$t$ Distribution

This is a family of Continuous Distributions

  • unimodal and bell-shaped, like Normal Distribution
  • centered at 0
  • has one parameter: degrees of freedom ($\text{df}$)


Origin

Origin (and usage):

  • arises when estimating the mean of normally distributed population when
  • sample size is small and population standard deviation is unknown


$t$-distribution vs Normal

  • for large $\text{df}$ ($\geqslant 100$) $t$-dist closely follows $N(0,1)$
  • but even for $\text{df} \geqslant 30$ it's already almost indistinguishable

2d36f1963cc54cd5be3534c691f68c1c.gif

  • for $t$ tails are thicker
    • so observations are more likely to fall beyond 2$\sigma$ from the mean (than under $N(0,1)$)
  • it's good for t-tests:
    • the thick tails are exactly the correction to deal with poorly estimated Standard Error


50205db619254cd9a7eded5d7579cabe.png

  • here, $\text{df}$ is the lowest, and it approaches the normal curse as $\text{df}$ grows
R code to produce the figure  
default.par = par()

x = seq(-4,4,0.1)
n = dnorm(x)

library(animation)

saveGIF({
  par(mar=c(0,0,0,0))
  
  for (i in 1:100) {
    plot(x, n, type='l', lty=2, col='grey')
    t = dt(x, df=i)
    lines(x, t, col='blue')
    text(1.5, 0.37, paste('df =', i))
    text(1.66, 0.35, format(sum(abs(n - t)))) 
  }
}, interval=0.1)

par(mar=c(0,0,0,0))
plot(x, n, type='l', lty=2, col='grey')

for (i in 1:7) {
  t = dt(x, df=i)
  lines(x, t, col=i)
}

par(default.par)



Sources