xlab = "Intervalos de clase")

Histograma de densidad en R

Formas de un histograma

Formas de un histograma

Formas de un histograma

¿Cuántos intervalos de clase?

Es conveniente tener presente las motivaciones para agrupar lo datos en intervalos de clase. Las más importantes son:

• Claridad en la descripción
• Facilidad de manipulación

Numero de datos	Intervalos de clase
20-50	7
50-70	10
70-100	12
más de 100	15

Gráfico de tallo

  The decimal point is 1 digit(s) to the right of the |

  1 | 89
  2 | 023344
  2 | 5666778999
  3 | 0123
  3 | 77
  4 | 0033
  4 | 78

¿Cómo interpretar un gráfico de tallo y de hojas?

  1 | 256                 # <-- 12, 15, 16
  2 | 149                 # <-- 21, 24, 29
  3 | 0123                # <-- 30, 31, 32, 33
  4 | 569                 # <-- 45, 46, 49
  5 | 028                 # <-- 50, 52, 58
  6 | 0345                # <-- 60, 63, 64, 65

gráfico de tallo

x<-c(18,22,24,26,27,29,30,32,37,40,
     43,47,19,23,24,26,27,29,31,33,
     37,40,43,48,20,23,25,26,28,29)
stem(x)

  The decimal point is 1 digit(s) to the right of the |

  1 | 89
  2 | 023344
  2 | 5666778999
  3 | 0123
  3 | 77
  4 | 0033
  4 | 78

Boxplot

Boxplot

Boxplot

Boxplot

Boxplot con datos atípicos

Valores atípicos

• Un valor atípico es una observación extrañamente grande o pequeña.
• Los valores atípicos pueden tener un efecto desproporcionado en los resultados estadísticos, como la media, lo que puede conducir a interpretaciones engañosas.
• Estos valores atípicos son observaciones que están a por lo menos 1.5 veces el rango intercuartil (Q3 – Q1) del borde de la caja.

Valores atípicos

• En R:
  • upper whisker = min(max(x), Q_3 + 1.5 * IQR)
  • lower whisker = max(min(x), Q_1 – 1.5 * IQR)

¿Cómo hacer un boxplot en R?

• Utilice la función

boxplot()

• Para el objeto x

boxplot(x, main="Sustituir por el título", ylab="Nombre del eje de las y",
        xlab="Nombre del eje de las x")

¿Cómo hacer un boxplot en R?ç

boxplot(x, main="Sustituir por el título", ylab="Nombre del eje de las y",
        xlab="Nombre del eje de las x")

• Interprete

• ¿A partir de que números se consideraría un valor extremo?

¿Cómo hacer un boxplot en R?

Gráfico de violín

Este tipo de gráfico combina elementos de un gráfico de caja (box plot) y un gráfico de densidad de probabilidad.

• Forma de violín: La característica más distintiva de un gráfico de violín es su forma de “violín”, que se forma alrededor de la distribución de los datos.
• Marcas de dispersión: Dentro del violín, a menudo se incluyen marcas o puntos que indican la posición de los valores individuales del conjunto de datos.

Gráfico de violín

Este tipo de gráfico combina elementos de un gráfico de caja (box plot) y un gráfico de densidad de probabilidad.

• Línea central: En el centro del violín, a menudo se dibuja una línea o una marca que representa la mediana de los datos.
• Densidad de probabilidad: La forma del violín se basa en la densidad de probabilidad de los datos en diferentes partes de la distribución.

Gráfico de violín

install.packages("vioplot")

Gráfico de violín

vioplot::vioplot(x, main="Mi primer gráfico de violín")

Gráfico de violín

Gráfico de violín de una distribució normal

x <- rnorm(n=5000, mean=10, sd=3)
vioplot::vioplot(x, main="Gráfico de violín de una distribución normal",
                 xlab="Nomrbe del eje de las x", 
                 ylab="Nombre del eje de la y",
                 col="cyan4")

Gráfico de violín de una distribució normal

Funciones útiles en R

Summary

• La función summary permite obtener:

  • Valor mínimo
  • Cuartiles
  • Mediana
  • Media

  summary()

Ejemplo Summary

• Determine con el objeto x:

• Valor mínimo

  • Cuartiles
  • Mediana
  • Media

  summary(x)

     Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  -0.9219  8.0318 10.0342 10.0388 12.0079 21.7146 

Función sort

• Ayuda a ordenar los datos de un vector
• Un vector es la estructura de datos más sencilla en R. Un vector es una colección de uno o más datos del mismo tipo. Por ejemplo el objeto x

sort()

Función sort

sort(x)

   [1] -0.9218946 -0.2986701 -0.1781866  1.1357672  1.2327196  1.5466381
   [7]  1.7052821  1.8759270  1.9048366  1.9574194  2.0619243  2.1830107
  [13]  2.2656922  2.3584563  2.3653793  2.4085078  2.4699508  2.4853968
  [19]  2.5233107  2.6231006  2.6387834  2.7206742  2.7258527  2.7540071
  [25]  2.8038039  2.8613528  2.8739738  2.8982457  2.9009396  2.9018604
  [31]  2.9106627  2.9317864  2.9365928  2.9799222  2.9834930  3.0077454
  [37]  3.0376562  3.0376673  3.0801688  3.0821595  3.0932274  3.0962768
  [43]  3.1041910  3.1137285  3.1582197  3.2005468  3.2169359  3.2206424
  [49]  3.2249494  3.2249998  3.2290599  3.2461655  3.2520932  3.2587412
  [55]  3.2940140  3.2950316  3.3146087  3.3151561  3.3337591  3.3413389
  [61]  3.3484589  3.3502197  3.3629278  3.3862987  3.3937482  3.4098656
  [67]  3.4286759  3.4743036  3.4820410  3.5174114  3.5297180  3.5320487
  [73]  3.5673213  3.5728075  3.5811540  3.6034474  3.6164059  3.6204581
  [79]  3.6356509  3.6416761  3.6800011  3.6889109  3.6981410  3.7212302
  [85]  3.7275610  3.7603627  3.7619518  3.7831103  3.7861047  3.8050105
  [91]  3.8075074  3.8135275  3.8188293  3.8467996  3.8532416  3.8618324
  [97]  3.8686028  3.8751826  3.8794015  3.8861405  3.8936705  3.8973210
 [103]  3.9007981  3.9051683  3.9140061  3.9219196  3.9313574  3.9351605
 [109]  3.9676841  4.0104743  4.0128940  4.0232039  4.0293421  4.0386137
 [115]  4.0526984  4.0544522  4.0905268  4.0934775  4.1056432  4.1096660
 [121]  4.1195705  4.1394821  4.1504950  4.1615652  4.1648486  4.1785066
 [127]  4.1827450  4.2010833  4.2305509  4.2415578  4.2474929  4.2566384
 [133]  4.2609175  4.2769063  4.2935307  4.2958574  4.3045515  4.3073267
 [139]  4.3135311  4.3176330  4.3330010  4.3423601  4.3437203  4.3914907
 [145]  4.4311075  4.4400904  4.4491945  4.4495622  4.4662914  4.4673006
 [151]  4.4723454  4.4763867  4.4838430  4.4840673  4.4859771  4.4948620
 [157]  4.4996000  4.5082865  4.5376177  4.5396644  4.5411337  4.5441036
 [163]  4.5511842  4.5582413  4.5649192  4.5722814  4.5767344  4.5838106
 [169]  4.5864438  4.6069561  4.6194094  4.6226708  4.6285234  4.6315826
 [175]  4.6477860  4.6552979  4.6591476  4.6778275  4.6845927  4.6849295
 [181]  4.7011392  4.7516847  4.7566230  4.7718619  4.7774309  4.7895283
 [187]  4.8007726  4.8119624  4.8135742  4.8388028  4.8450464  4.8464904
 [193]  4.8692815  4.8720717  4.8744919  4.8779061  4.8874698  4.8888136
 [199]  4.8897870  4.8932262  4.8975126  4.9010247  4.9044901  4.9202488
 [205]  4.9218112  4.9241212  4.9265674  4.9352628  4.9363472  4.9391735
 [211]  4.9478496  4.9549717  4.9549982  4.9850533  4.9962183  4.9964517
 [217]  5.0162922  5.0382996  5.0408163  5.0434210  5.0503127  5.0512193
 [223]  5.0519941  5.0532597  5.0628644  5.0823489  5.1005170  5.1044496
 [229]  5.1097952  5.1141536  5.1220230  5.1372638  5.1577311  5.1608776
 [235]  5.1750038  5.1785071  5.1890987  5.1953383  5.2018594  5.2025442
 [241]  5.2305579  5.2414444  5.2450529  5.2472201  5.2492090  5.2493605
 [247]  5.2547825  5.2558920  5.2560490  5.2561099  5.2739788  5.2789732
 [253]  5.2815694  5.2876227  5.2929813  5.3006242  5.3042764  5.3115448
 [259]  5.3141484  5.3261060  5.3316804  5.3347201  5.3426855  5.3558208
 [265]  5.3563828  5.3610908  5.3638349  5.3656249  5.3657866  5.3660627
 [271]  5.3719915  5.3739139  5.3749476  5.3844649  5.4107226  5.4157838
 [277]  5.4202794  5.4354167  5.4358700  5.4366803  5.4374430  5.4433662
 [283]  5.4552261  5.4657185  5.4727260  5.4727314  5.4805148  5.4929919
 [289]  5.4992756  5.4995951  5.4998495  5.5057406  5.5089536  5.5106791
 [295]  5.5122604  5.5171600  5.5200865  5.5249453  5.5270498  5.5497734
 [301]  5.5516738  5.5668960  5.5672911  5.5710897  5.5767762  5.5851475
 [307]  5.5883172  5.5908412  5.5926095  5.5986415  5.6013233  5.6026288
 [313]  5.6156206  5.6172208  5.6174215  5.6189640  5.6221025  5.6245650
 [319]  5.6271041  5.6383248  5.6403395  5.6433336  5.6558939  5.6581751
 [325]  5.6601121  5.6740971  5.6834708  5.6895135  5.6922093  5.6938097
 [331]  5.6942231  5.7022260  5.7171031  5.7213790  5.7222697  5.7342695
 [337]  5.7346211  5.7346973  5.7391936  5.7468284  5.7492327  5.7493634
 [343]  5.7622206  5.7627483  5.7752977  5.7776111  5.7789258  5.7815038
 [349]  5.7826831  5.7852690  5.7873880  5.7880786  5.7895660  5.7901541
 [355]  5.7906429  5.7908493  5.7922690  5.7969643  5.8056142  5.8077951
 [361]  5.8099199  5.8126752  5.8172288  5.8250535  5.8267114  5.8269444
 [367]  5.8331459  5.8372801  5.8439099  5.8630451  5.8633246  5.8701785
 [373]  5.8710339  5.8760698  5.8781621  5.8863295  5.8904787  5.8937448
 [379]  5.9000592  5.9023302  5.9123939  5.9193005  5.9220447  5.9253326
 [385]  5.9344759  5.9354053  5.9394043  5.9429688  5.9478184  5.9478917
 [391]  5.9498535  5.9516316  5.9549528  5.9622384  5.9635423  5.9648085
 [397]  5.9659399  5.9757399  5.9773839  5.9794667  5.9842060  5.9847456
 [403]  5.9864997  5.9883646  5.9888350  5.9897998  5.9909769  5.9922839
 [409]  5.9932577  5.9936052  5.9936571  5.9942494  5.9970026  5.9983607
 [415]  6.0041789  6.0072213  6.0098493  6.0164460  6.0209088  6.0256132
 [421]  6.0307307  6.0309637  6.0375865  6.0388550  6.0408734  6.0439620
 [427]  6.0442316  6.0454217  6.0575757  6.0631977  6.0807184  6.0840531
 [433]  6.0882324  6.0940776  6.0962619  6.0987580  6.0989936  6.1005675
 [439]  6.1035615  6.1043390  6.1046271  6.1049825  6.1059283  6.1137715
 [445]  6.1197476  6.1234825  6.1353687  6.1393338  6.1399032  6.1429075
 [451]  6.1522577  6.1541063  6.1599717  6.1613215  6.1646598  6.1716900
 [457]  6.1747374  6.1798475  6.1812254  6.1820743  6.1897256  6.1899544
 [463]  6.1914144  6.1931093  6.2095936  6.2154451  6.2207535  6.2238671
 [469]  6.2245296  6.2287743  6.2294658  6.2330398  6.2343001  6.2370106
 [475]  6.2434923  6.2457339  6.2466204  6.2468412  6.2522061  6.2644348
 [481]  6.2657644  6.2676708  6.2678283  6.2688222  6.2710102  6.2711810
 [487]  6.2724296  6.2791666  6.2797570  6.2807782  6.2885338  6.2932242
 [493]  6.2987853  6.3019015  6.3075877  6.3098277  6.3099149  6.3146418
 [499]  6.3178304  6.3207504  6.3208773  6.3227465  6.3257882  6.3322007
 [505]  6.3363031  6.3364324  6.3371038  6.3387876  6.3400711  6.3416456
 [511]  6.3488107  6.3513113  6.3516652  6.3537232  6.3538339  6.3556384
 [517]  6.3583685  6.3585229  6.3624968  6.3648542  6.3668618  6.3674344
 [523]  6.3684357  6.3698534  6.3815837  6.3838116  6.3847406  6.3855299
 [529]  6.3855335  6.3872022  6.3988086  6.3989900  6.3998800  6.4035343
 [535]  6.4074360  6.4153029  6.4186813  6.4206794  6.4213544  6.4244472
 [541]  6.4277252  6.4310175  6.4414427  6.4424395  6.4442734  6.4460142
 [547]  6.4483144  6.4500535  6.4519199  6.4532444  6.4544388  6.4589032
 [553]  6.4658641  6.4723497  6.4769547  6.4779677  6.4815705  6.4820890
 [559]  6.4841690  6.4860369  6.4865490  6.4916681  6.4917657  6.4945539
 [565]  6.4948376  6.4990522  6.5018988  6.5023937  6.5041630  6.5089664
 [571]  6.5104744  6.5120285  6.5146062  6.5160749  6.5178571  6.5269910
 [577]  6.5279282  6.5368066  6.5428180  6.5432366  6.5440169  6.5445176
 [583]  6.5517922  6.5535090  6.5573405  6.5575877  6.5581432  6.5625278
 [589]  6.5672958  6.5690290  6.5700115  6.5801927  6.5836430  6.5888083
 [595]  6.5912706  6.5926431  6.5952614  6.6010031  6.6021943  6.6041283
 [601]  6.6045772  6.6065459  6.6084977  6.6086317  6.6131070  6.6150022
 [607]  6.6208160  6.6219081  6.6337749  6.6338233  6.6339223  6.6379226
 [613]  6.6443263  6.6461728  6.6471521  6.6520693  6.6546898  6.6568855
 [619]  6.6607596  6.6610494  6.6615506  6.6616732  6.6618880  6.6625445
 [625]  6.6635808  6.6682639  6.6709920  6.6724495  6.6749812  6.6773032
 [631]  6.6813719  6.6855702  6.6863007  6.6872462  6.6895771  6.6904692
 [637]  6.6916425  6.6922304  6.6970978  6.7035650  6.7047868  6.7060929
 [643]  6.7062990  6.7092258  6.7174126  6.7223149  6.7230158  6.7237609
 [649]  6.7248763  6.7251126  6.7303832  6.7339301  6.7341576  6.7350924
 [655]  6.7362140  6.7426055  6.7488155  6.7517767  6.7537068  6.7591422
 [661]  6.7600444  6.7689012  6.7690622  6.7693718  6.7699624  6.7723808
 [667]  6.7724075  6.7795171  6.7802319  6.7812154  6.7825854  6.7845761
 [673]  6.7847601  6.7856554  6.7877573  6.7879444  6.7885028  6.7919702
 [679]  6.7930160  6.7964125  6.7965556  6.8034005  6.8037460  6.8059244
 [685]  6.8205618  6.8294052  6.8323018  6.8326275  6.8350938  6.8418325
 [691]  6.8508843  6.8526027  6.8528295  6.8577898  6.8601498  6.8606841
 [697]  6.8658142  6.8677884  6.8697398  6.8722930  6.8736498  6.8771074
 [703]  6.8780722  6.8786799  6.8826083  6.8842739  6.8848417  6.8877573
 [709]  6.8891686  6.8918226  6.8938826  6.8969014  6.9014209  6.9042994
 [715]  6.9073175  6.9086696  6.9137378  6.9148261  6.9157325  6.9161902
 [721]  6.9187006  6.9202372  6.9211498  6.9238130  6.9292848  6.9355537
 [727]  6.9366568  6.9368628  6.9383808  6.9407375  6.9410483  6.9429026
 [733]  6.9442974  6.9465621  6.9476720  6.9486321  6.9507164  6.9520131
 [739]  6.9522853  6.9526525  6.9528927  6.9542173  6.9544766  6.9563691
 [745]  6.9573281  6.9578875  6.9579311  6.9581211  6.9583446  6.9627594
 [751]  6.9709715  6.9723762  6.9750210  6.9762964  6.9766064  6.9776408
 [757]  6.9805165  6.9837138  6.9874331  6.9877873  6.9878310  6.9903084
 [763]  6.9917225  6.9934537  6.9946902  6.9954793  7.0046942  7.0055341
 [769]  7.0068000  7.0082366  7.0085796  7.0097955  7.0122609  7.0160311
 [775]  7.0198986  7.0221208  7.0224637  7.0241376  7.0250328  7.0286290
 [781]  7.0289758  7.0315638  7.0320031  7.0332912  7.0336460  7.0376786
 [787]  7.0386012  7.0451063  7.0460215  7.0469190  7.0474187  7.0481025
 [793]  7.0482820  7.0498738  7.0530937  7.0532053  7.0532713  7.0541010
 [799]  7.0561706  7.0566396  7.0629614  7.0644264  7.0690127  7.0703532
 [805]  7.0717324  7.0747718  7.0796441  7.0829278  7.0846466  7.0861471
 [811]  7.0898361  7.0900651  7.0915603  7.0918895  7.0971982  7.0973546
 [817]  7.0989856  7.1001203  7.1008101  7.1045339  7.1047641  7.1062290
 [823]  7.1071038  7.1075725  7.1089967  7.1109002  7.1116743  7.1120584
 [829]  7.1126698  7.1154027  7.1206091  7.1215008  7.1263958  7.1268884
 [835]  7.1309599  7.1311127  7.1377424  7.1383054  7.1392322  7.1416630
 [841]  7.1442326  7.1466352  7.1486149  7.1491223  7.1496486  7.1504298
 [847]  7.1506079  7.1550132  7.1551635  7.1552427  7.1565379  7.1587367
 [853]  7.1597159  7.1605236  7.1627430  7.1644379  7.1658935  7.1668229
 [859]  7.1683602  7.1709977  7.1736846  7.1771006  7.1791900  7.1803962
 [865]  7.1805367  7.1830958  7.1929267  7.1957174  7.1997093  7.2014940
 [871]  7.2041071  7.2053640  7.2075934  7.2081982  7.2162858  7.2175657
 [877]  7.2182329  7.2225131  7.2227937  7.2235077  7.2249238  7.2264735
 [883]  7.2346864  7.2355182  7.2392259  7.2448697  7.2459584  7.2465494
 [889]  7.2547649  7.2572892  7.2580268  7.2588270  7.2618954  7.2619077
 [895]  7.2620415  7.2621486  7.2647891  7.2665120  7.2697778  7.2727734
 [901]  7.2785354  7.2807647  7.2812170  7.2820083  7.2858608  7.2904030
 [907]  7.2904360  7.2975421  7.2990037  7.3001238  7.3004807  7.3017971
 [913]  7.3029940  7.3084952  7.3094720  7.3109768  7.3122185  7.3136573
 [919]  7.3199198  7.3206023  7.3221009  7.3229707  7.3242883  7.3262451
 [925]  7.3295877  7.3312263  7.3316641  7.3362310  7.3375078  7.3420009
 [931]  7.3427376  7.3430430  7.3454523  7.3458417  7.3486434  7.3578440
 [937]  7.3641621  7.3694491  7.3723650  7.3748817  7.3808762  7.3859049
 [943]  7.3859932  7.3861099  7.3872884  7.3907093  7.3909984  7.3986557
 [949]  7.4036747  7.4106332  7.4125316  7.4136051  7.4141181  7.4159072
 [955]  7.4215318  7.4234716  7.4258993  7.4310765  7.4318038  7.4325258
 [961]  7.4355399  7.4397310  7.4397678  7.4476632  7.4494889  7.4517147
 [967]  7.4523895  7.4578415  7.4583554  7.4605165  7.4605527  7.4623576
 [973]  7.4627489  7.4702984  7.4707648  7.4731465  7.4740016  7.4766277
 [979]  7.4806573  7.4828188  7.4899523  7.4914158  7.4922529  7.4925368
 [985]  7.4935575  7.5009029  7.5022391  7.5030287  7.5046817  7.5085698
 [991]  7.5090612  7.5145091  7.5201047  7.5255161  7.5264191  7.5266819
 [997]  7.5292185  7.5318275  7.5323978  7.5328821  7.5352474  7.5370030
[1003]  7.5392766  7.5405160  7.5439522  7.5459976  7.5460315  7.5465839
[1009]  7.5467997  7.5506839  7.5513181  7.5539141  7.5553169  7.5592676
[1015]  7.5594386  7.5598026  7.5604007  7.5641653  7.5682402  7.5691233
[1021]  7.5710569  7.5744532  7.5801843  7.5816045  7.5818123  7.5836803
[1027]  7.5837920  7.5894978  7.5932654  7.5933116  7.5933502  7.5933683
[1033]  7.5941764  7.5952580  7.5977066  7.5991633  7.5997489  7.6035451
[1039]  7.6037098  7.6045187  7.6089337  7.6138104  7.6144191  7.6153556
[1045]  7.6159354  7.6165481  7.6180955  7.6234123  7.6267550  7.6268193
[1051]  7.6268962  7.6295261  7.6301144  7.6319337  7.6328269  7.6357016
[1057]  7.6367876  7.6402118  7.6410857  7.6430995  7.6471096  7.6475173
[1063]  7.6506093  7.6550997  7.6554191  7.6558554  7.6580771  7.6595698
[1069]  7.6631310  7.6633621  7.6660952  7.6679217  7.6716290  7.6737688
[1075]  7.6743877  7.6770845  7.6780317  7.6789554  7.6790522  7.6794325
[1081]  7.6838793  7.6852304  7.6860316  7.6861195  7.6894462  7.6968713
[1087]  7.7034119  7.7048229  7.7049016  7.7060852  7.7096134  7.7105925
[1093]  7.7106211  7.7153126  7.7157649  7.7167520  7.7226761  7.7298466
[1099]  7.7304209  7.7304407  7.7317843  7.7318645  7.7320795  7.7321152
[1105]  7.7323230  7.7329808  7.7377077  7.7391744  7.7406308  7.7419568
[1111]  7.7427598  7.7500836  7.7518137  7.7532170  7.7541548  7.7550529
[1117]  7.7555259  7.7691869  7.7694909  7.7701950  7.7755128  7.7774289
[1123]  7.7809741  7.7843451  7.7854124  7.7862821  7.7871185  7.7885048
[1129]  7.7907184  7.7925068  7.7942003  7.7950727  7.7979333  7.8039849
[1135]  7.8110897  7.8116183  7.8133363  7.8175271  7.8226806  7.8228384
[1141]  7.8234436  7.8236169  7.8298645  7.8309528  7.8310953  7.8313989
[1147]  7.8327925  7.8351590  7.8404615  7.8410895  7.8455406  7.8481940
[1153]  7.8497512  7.8501245  7.8505943  7.8518078  7.8554714  7.8581314
[1159]  7.8598715  7.8621626  7.8636820  7.8670900  7.8677631  7.8699599
[1165]  7.8711248  7.8763954  7.8801837  7.8817122  7.8865431  7.8868308
[1171]  7.8880389  7.8891627  7.8895215  7.8905725  7.8947554  7.9006021
[1177]  7.9039895  7.9067814  7.9073771  7.9088492  7.9113110  7.9122143
[1183]  7.9150531  7.9164560  7.9212278  7.9223169  7.9249682  7.9254388
[1189]  7.9254555  7.9319507  7.9337531  7.9338089  7.9343203  7.9349624
[1195]  7.9371690  7.9373709  7.9376108  7.9379512  7.9380667  7.9430082
[1201]  7.9442537  7.9486715  7.9534890  7.9539730  7.9549339  7.9606704
[1207]  7.9660383  7.9698479  7.9700479  7.9723835  7.9724456  7.9726437
[1213]  7.9749228  7.9756175  7.9765696  7.9793057  7.9795154  7.9805770
[1219]  7.9811452  7.9820956  7.9849041  7.9869433  7.9876873  7.9881423
[1225]  7.9895140  7.9900415  7.9909525  7.9930340  7.9964654  7.9967107
[1231]  7.9982446  7.9994432  8.0004617  8.0010289  8.0020197  8.0040841
[1237]  8.0043167  8.0053932  8.0123367  8.0150806  8.0200737  8.0235459
[1243]  8.0248410  8.0254237  8.0254466  8.0256753  8.0269669  8.0269804
[1249]  8.0278281  8.0313238  8.0319132  8.0321072  8.0353185  8.0373964
[1255]  8.0378124  8.0402141  8.0404100  8.0422692  8.0460996  8.0462629
[1261]  8.0477523  8.0500898  8.0534776  8.0542059  8.0561166  8.0568989
[1267]  8.0595719  8.0603224  8.0607211  8.0641751  8.0657565  8.0693472
[1273]  8.0719585  8.0726761  8.0727098  8.0730032  8.0738311  8.0759051
[1279]  8.0774345  8.0802780  8.0812881  8.0815506  8.0820874  8.0878440
[1285]  8.0887436  8.0911147  8.0957757  8.0959050  8.0963614  8.0975240
[1291]  8.0986373  8.1016997  8.1017903  8.1035490  8.1081567  8.1089936
[1297]  8.1106399  8.1108786  8.1154182  8.1171958  8.1177631  8.1197987
[1303]  8.1244804  8.1256057  8.1266695  8.1267971  8.1275563  8.1297237
[1309]  8.1337727  8.1347971  8.1361163  8.1379982  8.1380822  8.1383792
[1315]  8.1388416  8.1392497  8.1405741  8.1425255  8.1435275  8.1435498
[1321]  8.1445539  8.1450705  8.1459106  8.1483622  8.1506647  8.1512240
[1327]  8.1519211  8.1583823  8.1584361  8.1589634  8.1620884  8.1625184
[1333]  8.1638810  8.1667792  8.1677003  8.1682245  8.1684742  8.1698483
[1339]  8.1705695  8.1709564  8.1723008  8.1727498  8.1735306  8.1740536
[1345]  8.1770066  8.1795055  8.1812746  8.1831967  8.1842831  8.1857203
[1351]  8.1858549  8.1864630  8.1886017  8.1910614  8.1976228  8.2009615
[1357]  8.2030187  8.2031726  8.2042208  8.2044967  8.2056459  8.2087079
[1363]  8.2093846  8.2100040  8.2105092  8.2121100  8.2145292  8.2155203
[1369]  8.2182920  8.2310687  8.2329470  8.2331093  8.2343242  8.2388789
[1375]  8.2416810  8.2434913  8.2441453  8.2449193  8.2472110  8.2501881
[1381]  8.2514624  8.2524575  8.2550408  8.2556719  8.2564597  8.2566511
[1387]  8.2583683  8.2585775  8.2585818  8.2645354  8.2649809  8.2680318
[1393]  8.2682565  8.2687972  8.2715679  8.2720476  8.2721255  8.2726733
[1399]  8.2729987  8.2781502  8.2786263  8.2812389  8.2816249  8.2827788
[1405]  8.2829918  8.2861642  8.2862315  8.2872460  8.2903618  8.2904330
[1411]  8.2907020  8.2921592  8.2967213  8.2983176  8.2996989  8.3029333
[1417]  8.3045412  8.3050232  8.3061571  8.3079414  8.3090301  8.3094123
[1423]  8.3107826  8.3113638  8.3136388  8.3150072  8.3164431  8.3173483
[1429]  8.3242320  8.3244336  8.3331674  8.3344568  8.3348612  8.3356779
[1435]  8.3382326  8.3384356  8.3422974  8.3424353  8.3427993  8.3469591
[1441]  8.3470128  8.3471570  8.3474550  8.3496197  8.3522056  8.3538943
[1447]  8.3543817  8.3548364  8.3565738  8.3584220  8.3617832  8.3618469
[1453]  8.3649495  8.3671473  8.3705511  8.3712542  8.3725812  8.3736643
[1459]  8.3827669  8.3867080  8.3917890  8.3934031  8.3938040  8.3941156
[1465]  8.3978096  8.3978361  8.3998148  8.4060387  8.4093036  8.4106987
[1471]  8.4129851  8.4159254  8.4167136  8.4176795  8.4182358  8.4184411
[1477]  8.4192776  8.4193226  8.4212157  8.4233264  8.4242803  8.4247193
[1483]  8.4253774  8.4307682  8.4317237  8.4351610  8.4363593  8.4378403
[1489]  8.4419350  8.4465590  8.4494281  8.4498876  8.4500066  8.4531242
[1495]  8.4543975  8.4551911  8.4572499  8.4616042  8.4637328  8.4669584
[1501]  8.4675528  8.4726624  8.4733994  8.4765737  8.4767911  8.4768791
[1507]  8.4807790  8.4824621  8.4838760  8.4845866  8.4851005  8.4854233
[1513]  8.4917455  8.4922275  8.4936987  8.4942066  8.4958795  8.4966249
[1519]  8.4981785  8.5032727  8.5052310  8.5064343  8.5085750  8.5160675
[1525]  8.5197177  8.5207485  8.5212052  8.5213855  8.5218170  8.5222290
[1531]  8.5224175  8.5228266  8.5245264  8.5248164  8.5254969  8.5321772
[1537]  8.5327874  8.5332221  8.5354616  8.5414112  8.5415150  8.5419448
[1543]  8.5430111  8.5452398  8.5454472  8.5474104  8.5482597  8.5530408
[1549]  8.5546369  8.5548744  8.5551946  8.5575619  8.5597270  8.5613022
[1555]  8.5650109  8.5663232  8.5693327  8.5726621  8.5737973  8.5742743
[1561]  8.5756263  8.5770596  8.5777432  8.5797456  8.5832664  8.5897902
[1567]  8.5898220  8.5904492  8.5905358  8.5915080  8.5952182  8.5967308
[1573]  8.5988992  8.5994762  8.5997113  8.6003992  8.6008940  8.6011662
[1579]  8.6014278  8.6014929  8.6015395  8.6025106  8.6087543  8.6165069
[1585]  8.6222642  8.6229690  8.6233581  8.6242453  8.6261223  8.6273064
[1591]  8.6286293  8.6291625  8.6304183  8.6312669  8.6318491  8.6318605
[1597]  8.6321580  8.6342674  8.6349137  8.6354751  8.6396601  8.6398764
[1603]  8.6416287  8.6480092  8.6504776  8.6507920  8.6537567  8.6541072
[1609]  8.6542120  8.6548272  8.6561920  8.6606397  8.6607143  8.6661737
[1615]  8.6694224  8.6695784  8.6696188  8.6732256  8.6752494  8.6769096
[1621]  8.6771611  8.6781767  8.6787062  8.6806276  8.6819541  8.6830778
[1627]  8.6867814  8.6869395  8.6947999  8.6950960  8.6954901  8.6960606
[1633]  8.6961487  8.6975312  8.7010986  8.7038483  8.7120882  8.7130433
[1639]  8.7147422  8.7150369  8.7163309  8.7169720  8.7177808  8.7186153
[1645]  8.7187385  8.7201416  8.7208062  8.7208864  8.7243768  8.7246964
[1651]  8.7248085  8.7257480  8.7275683  8.7284178  8.7284470  8.7327635
[1657]  8.7365018  8.7421439  8.7445373  8.7450451  8.7456911  8.7475009
[1663]  8.7506379  8.7507185  8.7541820  8.7550444  8.7553813  8.7555105
[1669]  8.7586042  8.7597896  8.7607655  8.7612413  8.7631149  8.7638120
[1675]  8.7655917  8.7657814  8.7662183  8.7663569  8.7673943  8.7678074
[1681]  8.7685762  8.7720800  8.7738304  8.7756828  8.7767799  8.7779953
[1687]  8.7841636  8.7848011  8.7867306  8.7887993  8.7890351  8.7925105
[1693]  8.7947664  8.7966182  8.7968534  8.7972541  8.7973231  8.8027695
[1699]  8.8048340  8.8081194  8.8083075  8.8109110  8.8132788  8.8143252
[1705]  8.8149743  8.8223203  8.8238267  8.8264701  8.8276753  8.8277960
[1711]  8.8284072  8.8296260  8.8302117  8.8307715  8.8320519  8.8327431
[1717]  8.8347775  8.8356989  8.8377241  8.8387775  8.8429853  8.8439484
[1723]  8.8467128  8.8503159  8.8539847  8.8567009  8.8573320  8.8587183
[1729]  8.8590179  8.8610412  8.8623993  8.8642470  8.8648043  8.8677848
[1735]  8.8678049  8.8679633  8.8687928  8.8692495  8.8698732  8.8737047
[1741]  8.8745256  8.8818500  8.8841050  8.8857530  8.8872299  8.8873522
[1747]  8.8876518  8.8918562  8.8925094  8.8952904  8.8958421  8.8971408
[1753]  8.9020547  8.9040334  8.9048692  8.9072765  8.9079696  8.9079837
[1759]  8.9097584  8.9117996  8.9118175  8.9122817  8.9135652  8.9210254
[1765]  8.9225799  8.9286929  8.9288185  8.9327883  8.9337213  8.9344764
[1771]  8.9345354  8.9349220  8.9353962  8.9361708  8.9365461  8.9387503
[1777]  8.9425466  8.9439329  8.9452617  8.9454336  8.9465984  8.9467313
[1783]  8.9489391  8.9491290  8.9513584  8.9565551  8.9576243  8.9600946
[1789]  8.9606827  8.9623936  8.9638756  8.9650851  8.9653406  8.9675395
[1795]  8.9693801  8.9696717  8.9718901  8.9724702  8.9731847  8.9736258
[1801]  8.9744143  8.9746643  8.9773000  8.9781448  8.9824597  8.9858724
[1807]  8.9873328  8.9878458  8.9896125  8.9900261  8.9905154  8.9909690
[1813]  8.9927516  8.9939630  8.9963034  8.9984326  8.9988520  9.0004807
[1819]  9.0016281  9.0026590  9.0029776  9.0038815  9.0047321  9.0057790
[1825]  9.0071760  9.0115784  9.0146981  9.0164368  9.0222053  9.0235856
[1831]  9.0251126  9.0266718  9.0300473  9.0337665  9.0339579  9.0358726
[1837]  9.0416322  9.0428758  9.0432181  9.0443133  9.0444541  9.0540688
[1843]  9.0542513  9.0564530  9.0599812  9.0645828  9.0653554  9.0678267
[1849]  9.0680876  9.0713132  9.0715998  9.0739482  9.0760229  9.0773366
[1855]  9.0781075  9.0788737  9.0794365  9.0800233  9.0803429  9.0806920
[1861]  9.0812099  9.0813916  9.0815342  9.0819262  9.0844691  9.0883223
[1867]  9.0892114  9.0899607  9.0903732  9.0951422  9.0968445  9.0976443
[1873]  9.0985095  9.0985810  9.0987854  9.0999138  9.1012793  9.1019779
[1879]  9.1029629  9.1032412  9.1050392  9.1058504  9.1059383  9.1071678
[1885]  9.1163861  9.1180124  9.1187429  9.1191054  9.1195908  9.1205297
[1891]  9.1244525  9.1317421  9.1329410  9.1342612  9.1357493  9.1375838
[1897]  9.1377371  9.1381580  9.1385235  9.1392452  9.1417609  9.1432420
[1903]  9.1437367  9.1446016  9.1507436  9.1538759  9.1570752  9.1571799
[1909]  9.1579763  9.1582783  9.1600817  9.1610184  9.1633175  9.1674833
[1915]  9.1680960  9.1685653  9.1687484  9.1694361  9.1707121  9.1711625
[1921]  9.1752524  9.1780890  9.1804920  9.1871282  9.1881110  9.1881840
[1927]  9.1891552  9.1964144  9.1965688  9.2018816  9.2032296  9.2053682
[1933]  9.2078291  9.2093366  9.2094526  9.2098250  9.2116116  9.2134771
[1939]  9.2157366  9.2166818  9.2200655  9.2204475  9.2217218  9.2220533
[1945]  9.2222467  9.2237146  9.2238436  9.2269680  9.2273804  9.2280185
[1951]  9.2331042  9.2364801  9.2375541  9.2380612  9.2382467  9.2391013
[1957]  9.2392370  9.2411392  9.2423211  9.2453752  9.2462959  9.2508914
[1963]  9.2528116  9.2537292  9.2573870  9.2614008  9.2615404  9.2631488
[1969]  9.2635754  9.2637523  9.2640042  9.2654989  9.2658884  9.2677252
[1975]  9.2681974  9.2700820  9.2712997  9.2719378  9.2735581  9.2746544
[1981]  9.2782225  9.2795496  9.2798690  9.2813677  9.2823100  9.2839763
[1987]  9.2846712  9.2890059  9.2915249  9.2920669  9.2932214  9.2950206
[1993]  9.2965536  9.3010336  9.3011299  9.3033011  9.3052731  9.3081262
[1999]  9.3102285  9.3106707  9.3117076  9.3141061  9.3146972  9.3176988
[2005]  9.3183685  9.3226692  9.3231449  9.3238593  9.3258598  9.3274649
[2011]  9.3315496  9.3329742  9.3343717  9.3363727  9.3383237  9.3386871
[2017]  9.3410766  9.3441758  9.3444960  9.3446464  9.3446871  9.3449490
[2023]  9.3470970  9.3486272  9.3496196  9.3499715  9.3509538  9.3513490
[2029]  9.3524569  9.3558371  9.3563949  9.3578566  9.3586268  9.3644568
[2035]  9.3648147  9.3667028  9.3667545  9.3704426  9.3740034  9.3768390
[2041]  9.3782665  9.3792029  9.3797975  9.3810716  9.3811199  9.3814455
[2047]  9.3817292  9.3817706  9.3820535  9.3836455  9.3837558  9.3838953
[2053]  9.3866782  9.3872092  9.3877914  9.3910418  9.3945039  9.3955711
[2059]  9.3961090  9.4006475  9.4011097  9.4017816  9.4027531  9.4031005
[2065]  9.4112225  9.4136321  9.4143436  9.4167902  9.4192387  9.4228340
[2071]  9.4234091  9.4242516  9.4267151  9.4268653  9.4322032  9.4340517
[2077]  9.4386580  9.4391522  9.4391777  9.4394901  9.4400820  9.4402942
[2083]  9.4425711  9.4429735  9.4434751  9.4449053  9.4449807  9.4477035
[2089]  9.4485598  9.4488730  9.4510132  9.4515681  9.4526098  9.4533217
[2095]  9.4591426  9.4591819  9.4600690  9.4604830  9.4612765  9.4641595
[2101]  9.4646731  9.4677656  9.4681379  9.4695221  9.4712355  9.4748969
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[2227]  9.6425409  9.6432817  9.6454374  9.6470195  9.6474807  9.6511060
[2233]  9.6527572  9.6532382  9.6532535  9.6575423  9.6589834  9.6605276
[2239]  9.6607866  9.6612112  9.6614805  9.6617149  9.6619038  9.6653184
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[2263]  9.6794290  9.6796237  9.6813303  9.6814977  9.6822307  9.6822578
[2269]  9.6841143  9.6856088  9.6867490  9.6890494  9.6901766  9.6926846
[2275]  9.6942307  9.6942537  9.6944569  9.6955036  9.6966635  9.6982543
[2281]  9.7006758  9.7026809  9.7034563  9.7090010  9.7090548  9.7107473
[2287]  9.7121861  9.7122279  9.7150983  9.7169627  9.7180623  9.7196880
[2293]  9.7208969  9.7211378  9.7228598  9.7230215  9.7242506  9.7345143
[2299]  9.7356768  9.7365307  9.7382466  9.7399027  9.7400403  9.7405971
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[2311]  9.7610727  9.7615277  9.7627431  9.7640622  9.7652506  9.7665333
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[2323]  9.7780640  9.7785760  9.7809584  9.7839357  9.7855033  9.7869666
[2329]  9.7875125  9.7875734  9.7877215  9.7886413  9.7899814  9.7907192
[2335]  9.7917070  9.7954951  9.7987275  9.7988693  9.8014046  9.8017198
[2341]  9.8065226  9.8084595  9.8084646  9.8091407  9.8105342  9.8117362
[2347]  9.8124993  9.8132092  9.8132124  9.8153119  9.8200504  9.8205631
[2353]  9.8218863  9.8230118  9.8239063  9.8241369  9.8271392  9.8281116
[2359]  9.8287207  9.8297160  9.8327450  9.8333306  9.8340469  9.8349982
[2365]  9.8384654  9.8397218  9.8401492  9.8490695  9.8505667  9.8517008
[2371]  9.8546346  9.8551526  9.8552902  9.8555707  9.8581803  9.8587623
[2377]  9.8588960  9.8608454  9.8623060  9.8653882  9.8666824  9.8671048
[2383]  9.8706066  9.8766525  9.8782089  9.8791289  9.8808006  9.8808126
[2389]  9.8832223  9.8843769  9.8853979  9.8862023  9.8867408  9.8876499
[2395]  9.8880595  9.8898866  9.8922422  9.8923984  9.8943850  9.8961534
[2401]  9.8970096  9.8970117  9.8970609  9.8981376  9.8998232  9.9000680
[2407]  9.9001680  9.9050034  9.9073685  9.9082731  9.9115114  9.9129736
[2413]  9.9139943  9.9141080  9.9141309  9.9159413  9.9169538  9.9189628
[2419]  9.9189726  9.9191273  9.9198262  9.9217619  9.9232260  9.9244921
[2425]  9.9263687  9.9272208  9.9295287  9.9353078  9.9374079  9.9375862
[2431]  9.9402739  9.9448253  9.9458229  9.9460866  9.9465329  9.9484056
[2437]  9.9486799  9.9507379  9.9520808  9.9535133  9.9553503  9.9559949
[2443]  9.9569438  9.9578636  9.9579327  9.9585206  9.9609226  9.9614409
[2449]  9.9627217  9.9627580  9.9641913  9.9671522  9.9708770  9.9724641
[2455]  9.9736795  9.9773518  9.9792434  9.9822662  9.9836391  9.9848617
[2461]  9.9851950  9.9880348  9.9881433  9.9892264  9.9902000  9.9910915
[2467]  9.9934777  9.9937885  9.9946137  9.9950126  9.9951633  9.9977137
[2473]  9.9985982  9.9999618 10.0007955 10.0018193 10.0018783 10.0032296
[2479] 10.0037906 10.0070093 10.0075956 10.0088162 10.0091325 10.0109102
[2485] 10.0144857 10.0148196 10.0158312 10.0165256 10.0183178 10.0187832
[2491] 10.0197813 10.0227277 10.0238716 10.0252290 10.0266075 10.0290039
[2497] 10.0291986 10.0299585 10.0300645 10.0339209 10.0345082 10.0355759
[2503] 10.0371259 10.0418195 10.0438398 10.0473007 10.0481610 10.0520248
[2509] 10.0586793 10.0595128 10.0602631 10.0645997 10.0654034 10.0656195
[2515] 10.0665211 10.0673939 10.0683549 10.0683603 10.0689627 10.0733955
[2521] 10.0749812 10.0763504 10.0763750 10.0769159 10.0771423 10.0782778
[2527] 10.0803197 10.0805077 10.0861748 10.0876419 10.0892238 10.0924803
[2533] 10.0927080 10.0933805 10.0950500 10.0970703 10.0978550 10.0985990
[2539] 10.0992465 10.0995220 10.1006577 10.1018731 10.1076359 10.1126960
[2545] 10.1131435 10.1132120 10.1171767 10.1237745 10.1240052 10.1250153
[2551] 10.1250942 10.1268601 10.1270670 10.1282041 10.1283252 10.1289316
[2557] 10.1325332 10.1348280 10.1350619 10.1363275 10.1363277 10.1395480
[2563] 10.1412475 10.1412983 10.1413067 10.1447318 10.1483746 10.1489418
[2569] 10.1494031 10.1501618 10.1508359 10.1510639 10.1510867 10.1514089
[2575] 10.1515894 10.1557705 10.1583209 10.1584924 10.1591557 10.1608170
[2581] 10.1651324 10.1653326 10.1666596 10.1679234 10.1680582 10.1694661
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[2593] 10.1770569 10.1771776 10.1780660 10.1787775 10.1799839 10.1832539
[2599] 10.1848543 10.1885737 10.1889018 10.1907249 10.1911615 10.1946825
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[2611] 10.2064055 10.2095182 10.2109996 10.2123750 10.2144056 10.2144494
[2617] 10.2148400 10.2153848 10.2182997 10.2190448 10.2193499 10.2212190
[2623] 10.2223688 10.2228092 10.2232619 10.2254903 10.2287313 10.2292944
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[2635] 10.2371204 10.2376173 10.2431023 10.2447879 10.2469946 10.2511268
[2641] 10.2577230 10.2596170 10.2605075 10.2605905 10.2612986 10.2618453
[2647] 10.2639557 10.2651648 10.2659612 10.2663610 10.2681267 10.2687717
[2653] 10.2716068 10.2724581 10.2728963 10.2731337 10.2755646 10.2759974
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[2695] 10.3064497 10.3068634 10.3075378 10.3084470 10.3092512 10.3115877
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[2827] 10.5001397 10.5006654 10.5036657 10.5040557 10.5040649 10.5079311
[2833] 10.5091862 10.5095682 10.5124179 10.5157004 10.5164420 10.5174899
[2839] 10.5178772 10.5200084 10.5209141 10.5221323 10.5227629 10.5228920
[2845] 10.5242511 10.5252910 10.5266209 10.5269319 10.5278241 10.5287591
[2851] 10.5299761 10.5307726 10.5351859 10.5366786 10.5370307 10.5380560
[2857] 10.5396482 10.5400101 10.5400927 10.5424790 10.5443010 10.5455927
[2863] 10.5465678 10.5472972 10.5506557 10.5516440 10.5530149 10.5549211
[2869] 10.5550994 10.5566543 10.5567454 10.5573655 10.5599925 10.5616410
[2875] 10.5627986 10.5641567 10.5647716 10.5659618 10.5679953 10.5685719
[2881] 10.5745149 10.5751122 10.5756316 10.5759804 10.5787911 10.5804196
[2887] 10.5807124 10.5818998 10.5849297 10.5869853 10.5873023 10.5873804
[2893] 10.5875689 10.5899912 10.5910342 10.5923720 10.5935513 10.5936122
[2899] 10.5951926 10.5964692 10.5979336 10.5986472 10.5988859 10.6010744
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[2911] 10.6084380 10.6114619 10.6114903 10.6141234 10.6149230 10.6155273
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[2923] 10.6257517 10.6275496 10.6288442 10.6318803 10.6366118 10.6396596
[2929] 10.6411055 10.6426989 10.6449955 10.6459204 10.6462267 10.6464780
[2935] 10.6472651 10.6505871 10.6533856 10.6539626 10.6581849 10.6581968
[2941] 10.6611618 10.6616349 10.6631143 10.6639142 10.6672123 10.6679203
[2947] 10.6696247 10.6732399 10.6748250 10.6748465 10.6766969 10.6784397
[2953] 10.6786165 10.6833199 10.6843048 10.6861871 10.6872293 10.6872485
[2959] 10.6873871 10.6874333 10.6878890 10.6892628 10.6894414 10.6897350
[2965] 10.6916069 10.6926777 10.6941730 10.6956853 10.6965901 10.6978371
[2971] 10.6984853 10.6999376 10.7026122 10.7034535 10.7036735 10.7042637
[2977] 10.7042726 10.7084866 10.7096052 10.7101586 10.7106642 10.7107924
[2983] 10.7115009 10.7118094 10.7122202 10.7125374 10.7153165 10.7157432
[2989] 10.7186319 10.7204454 10.7212375 10.7220553 10.7259271 10.7288002
[2995] 10.7310126 10.7320657 10.7333159 10.7338990 10.7349591 10.7351564
[3001] 10.7352318 10.7355786 10.7371825 10.7375078 10.7437557 10.7528716
[3007] 10.7550606 10.7578055 10.7581226 10.7581843 10.7583362 10.7596722
[3013] 10.7599462 10.7601292 10.7607907 10.7615115 10.7631915 10.7655272
[3019] 10.7672002 10.7684201 10.7711841 10.7719825 10.7734072 10.7745689
[3025] 10.7762773 10.7766621 10.7767339 10.7773685 10.7774347 10.7781327
[3031] 10.7797474 10.7817558 10.7834747 10.7867426 10.7888450 10.7894710
[3037] 10.7912578 10.7922212 10.7944002 10.7947198 10.7969590 10.7979050
[3043] 10.8101903 10.8106188 10.8134832 10.8135724 10.8136065 10.8192395
[3049] 10.8212353 10.8214604 10.8238243 10.8238904 10.8256428 10.8284433
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[3073] 10.8611438 10.8621766 10.8671733 10.8684558 10.8700786 10.8740721
[3079] 10.8754846 10.8770957 10.8781888 10.8794232 10.8794451 10.8812360
[3085] 10.8837521 10.8853144 10.8854954 10.8874002 10.8900494 10.8904794
[3091] 10.8965482 10.8965652 10.8972116 10.8994902 10.9017155 10.9031397
[3097] 10.9035240 10.9039391 10.9053081 10.9055315 10.9056600 10.9058581
[3103] 10.9076711 10.9089056 10.9095121 10.9117163 10.9181426 10.9183622
[3109] 10.9199910 10.9201049 10.9202648 10.9206258 10.9238219 10.9245610
[3115] 10.9253240 10.9283923 10.9287562 10.9289579 10.9315757 10.9326964
[3121] 10.9328008 10.9333429 10.9335592 10.9347653 10.9423957 10.9441331
[3127] 10.9461064 10.9465503 10.9480729 10.9493681 10.9514422 10.9545177
[3133] 10.9565830 10.9572532 10.9578400 10.9587400 10.9602165 10.9610899
[3139] 10.9613308 10.9620729 10.9621681 10.9633551 10.9642821 10.9653168
[3145] 10.9692907 10.9726754 10.9746419 10.9795917 10.9805130 10.9818961
[3151] 10.9836342 10.9838515 10.9856219 10.9867147 10.9883681 10.9894034
[3157] 10.9910658 10.9975522 10.9982730 11.0002968 11.0013114 11.0027628
[3163] 11.0069204 11.0085501 11.0105346 11.0120874 11.0123710 11.0126469
[3169] 11.0150372 11.0153129 11.0188896 11.0193025 11.0194464 11.0205297
[3175] 11.0230770 11.0232852 11.0236449 11.0236841 11.0263564 11.0279553
[3181] 11.0288844 11.0304988 11.0315547 11.0396652 11.0449541 11.0454734
[3187] 11.0469751 11.0479950 11.0481901 11.0497678 11.0505296 11.0519851
[3193] 11.0544820 11.0550654 11.0551686 11.0558243 11.0601494 11.0638442
[3199] 11.0649939 11.0658109 11.0702055 11.0742886 11.0743028 11.0749182
[3205] 11.0762802 11.0828847 11.0861885 11.0871169 11.0891943 11.0913788
[3211] 11.0923696 11.0944224 11.0946354 11.0979837 11.0994193 11.1000789
[3217] 11.1033546 11.1034627 11.1065112 11.1072060 11.1088728 11.1097810
[3223] 11.1114098 11.1147295 11.1175127 11.1192254 11.1221416 11.1230627
[3229] 11.1243963 11.1254718 11.1317501 11.1321790 11.1337617 11.1364220
[3235] 11.1367438 11.1373366 11.1415301 11.1456230 11.1456707 11.1460318
[3241] 11.1476462 11.1478368 11.1494111 11.1508071 11.1508139 11.1508998
[3247] 11.1522669 11.1564197 11.1568645 11.1617922 11.1645924 11.1671611
[3253] 11.1689887 11.1691073 11.1699315 11.1699791 11.1700520 11.1701923
[3259] 11.1797779 11.1849019 11.1861260 11.1868790 11.1894631 11.1896739
[3265] 11.1905048 11.1912885 11.1934611 11.1941324 11.1950865 11.1954003
[3271] 11.1955252 11.1971139 11.1975229 11.1978987 11.2033032 11.2033134
[3277] 11.2034533 11.2068485 11.2076524 11.2106722 11.2110773 11.2152171
[3283] 11.2175310 11.2176773 11.2179344 11.2182459 11.2194300 11.2206079
[3289] 11.2237213 11.2240129 11.2308749 11.2309148 11.2311821 11.2318554
[3295] 11.2334510 11.2337662 11.2346302 11.2350325 11.2411236 11.2430039
[3301] 11.2443042 11.2452357 11.2467129 11.2474273 11.2495401 11.2520291
[3307] 11.2534671 11.2545575 11.2545723 11.2569511 11.2588005 11.2612043
[3313] 11.2618742 11.2640066 11.2644745 11.2672661 11.2676082 11.2676565
[3319] 11.2705172 11.2707877 11.2711949 11.2723451 11.2732216 11.2765691
[3325] 11.2766857 11.2807972 11.2857011 11.2867992 11.2870986 11.2951809
[3331] 11.2952010 11.2953074 11.2953401 11.2970979 11.2971485 11.2976103
[3337] 11.2977677 11.3044302 11.3058945 11.3067973 11.3085984 11.3105017
[3343] 11.3120973 11.3135936 11.3138508 11.3143428 11.3152058 11.3164934
[3349] 11.3178233 11.3183098 11.3185379 11.3193423 11.3197533 11.3201600
[3355] 11.3208953 11.3215401 11.3240320 11.3280058 11.3291196 11.3317895
[3361] 11.3340283 11.3379182 11.3389622 11.3400566 11.3401915 11.3409972
[3367] 11.3446592 11.3467061 11.3492549 11.3517670 11.3534445 11.3559385
[3373] 11.3572928 11.3575976 11.3653917 11.3671546 11.3713046 11.3760391
[3379] 11.3778155 11.3778411 11.3779437 11.3787495 11.3828880 11.3835993
[3385] 11.3847110 11.3886015 11.3910299 11.3924827 11.3924832 11.3935729
[3391] 11.3946105 11.3967682 11.3991894 11.4028492 11.4042446 11.4047590
[3397] 11.4065805 11.4090631 11.4100904 11.4103676 11.4111788 11.4130681
[3403] 11.4140262 11.4154357 11.4180116 11.4199850 11.4246305 11.4273568
[3409] 11.4283679 11.4287250 11.4325245 11.4331923 11.4347813 11.4352810
[3415] 11.4361034 11.4385847 11.4409803 11.4420054 11.4424917 11.4432295
[3421] 11.4440651 11.4448861 11.4493128 11.4535152 11.4557411 11.4561496
[3427] 11.4577922 11.4595250 11.4596992 11.4627240 11.4631563 11.4651514
[3433] 11.4677354 11.4695168 11.4703804 11.4743549 11.4760911 11.4780068
[3439] 11.4783417 11.4787002 11.4787652 11.4788054 11.4838130 11.4860466
[3445] 11.4869025 11.4878904 11.4893757 11.4904771 11.4946184 11.4964239
[3451] 11.4985525 11.4998008 11.5006355 11.5057486 11.5075441 11.5105112
[3457] 11.5127733 11.5175541 11.5179745 11.5215594 11.5221848 11.5231192
[3463] 11.5241322 11.5267978 11.5274311 11.5300284 11.5310523 11.5319288
[3469] 11.5326395 11.5331983 11.5341134 11.5356629 11.5358794 11.5367631
[3475] 11.5393464 11.5398767 11.5414574 11.5426525 11.5439638 11.5449067
[3481] 11.5455295 11.5464165 11.5464779 11.5465180 11.5468116 11.5501781
[3487] 11.5502687 11.5521786 11.5547561 11.5578935 11.5583292 11.5592764
[3493] 11.5598776 11.5620890 11.5635063 11.5644130 11.5665518 11.5667107
[3499] 11.5669024 11.5677533 11.5697320 11.5706217 11.5708371 11.5719408
[3505] 11.5739331 11.5741954 11.5750365 11.5751838 11.5756136 11.5783448
[3511] 11.5790277 11.5794758 11.5796302 11.5800329 11.5844321 11.5869081
[3517] 11.5877556 11.5879698 11.5894204 11.5900301 11.5920725 11.5928951
[3523] 11.5946503 11.5960012 11.5960128 11.5980890 11.6009796 11.6039053
[3529] 11.6064090 11.6099347 11.6100803 11.6104058 11.6111700 11.6149699
[3535] 11.6182984 11.6213533 11.6252632 11.6266914 11.6276281 11.6292073
[3541] 11.6302932 11.6312985 11.6341939 11.6356793 11.6360328 11.6442150
[3547] 11.6454191 11.6463500 11.6481042 11.6505715 11.6509418 11.6567135
[3553] 11.6571935 11.6594688 11.6608087 11.6615377 11.6627323 11.6688466
[3559] 11.6707974 11.6711023 11.6725605 11.6745481 11.6746641 11.6765572
[3565] 11.6769267 11.6863201 11.6872953 11.6883890 11.6907848 11.6917329
[3571] 11.6932867 11.6978307 11.7030725 11.7032231 11.7033184 11.7042063
[3577] 11.7058904 11.7071161 11.7074352 11.7089362 11.7101078 11.7112017
[3583] 11.7119005 11.7138336 11.7152895 11.7194863 11.7196873 11.7206191
[3589] 11.7206790 11.7225574 11.7237803 11.7242695 11.7247054 11.7252145
[3595] 11.7267419 11.7272852 11.7282412 11.7283546 11.7310475 11.7328924
[3601] 11.7338126 11.7339439 11.7346425 11.7365068 11.7379669 11.7419197
[3607] 11.7422256 11.7436450 11.7449848 11.7463191 11.7484980 11.7515108
[3613] 11.7527665 11.7539199 11.7559899 11.7561326 11.7563341 11.7568612
[3619] 11.7578850 11.7624634 11.7660819 11.7668042 11.7686308 11.7714026
[3625] 11.7730110 11.7734241 11.7738096 11.7757751 11.7764896 11.7778596
[3631] 11.7874103 11.7874290 11.7875165 11.7880782 11.7888490 11.7888673
[3637] 11.7905816 11.7913893 11.7927274 11.7930362 11.7930440 11.7951726
[3643] 11.7988669 11.7990292 11.7991856 11.7996072 11.8017512 11.8019791
[3649] 11.8045086 11.8108930 11.8156585 11.8157632 11.8165355 11.8167218
[3655] 11.8186758 11.8200078 11.8234806 11.8249240 11.8249794 11.8291692
[3661] 11.8302116 11.8311982 11.8312253 11.8317263 11.8355976 11.8397651
[3667] 11.8399913 11.8434043 11.8480098 11.8492766 11.8511025 11.8551431
[3673] 11.8576875 11.8584522 11.8595290 11.8619317 11.8628082 11.8646144
[3679] 11.8651917 11.8659934 11.8670094 11.8741576 11.8749085 11.8754159
[3685] 11.8796104 11.8805558 11.8833257 11.8855859 11.8857657 11.8864159
[3691] 11.8867257 11.8877927 11.8883171 11.8895033 11.8933512 11.8976752
[3697] 11.8980159 11.8989251 11.9019673 11.9022658 11.9066425 11.9087873
[3703] 11.9096611 11.9116499 11.9119494 11.9123417 11.9188662 11.9252587
[3709] 11.9253968 11.9256244 11.9277732 11.9328587 11.9334744 11.9348005
[3715] 11.9354210 11.9413495 11.9438658 11.9482163 11.9511829 11.9517668
[3721] 11.9518170 11.9519726 11.9566494 11.9567386 11.9581367 11.9591977
[3727] 11.9599158 11.9656649 11.9656994 11.9662332 11.9669278 11.9688145
[3733] 11.9704617 11.9748214 11.9764685 11.9792539 11.9816031 11.9827364
[3739] 11.9827625 11.9828814 11.9875396 11.9893553 11.9902849 11.9903035
[3745] 11.9909190 11.9914503 11.9979171 11.9997007 12.0072441 12.0078297
[3751] 12.0079724 12.0091848 12.0145616 12.0164660 12.0173476 12.0176045
[3757] 12.0235895 12.0249549 12.0252474 12.0264144 12.0280018 12.0348376
[3763] 12.0350845 12.0358870 12.0371549 12.0379382 12.0390547 12.0393775
[3769] 12.0398992 12.0413431 12.0441705 12.0469194 12.0498869 12.0500701
[3775] 12.0568587 12.0572749 12.0586782 12.0600023 12.0616907 12.0680177
[3781] 12.0715569 12.0746378 12.0763353 12.0787945 12.0793831 12.0800142
[3787] 12.0812275 12.0819861 12.0871211 12.0876828 12.0896299 12.0926883
[3793] 12.0941677 12.0993343 12.1004659 12.1048370 12.1072925 12.1156413
[3799] 12.1166119 12.1171769 12.1206602 12.1209175 12.1253096 12.1253228
[3805] 12.1271153 12.1271565 12.1294516 12.1297865 12.1308751 12.1313142
[3811] 12.1378063 12.1387499 12.1389984 12.1427469 12.1468738 12.1468778
[3817] 12.1476890 12.1476932 12.1538994 12.1551680 12.1560109 12.1597231
[3823] 12.1618808 12.1641009 12.1645086 12.1684546 12.1688767 12.1709651
[3829] 12.1716519 12.1798982 12.1823524 12.1865373 12.1879368 12.1889214
[3835] 12.1909014 12.1910080 12.1914597 12.1933882 12.1948114 12.2005212
[3841] 12.2013768 12.2021348 12.2026528 12.2052128 12.2062442 12.2064641
[3847] 12.2137923 12.2164712 12.2165035 12.2180419 12.2181301 12.2183545
[3853] 12.2189828 12.2210457 12.2213702 12.2229044 12.2233880 12.2319078
[3859] 12.2326945 12.2367462 12.2437129 12.2438616 12.2452225 12.2455868
[3865] 12.2462504 12.2503321 12.2511440 12.2512922 12.2556144 12.2557298
[3871] 12.2576390 12.2651167 12.2657086 12.2658824 12.2717141 12.2730607
[3877] 12.2743637 12.2758511 12.2775477 12.2780779 12.2806561 12.2820846
[3883] 12.2826209 12.2872861 12.2914413 12.2928010 12.2942473 12.2963763
[3889] 12.2976311 12.2985646 12.3056727 12.3062062 12.3075129 12.3100803
[3895] 12.3122479 12.3150300 12.3196556 12.3205982 12.3206638 12.3208418
[3901] 12.3223224 12.3276870 12.3277520 12.3321134 12.3340817 12.3381298
[3907] 12.3391176 12.3404981 12.3421891 12.3422764 12.3429607 12.3468429
[3913] 12.3499462 12.3527686 12.3558619 12.3567858 12.3614117 12.3677277
[3919] 12.3687341 12.3727599 12.3743976 12.3752334 12.3770199 12.3772911
[3925] 12.3796604 12.3811535 12.3830111 12.3887209 12.3888232 12.3909958
[3931] 12.3929744 12.3937165 12.3944955 12.3984538 12.4006478 12.4032707
[3937] 12.4038307 12.4039621 12.4086382 12.4092555 12.4097171 12.4126457
[3943] 12.4141626 12.4141751 12.4157372 12.4182422 12.4234735 12.4266582
[3949] 12.4281001 12.4439223 12.4478783 12.4488496 12.4502298 12.4504258
[3955] 12.4548112 12.4564437 12.4573641 12.4597178 12.4601713 12.4626708
[3961] 12.4634734 12.4675524 12.4687421 12.4706591 12.4730605 12.4733376
[3967] 12.4744810 12.4749391 12.4771398 12.4817526 12.4818909 12.4872957
[3973] 12.4893653 12.4987135 12.4995181 12.5008519 12.5012204 12.5075863
[3979] 12.5077053 12.5081242 12.5082840 12.5097129 12.5102539 12.5193503
[3985] 12.5216337 12.5229894 12.5241435 12.5271208 12.5283277 12.5285303
[3991] 12.5288928 12.5316350 12.5353798 12.5425528 12.5431679 12.5435688
[3997] 12.5456606 12.5459836 12.5465217 12.5487221 12.5496150 12.5521442
[4003] 12.5538379 12.5553634 12.5579550 12.5621634 12.5663408 12.5691003
[4009] 12.5702121 12.5725906 12.5755533 12.5781354 12.5789798 12.5805027
[4015] 12.5826671 12.5886738 12.5891546 12.5912307 12.5932547 12.5958918
[4021] 12.5960819 12.5966869 12.5978636 12.5994889 12.6013637 12.6048902
[4027] 12.6049039 12.6119954 12.6145684 12.6167960 12.6173889 12.6203232
[4033] 12.6256720 12.6273371 12.6295210 12.6298999 12.6373172 12.6402435
[4039] 12.6415352 12.6428160 12.6429957 12.6438897 12.6440402 12.6449101
[4045] 12.6497121 12.6566699 12.6572620 12.6574804 12.6633561 12.6637871
[4051] 12.6674857 12.6678771 12.6762725 12.6772191 12.6776145 12.6829197
[4057] 12.6831945 12.6866657 12.6895461 12.6899767 12.6911904 12.6939052
[4063] 12.6939302 12.6959362 12.6989088 12.7039718 12.7047577 12.7066067
[4069] 12.7080937 12.7096517 12.7098806 12.7099100 12.7113585 12.7127972
[4075] 12.7135344 12.7142223 12.7177193 12.7183544 12.7186302 12.7195069
[4081] 12.7212061 12.7259064 12.7262367 12.7278412 12.7283247 12.7343508
[4087] 12.7368547 12.7405602 12.7435852 12.7467056 12.7467255 12.7489504
[4093] 12.7496268 12.7515032 12.7554350 12.7659921 12.7660715 12.7661008
[4099] 12.7705191 12.7719266 12.7729971 12.7739530 12.7752144 12.7791046
[4105] 12.7816206 12.7820196 12.7833476 12.7873691 12.7875308 12.7882440
[4111] 12.7984963 12.8035894 12.8053533 12.8113263 12.8119704 12.8129481
[4117] 12.8145676 12.8181116 12.8192313 12.8271844 12.8275815 12.8280009
[4123] 12.8296565 12.8313743 12.8320757 12.8330233 12.8348303 12.8371977
[4129] 12.8380630 12.8389587 12.8443636 12.8477285 12.8492363 12.8493325
[4135] 12.8503009 12.8512248 12.8547186 12.8598290 12.8638280 12.8690323
[4141] 12.8730214 12.8750358 12.8752124 12.8836919 12.8855765 12.8880723
[4147] 12.8926577 12.8954623 12.8959053 12.8959896 12.8973187 12.8993951
[4153] 12.9030278 12.9081447 12.9121513 12.9122322 12.9161568 12.9177633
[4159] 12.9238128 12.9260630 12.9264697 12.9272270 12.9359785 12.9390453
[4165] 12.9464681 12.9509901 12.9545687 12.9563805 12.9570322 12.9601698
[4171] 12.9622656 12.9665135 12.9710686 12.9741184 12.9746197 12.9769658
[4177] 12.9779382 12.9788041 12.9825487 12.9851181 12.9890029 12.9919045
[4183] 12.9933330 12.9983869 12.9995392 12.9996588 13.0001541 13.0001637
[4189] 13.0010495 13.0019441 13.0030086 13.0044530 13.0045870 13.0054154
[4195] 13.0070297 13.0070664 13.0071135 13.0157338 13.0184569 13.0194958
[4201] 13.0219414 13.0220001 13.0224495 13.0225721 13.0279688 13.0305392
[4207] 13.0318661 13.0324958 13.0326720 13.0345726 13.0346840 13.0393298
[4213] 13.0415606 13.0424389 13.0459956 13.0460690 13.0501574 13.0513318
[4219] 13.0521008 13.0595032 13.0609517 13.0627563 13.0665027 13.0713954
[4225] 13.0729509 13.0763185 13.0765086 13.0784705 13.0796178 13.0810043
[4231] 13.0812268 13.0826119 13.0831898 13.0847962 13.0849922 13.0859416
[4237] 13.0861194 13.0880905 13.0900644 13.0914196 13.0925873 13.0941926
[4243] 13.0954842 13.0964268 13.0990687 13.0999744 13.1017260 13.1017479
[4249] 13.1073117 13.1094737 13.1094823 13.1097397 13.1101813 13.1160073
[4255] 13.1177776 13.1198959 13.1208245 13.1216606 13.1217969 13.1221884
[4261] 13.1236421 13.1283681 13.1342511 13.1342554 13.1369605 13.1371311
[4267] 13.1411247 13.1424057 13.1447776 13.1449568 13.1458370 13.1517322
[4273] 13.1524176 13.1562232 13.1586708 13.1597011 13.1607737 13.1616573
[4279] 13.1631587 13.1633167 13.1656169 13.1678663 13.1685956 13.1709387
[4285] 13.1710638 13.1717309 13.1734599 13.1736223 13.1778324 13.1845587
[4291] 13.1876063 13.1910093 13.1914232 13.1917066 13.1994077 13.1994973
[4297] 13.2002327 13.2072732 13.2109870 13.2119792 13.2148556 13.2157957
[4303] 13.2160787 13.2162072 13.2173423 13.2214023 13.2323844 13.2360654
[4309] 13.2367933 13.2369650 13.2373771 13.2379416 13.2450527 13.2470330
[4315] 13.2485534 13.2486708 13.2506855 13.2514246 13.2531870 13.2534375
[4321] 13.2562066 13.2582574 13.2608725 13.2646059 13.2749688 13.2754694
[4327] 13.2772168 13.2816694 13.2845743 13.2881192 13.2884065 13.2888361
[4333] 13.2890443 13.2940806 13.3003099 13.3039330 13.3045672 13.3084589
[4339] 13.3160673 13.3177145 13.3198684 13.3229302 13.3232957 13.3250355
[4345] 13.3252134 13.3277867 13.3337499 13.3344311 13.3355217 13.3408516
[4351] 13.3416248 13.3443667 13.3479275 13.3488081 13.3503832 13.3511817
[4357] 13.3520470 13.3520799 13.3562302 13.3571504 13.3572668 13.3576077
[4363] 13.3590870 13.3619878 13.3647933 13.3705385 13.3734351 13.3744367
[4369] 13.3744586 13.3768589 13.3945965 13.3955679 13.4019622 13.4048399
[4375] 13.4051618 13.4139304 13.4141373 13.4151503 13.4191112 13.4224267
[4381] 13.4272348 13.4320921 13.4381911 13.4387095 13.4449683 13.4558334
[4387] 13.4605285 13.4619771 13.4654786 13.4692527 13.4693657 13.4718557
[4393] 13.4742996 13.4744175 13.4747637 13.4751362 13.4759189 13.4759445
[4399] 13.4767895 13.4777998 13.4868862 13.4870158 13.4916976 13.4928681
[4405] 13.4948217 13.4976247 13.5041723 13.5092843 13.5167202 13.5265623
[4411] 13.5275929 13.5318801 13.5329499 13.5370684 13.5398576 13.5443883
[4417] 13.5489405 13.5514709 13.5524202 13.5535298 13.5545311 13.5552702
[4423] 13.5559036 13.5649551 13.5679786 13.5698785 13.5700291 13.5747016
[4429] 13.5757845 13.5772056 13.5786456 13.5790613 13.5814660 13.5819264
[4435] 13.5824641 13.5837437 13.5838988 13.5910385 13.5984713 13.6066771
[4441] 13.6070845 13.6105930 13.6143044 13.6167466 13.6186696 13.6238470
[4447] 13.6239338 13.6270021 13.6308676 13.6320906 13.6333373 13.6341522
[4453] 13.6383975 13.6393569 13.6426013 13.6433367 13.6458398 13.6479764
[4459] 13.6511604 13.6530135 13.6585681 13.6681965 13.6712426 13.6739270
[4465] 13.6751127 13.6773807 13.6780148 13.6786222 13.6825766 13.6834428
[4471] 13.6836765 13.6868853 13.6869861 13.6884675 13.6973811 13.6981025
[4477] 13.7008027 13.7039871 13.7048641 13.7124920 13.7139868 13.7142496
[4483] 13.7178573 13.7217640 13.7239204 13.7382485 13.7431067 13.7437410
[4489] 13.7474640 13.7488687 13.7501368 13.7525032 13.7531552 13.7541395
[4495] 13.7572350 13.7572496 13.7575193 13.7605490 13.7658484 13.7668250
[4501] 13.7737729 13.7738078 13.7755681 13.7783416 13.7809830 13.7826430
[4507] 13.7869144 13.7912620 13.7964292 13.7976313 13.7996785 13.8041697
[4513] 13.8296197 13.8302756 13.8381155 13.8479503 13.8560727 13.8562792
[4519] 13.8571589 13.8587347 13.8609823 13.8664756 13.8747839 13.8756160
[4525] 13.8795633 13.8849686 13.8982767 13.9154883 13.9204103 13.9221859
[4531] 13.9226360 13.9328160 13.9376539 13.9408153 13.9489595 13.9572225
[4537] 13.9659614 13.9705942 13.9708648 13.9719701 13.9730174 13.9769878
[4543] 13.9797300 13.9797748 13.9842049 14.0056930 14.0060340 14.0085116
[4549] 14.0101636 14.0130105 14.0189316 14.0227780 14.0278796 14.0286854
[4555] 14.0382272 14.0387798 14.0436653 14.0447302 14.0458205 14.0471759
[4561] 14.0511405 14.0540803 14.0601423 14.0604612 14.0638497 14.0648670
[4567] 14.0649228 14.0688737 14.0745723 14.0757235 14.0796495 14.0869948
[4573] 14.0893191 14.0916373 14.0997868 14.1072899 14.1100470 14.1164275
[4579] 14.1252755 14.1285066 14.1296236 14.1360306 14.1371316 14.1379445
[4585] 14.1392060 14.1434462 14.1448739 14.1471633 14.1504338 14.1518722
[4591] 14.1529730 14.1565425 14.1653118 14.1701838 14.1725149 14.1751253
[4597] 14.1788062 14.1826036 14.1975447 14.2010793 14.2037523 14.2152527
[4603] 14.2205053 14.2234617 14.2247582 14.2257927 14.2259957 14.2290419
[4609] 14.2297346 14.2329630 14.2351890 14.2376159 14.2411185 14.2462911
[4615] 14.2467576 14.2509505 14.2542363 14.2543185 14.2552134 14.2777276
[4621] 14.2805275 14.2809963 14.2823193 14.2852806 14.2861998 14.2900564
[4627] 14.2948465 14.3001620 14.3026780 14.3038877 14.3109797 14.3122976
[4633] 14.3205925 14.3265145 14.3338895 14.3346763 14.3397656 14.3419986
[4639] 14.3444548 14.3538779 14.3615385 14.3619809 14.3666181 14.3674053
[4645] 14.3697177 14.3701723 14.3748446 14.3819102 14.3946616 14.3959478
[4651] 14.4105826 14.4140408 14.4193292 14.4216094 14.4313164 14.4326525
[4657] 14.4419898 14.4446802 14.4487754 14.4489980 14.4518865 14.4535875
[4663] 14.4796452 14.4798077 14.4945255 14.4946807 14.4950648 14.5011799
[4669] 14.5020426 14.5037782 14.5048644 14.5099243 14.5138095 14.5238349
[4675] 14.5243520 14.5346504 14.5427673 14.5537165 14.5562248 14.5706674
[4681] 14.5735988 14.5748752 14.5767547 14.5864472 14.5918130 14.5958589
[4687] 14.5976308 14.6072521 14.6286957 14.6338228 14.6432868 14.6476302
[4693] 14.6564588 14.6568048 14.6596004 14.6624508 14.6689713 14.6716442
[4699] 14.6732474 14.6754620 14.6780509 14.6795905 14.6854096 14.6941466
[4705] 14.6984830 14.7016175 14.7026318 14.7028261 14.7209971 14.7224119
[4711] 14.7260008 14.7262641 14.7264198 14.7282096 14.7345768 14.7385795
[4717] 14.7386655 14.7591034 14.7691530 14.7700162 14.7754080 14.7872329
[4723] 14.8033642 14.8124874 14.8141896 14.8232422 14.8325272 14.8370330
[4729] 14.8375182 14.8378663 14.8380135 14.8420250 14.8490922 14.8520913
[4735] 14.8576110 14.8740659 14.8744649 14.8880231 14.8951861 14.8967798
[4741] 14.8982088 14.9037594 14.9082848 14.9092252 14.9099091 14.9165673
[4747] 14.9218203 14.9367771 14.9394922 14.9465841 14.9573654 14.9581805
[4753] 14.9683106 14.9723225 14.9752697 14.9871046 14.9895989 14.9944398
[4759] 15.0036875 15.0166838 15.0186023 15.0262410 15.0289807 15.0298227
[4765] 15.0338302 15.0359658 15.0573745 15.0641677 15.0659808 15.0776048
[4771] 15.0781830 15.0949264 15.0952902 15.0953119 15.1052938 15.1243212
[4777] 15.1255204 15.1273635 15.1282605 15.1295338 15.1334283 15.1362581
[4783] 15.1457794 15.1551983 15.1565280 15.1571637 15.1672327 15.1684281
[4789] 15.1695128 15.1841684 15.1882735 15.1927299 15.1965789 15.2047805
[4795] 15.2131840 15.2192821 15.2196989 15.2471300 15.2493737 15.2559006
[4801] 15.2562260 15.2672886 15.2684403 15.2699156 15.2708118 15.2844852
[4807] 15.2869847 15.2902317 15.3097951 15.3164700 15.3210792 15.3261652
[4813] 15.3263202 15.3271389 15.3312100 15.3359947 15.3397871 15.3403049
[4819] 15.3419667 15.3479754 15.3567533 15.3589489 15.3609259 15.3738379
[4825] 15.3781568 15.3820535 15.3890061 15.3909497 15.4003579 15.4123057
[4831] 15.4198569 15.4222086 15.4282932 15.4438776 15.4439083 15.4478594
[4837] 15.4525324 15.4554152 15.4612875 15.4617193 15.4639227 15.4680230
[4843] 15.4822214 15.4825070 15.4872693 15.5091403 15.5323215 15.5345164
[4849] 15.5368825 15.5457917 15.5537540 15.5565559 15.5656394 15.5805577
[4855] 15.5825359 15.5992401 15.6061015 15.6189356 15.6334175 15.6575922
[4861] 15.6681300 15.6690653 15.6786103 15.6802377 15.7097645 15.7100557
[4867] 15.7231855 15.7286073 15.7344454 15.7456991 15.7588453 15.7647823
[4873] 15.7698539 15.7741090 15.7797609 15.7905587 15.7913583 15.8123940
[4879] 15.8400004 15.8601247 15.8768581 15.8854901 15.8960658 15.9053880
[4885] 15.9121372 15.9181516 15.9192291 15.9207439 15.9220447 15.9338816
[4891] 15.9370505 15.9562621 15.9815344 15.9900873 16.0201502 16.0232773
[4897] 16.0347217 16.0402252 16.0609092 16.0895943 16.0965689 16.1186051
[4903] 16.1304441 16.1334973 16.1345469 16.1456014 16.1537537 16.1623289
[4909] 16.1762881 16.1999693 16.2214578 16.2267133 16.2573039 16.2855921
[4915] 16.2894463 16.2961118 16.3011658 16.3056647 16.3084932 16.3120504
[4921] 16.3269029 16.3364735 16.3521765 16.3580115 16.3638598 16.3793918
[4927] 16.4082274 16.4168582 16.4322974 16.4406293 16.4594794 16.4634214
[4933] 16.4741032 16.4862653 16.4924662 16.4932352 16.5026965 16.5100369
[4939] 16.5135970 16.5544317 16.6036039 16.6061192 16.6260584 16.6467123
[4945] 16.6629920 16.7015086 16.7068102 16.7238192 16.7873097 16.8110663
[4951] 16.8336375 16.8356145 16.8624233 16.8760549 16.9371462 16.9773948
[4957] 16.9917909 17.0188799 17.0526029 17.0710984 17.1016879 17.1062616
[4963] 17.1362985 17.1887287 17.2166508 17.2528501 17.2892731 17.3152612
[4969] 17.3448231 17.3870505 17.5153157 17.6188173 17.6846070 17.7108944
[4975] 17.8115085 17.8533770 17.8920888 17.9451848 17.9654279 17.9812062
[4981] 18.1062997 18.1162707 18.1336216 18.1562876 18.1666327 18.2208342
[4987] 18.3135688 18.3407491 18.4673863 18.5187772 18.5693098 18.7911800
[4993] 18.8090445 18.9536360 19.2536029 19.5845364 19.6748008 20.1131085
[4999] 20.5416237 21.7146379

Bibliografía y lecturas

Bibliografía

1. Álvarez Cáseres R (2007). ESTADÍSTICA APLICADA A LAS CIENCIAS DE LA SALUD. Díaz-Santos, España
2. Celis de la Rosa, A. de J., & Labrada Martagón, V. (2014). Bioestadística (3ra ed.). Guadalajara, México: Editorial El Manual Moderno.
3. Babak S. (2012). Biostatistics with R: An Introduction to Statistics Through Biological Data (Use R!). Springer New York Dordrecht Heidelberg London

Más información

• https://www.r-bloggers.com/whisker-of-boxplot/
• https://bookdown.org/gboccardo/manual-ED-UCH/estadistica-descriptiva-con-rstudio.html
• https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2770212/
• https://ncbi.nlm.nih.gov/pubmed/18450044

Ejercicios clase

Instrucciones

• Con los siguientes vectores estime:
  • Cuartiles
  • Rango intercuartil
    
  • Percentil 85, 77, 44
  • Realice un boxplot e interprete
• Solo puede crear objetos y usar la función boxplot y sort
• Utilice la formula de la clase para los percentiles: \(P(n +1)\) cambie el argunmento type=6

Vector

Talla (1520.00, 152.00, 156.00, 158.00, 156.00, 161.00, 158.00, 152.00, 154.00, 150.0, 165.00, 148.00, 164.00, 153.00, 163.00, 154.00, 153.00, 155.00, 153.00, 164.00, 160.00, 158.00, 156.00)

Instrucciones

• Con los siguientes vectores estime:
  • Cuartiles
  • Rango intercuartil
    
  • Percentil 85, 77, 44
  • Realice un boxplot e interprete
• Solo puede crear objetos y usar la función boxplot y sort
• Utilice la formula de la clase: \(P(n +1)\)

Vector

• Citocina (8.14, 36.95, 423.85, 104.86, 22.47, 245.53,47.74,424.18, 438.99,170.02,22.13,915.90, 190.96, 146.04,169.63 ,49.68,168.10, 163.10,317.60 ,603.76, 141.51,12.90,40.42,30.79,190.18,192.13,43.90,131.04, 50.89,484.36,25.22,11.54, 125.46, 14.60, 8.71,23.84,198.76,17.74,14.95,27.62, 146.00,16.65,29.35,36.26,9.16,230.74, 14.95, 20.07, 19.39)

Instrucciones

• Con los vectores descritos anteriormente y utilizando funciones de R:
  • Cuartiles
  • Rango intercuartil
    
  • Percentil 85, 77, 44
  • Realice un boxplot e interprete

Solución para la variable Talla

    0%    25%    50%    75%    85%    77%    44% 
148.00 153.00 156.00 161.00 164.00 161.96 155.56 

Solución para la variable Talla en R

quantile(Talla, c(0.0, 0.25, 0.50, 0.75, 0.85,0.77,0.44))#No cambie el argumento type

    0%    25%    50%    75%    85%    77%    44% 
148.00 153.00 156.00 160.50 163.70 160.94 155.68 

Solución para la variable Talla

boxplot(Talla, 
        main="Boxplot de la Talla de un grupo de personas", ylab="cm")

Solución para la variable Talla

Solución para la variable Citocina

Citocina<-c (8.14, 36.95, 423.85, 104.86, 22.47, 245.53,47.74,424.18,
             438.99,170.02,22.13,915.90, 190.96, 146.04,169.63,
             49.68,168.10, 163.10,317.60 ,603.76,
             141.51,12.90,40.42,30.79,190.18,192.13,43.90,131.04,
             50.89,484.36,25.22,11.54, 125.46, 14.60,
             8.71,23.84,198.76,17.74,14.95,27.62,
             146.00,16.65,29.35,36.26,9.16,230.74, 14.95, 20.07, 19.39)
quantile(Citocina, c(0.0, 0.25, 0.50, 0.75, 0.85,0.77,0.44), type = 6)

Solución para la variable Citocina

     0%     25%     50%     75%     85%     77%     44% 
  8.140  21.100  49.680 190.570 281.565 191.545  40.420 

Solución para la variable Citocina en R

quantile(Citocina, c(0.0, 0.25, 0.50, 0.75, 0.85,0.77,0.44))#No cambie el argumento type

      0%      25%      50%      75%      85%      77%      44% 
  8.1400  22.1300  49.6800 190.1800 242.5720 190.9288  40.8376 

Solución para la variable Talla

boxplot(Citocina, 
        main="Boxplot de las concentraciones de una Citocina en un grupo de personas", 
        ylab="ng/mL")

Solución para la variable Talla