Good teaching activities for variability and standardized distributions for undergrad stats As the question states, I am looking for some warmup activities for teaching undergrads practical examples of variation and standardized distributions. I think the first part is relatively easy to teach. I was thinking of selecting a small random sample of students, picking out about 5 or 6 of their heights, then asking them if they think it represents the entire class. I would probably draw it on the board like below so it makes the data more "real" (the red line is the mean and the black dots are raw values):

However, I'm not sure if there are any practical examples for teaching standardization in class that immediately come to mind. I was thinking of taking two things that are on different scales and estimating their z scores in some way, but not sure if there is a more fun way of doing that. Are there any activities that you would recommend?

Just to give you some basic ideas (if other things come to my mind I'll update the answer), I would show how standardization:

*

*makes the distribution of different (normally distributed) variables similar to each other. See figure.

*Preserves correlation but the covariance matrix is not the same (run code). You can show how this is justified by the corresponding formulas.

Apart from this, I suggest having a look on Teaching Statistics which is academic journal specialized in teaching statistics.

Code:
set.seed(123)

v1 <- rnorm(1000, mean = 20, sd=2)
v2 <- rnorm(1000, mean = 30, sd=4)
v3 <- rnorm(1000, mean = 25, sd=0.5)
cov(cbind(v1,v2,v3))
cor(cbind(v1,v2,v3))


s1 <- scale(v1)[,1]; s2 <- scale(v2)[,1]; s3 <- scale(v3)[,1]
cov(cbind(v1,v2,v3))
cor(cbind(s1,s2,s3))


par(mfrow = c(3, 1))
h1 <- hist(v1, plot = F)
hs1 <- hist(s1, plot = F)
h2 <- hist(v2,  plot = F)
hs2 <- hist(s2,  plot = F)
h3 <- hist(v3,  plot = F)
hs3 <- hist(s3,  plot = F)

plot(h1, col="lightblue", xlim=c(-10,50),ylim=c(0,0.5),freq = F)
abline(v = quantile(v1,probs=c(0.05, 0.5, 0.95)), col="blue",cex=2)
plot(hs1, col="green", add=T,freq = F)
abline(v = quantile(s1,probs=c(0.05, 0.5, 0.95)), col="red",cex=2)


plot(h2, col="lightblue", xlim=c(-10,50),ylim=c(0,0.5),freq = F)
abline(v = quantile(v2,probs=c(0.05, 0.5, 0.95)), col="blue",cex=2)
plot(hs2, col="green", add=T,freq = F)
abline(v = quantile(s2,probs=c(0.05, 0.5, 0.95)), col="red",cex=2)


plot(h3, col="lightblue", xlim=c(-10,50), ylim=c(0,0.5),freq = F)
abline(v = quantile(v3,probs=c(0.05, 0.5, 0.95)), col="blue",cex=2)
plot(hs3, col="green", add=T,freq = F)
abline(v = quantile(s3,probs=c(0.05, 0.5, 0.95)), col="red",cex=2)

