For example, can I take samples of size ten with replacement from a sample of size 100? I'm trying to teach my high school statistics students about bootstrapping and I want to use m and ms candy, but taking multiple samples of 100 from the original sample would take a very long time so I wanted to see if I can have them do 10 samples of the original sample of 100 with replacement. Is that still bootstrapping?


1 Answer 1


Try it and see what happens.

N <- 100
B <- 1000
n <- 10
x <- rnorm(N, 0, 1)
xbar100 <- xbar10 <- rep(NA, B)
for (i in 1:B){
    xbar100[i] <- mean(sample(x, N, replace = T))
    xbar10[i]  <- mean(sample(x, n, replace = T))
sd(xbar100) # 0.0981310715978676
sd(xbar10)  # 0.307451079371199

While this isn’t the best way to calculate bootstrap standard errors, the fact that bootstrapping with the original sample size gives a standard error close to the true value of $0.1$ while bootstrap sampling ten observations gives a standard error three times bigger shows the issues with this.

Perhaps a bigger demonstration of why you shouldn’t change the sample size would be seen by changing the code to give n <- 1000000 and shrink the standard error to almost zero.


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