# When doing bootstrapping, can you take smaller samples of a sample with replacement?

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?

• No. The main aim of the Bootstrap is to estimate uncertainty, which is massively being impacted by the sample size. Thus, it is important to keep the sample size fixed. Dec 10, 2023 at 14:26
• Does this answer your question? Can we use bootstrap samples that are smaller than original sample? Dec 10, 2023 at 15:29

Try it and see what happens.

set.seed(2023)
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.