#Statistique Générale Séance 1

#################################
#Exemple
#################################
n=20
tour=85
sigma=2

#################################
#Prenons 20 personnes au hasard
#################################
a=rnorm(20,85,2)
b=rep(0,20)
plot(a,b,cex = .5, col = "dark red")

#Résumons le jeu de données
summary(a)
mean(a)

#Trions nos observations 
sort(a)

#Notion de médiane
mediane = (sort(a)[10]+sort(a)[11])/2

abline(v = mediane, col = "red") 


#################################
#Prenons 21 personnes au hasard
#################################
a=rnorm(21,85,2)
b=rep(0,21)
plot(a,b,cex = .5, col = "dark red")

summary(a)
mean(a)
sort(a)
mediane = sort(a)[11]

abline(v = mediane, col = "red") 

#################################
#Boîte à  moustaches
#################################
par(mfrow=c(2,1))

a=rnorm(21,85,2)
b=rep(0,21)
mediane = sort(a)[11]
plot(a,b,cex = .5, col = "dark red")
abline(v = mediane, col = "red")

boxplot(a,horizontal=T)
abline(v = mediane, col = "red")

#################################
#Histogramme, n=200
#################################
par(mfrow=c(2,1))

a=rnorm(200,85,2)
b=rep(0,200)
plot(a,b,cex = .5, col = "dark red")

hist(a)

#################################
#Diagramme en bâtons
#################################
library(FactoMineR)
library(Rcmdr)
library(colorspace, pos=4)

data(tea)
barplot(table(tea$How), xlab="How", ylab="Frequency")

#################################
#Diagramme en camembert
#################################
pie(table(tea$How), labels=levels(tea$How), main="How", 
  col=rainbow_hcl(length(levels(tea$How))))

#################################
#Distribution normale
#################################
x=seq(-4,4,length=200)
y=dnorm(x)
plot(x,y,type="l", lwd=2, col="blue")
abline(v = 1.96, col = "grey")
abline(v = -1.96, col = "grey")

x=seq(-4,4,length=200)
y=dnorm(x)
plot(x,y,type="l", lwd=2, col="blue")
x=seq(-1.96,1.96,length=200)
y=dnorm(x)
polygon(c(-2,x,2),c(0,y,0),col="gray")

#Pourcentage des réalisations entre -1.96 et 1.96  
pnorm(1.96,mean=0,sd=1)-pnorm(-1.96,mean=0,sd=1)

#################################
#Notion d'écart-type
#################################

#Sur 200 réalisations, le nombre de réalisations inf. à -1.96 et sup. à 1.96
#A essayer sans modération
a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

#Regardons ce qui se passe sur 20 échantillons
#On va créer un vecteur pour stocker ce nombre de réalisations en dehors des bornes 
ecarttype=vector(mode = "numeric", length = 20)

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[1]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[2]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[3]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[4]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[5]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[6]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[7]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[8]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[9]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[10]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[11]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[12]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[13]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[14]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[15]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[16]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[17]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[18]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[19]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[20]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))

#On calcule la moyenne de ce nombre de réalisations
mean(ecarttype)

#Le même en "programmant"

for (i in 1:20){
a=rnorm(200,85,2)
sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
ecarttype[i]=sum(a<mean(a)-1.96*sd(a))+sum(a>mean(a)+1.96*sd(a))
}

mean(ecarttype)



#################################
#Centrage
#################################
par(mfrow=c(2,1))

a=rnorm(20,85,2)
b=rep(0,20)
plot(a,b,cex = .5, col = "dark red")
abline(v = mean(a), col = "grey")

acentre=a-mean(a)
plot(acentre,b,cex = .5, col = "dark red")
abline(v = 0, col = "grey")

#################################
#Réduction
#################################
par(mfrow=c(1,1))

b=rep(0,20)
a=rnorm(20,85,2)
acentre=a-mean(a)
plot(acentre,b,cex = .5, col = "dark red")
abline(v = 0, col = "grey")

areduit=(a-mean(a))/sd(a)
b=rep(0.5,20)
points(areduit,b,cex = .5, col = "blue")

#################################
#La moyenne des moyennes tend vers la vraie moyenne
#################################
par(mfrow=c(1,1))

#A faire tourner
a=rnorm(20,85,2)
b=rep(1,20)
plot(a,b,cex = .5, col = "dark red")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "dark red")
abline(v = 85, col = "grey")

#Regardons ce qui se passe sur 11 échantillons
#On va créer un vecteur pour stocker ce nombre de réalisations en dehors des bornes 
moyennes=vector(mode = "numeric", length = 11)

a=rnorm(20,85,2)
b=rep(1,20)
moyennes[1]=mean(a)
plot(a,b,cex = .5, col = "dark red")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "dark red")

a=rnorm(20,85,2)
b=rep(1.025,20)
moyennes[2]=mean(a)
points(a,b,cex = .5, col = "pink")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "pink")

a=rnorm(20,85,2)
b=rep(1.05,20)
moyennes[3]=mean(a)
points(a,b,cex = .5)
points(x=mean(a),y=0.8,pch=24,cex=0.5)

a=rnorm(20,85,2)
b=rep(1.075,20)
moyennes[4]=mean(a)
points(a,b,cex = .5, col = "green")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "green")

a=rnorm(20,85,2)
b=rep(1.1,20)
moyennes[5]=mean(a)
points(a,b,cex = .5, col = "purple")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "purple")

a=rnorm(20,85,2)
b=rep(1.125,20)
moyennes[6]=mean(a)
points(a,b,cex = .5, col = "tomato")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "tomato")

a=rnorm(20,85,2)
b=rep(1.15,20)
moyennes[7]=mean(a)
points(a,b,cex = .5, col = "plum")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "plum")

a=rnorm(20,85,2)
b=rep(1.175,20)
moyennes[8]=mean(a)
points(a,b,cex = .5, col = "gold")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "gold")

a=rnorm(20,85,2)
b=rep(1.2,20)
moyennes[9]=mean(a)
points(a,b,cex = .5, col = "dimgrey")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "dimgrey")

a=rnorm(20,85,2)
b=rep(1.225,20)
moyennes[10]=mean(a)
points(a,b,cex = .5, col = "aquamarine")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "aquamarine")

a=rnorm(20,85,2)
b=rep(1.25,20)
moyennes[11]=mean(a)
points(a,b,cex = .5, col = "firebrick1")
points(x=mean(a),y=0.8,pch=24,cex=0.5,col = "firebrick1")

mean(moyennes)

abline(v = mean(moyennes), col = "red")

#Par simulations
moyennes=vector(mode = "numeric", length = 100)
for (i in 1:100){
a=rnorm(20,85,2)
moyennes[i]=mean(a)
}
mean(moyennes)

#################################
#Si la distribution mère n'est pas normale?
#################################
a=rchisq(100, df=5)
b=rep(0,100)
plot(a,b,cex = .5, col = "firebrick1")
hist(a)

.x <- seq(0.158, 22.105, length.out=100)
plot(.x, dchisq(.x, df=5), xlab="x", ylab="Density", main=paste("ChiSquared Distribution:  Degrees of freedom=5"),type="l")
abline(h=0, col="gray")

#Simulons 150 échantillons suivant une loi du Chi-deux

moyennes=vector(mode = "numeric", length = 150)
for (i in 1:150){
a=rchisq(150, df=5)
moyennes[i]=mean(a)
}
hist(moyennes)