The Rule of 5

Douglas Hubbard’s Rule of 5: There is a 93.8% chance that the median of a population is between the smallest and largest values in any random sample of five from that population.

We could derive 93.8 by using equations, or we could run monte carlo simulations.

One sample of 5 from a distribution:

n <- 1000
mean <- 300
sd <- 24
population <- rnorm(n, mean, sd)

draw <- sample(population, size = 5, replace = F)
low <- min(draw)
high <- max(draw)

contains_true <- NULL

if(low < mean & high > mean){
  contains_true <- 'yes'
}else{
  contains_true <- 'no'
}

print(c(low, 
        high,
        contains_true))

[1] "264.729496040933" "324.403709124486" "yes"             

Now iterate the same scheme many times. Does the median fall between the low and high values 93% of the time?

simulations <- 5000
storeit <- numeric(simulations)

for(i in 1:simulations){
  
  n <- 1000
  mean <- 300
  sd <- 24
  population <- rnorm(n, mean, sd)
  
  draw <- sample(population, size = 5, replace = F)
  low <- min(draw)
  high <- max(draw)
  
  contains_true <- NULL
  
  if(low < mean & high > mean){
    contains_true <- 'yes'
  }else{
    contains_true <- 'no'
  }
  
  storeit[i] <- contains_true
  
}

sum(storeit == 'yes') / simulations

[1] 0.9444

Yes.

Bo\(^2\)m =)