t estándar

La t de student tiene media 0, varianza \(\sigma^2_t=\frac{n-1}{n-3}\), y asimetría 0.

n=5
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c1=c(n,max(d1))
n=6
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c2=c(n,max(d1))
n=7
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c3=c(n,max(d1))
n=8
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c4=c(n,max(d1))
n=9
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c5=c(n,max(d1))
n=10
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c6=c(n,max(d1))
n=11
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c7=c(n,max(d1))
n=12
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c8=c(n,max(d1))
n=13
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c9=c(n,max(d1))
n=14
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c10=c(n,max(d1))
n=15
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c11=c(n,max(d1))
n=16
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c12=c(n,max(d1))
n=17
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c13=c(n,max(d1))
n=18
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c14=c(n,max(d1))
n=19
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c15=c(n,max(d1))
n=30
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c16=c(n,max(d1))
n=40
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c17=c(n,max(d1))
n=50
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c18=c(n,max(d1))
n=60
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c19=c(n,max(d1))
n=70
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c20=c(n,max(d1))
n=80
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c21=c(n,max(d1))
n=90
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c22=c(n,max(d1))
n=100
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c23=c(n,max(d1))
n=110
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c24=c(n,max(d1))
n=120
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c25=c(n,max(d1))
n=130
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c26=c(n,max(d1))
n=140
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c27=c(n,max(d1))
n=150
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c28=c(n,max(d1))
n=160
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c29=c(n,max(d1))
n=170
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c30=c(n,max(d1))
n=180
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c31=c(n,max(d1))
n=190
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c32=c(n,max(d1))
n=200
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c33=c(n,max(d1))
n=210
t=seq(from=-10,to=10,by=0.001)
d=pt(t,n-1)-pnorm(t/((n-1)/(n-3))^0.5,mean=0,sd=1)
d1=abs(d)
c34=c(n,max(d1))
r=rbind(c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c30,c31,c32,c33,c34)
plot(r[,1],r[,2])

Para más de 29 grados de libertad, la t de student estandarizada sigue muy aproximadamente una distribución normal, por lo que, en muestras, provenientes de poblaciones normales, de más de 30 datos conviene usar una aproximación de la t, en su forma estándar, por la normal.