Correlation & Z-Scores

If you restrict the range of one variable, does the correlation remain unchanged?

#### correlation unchanged? ####
set.seed(100)  # to make sure we can reproduce same values
x <- rnorm(100, mean=10, sd=5)
y <- x + rnorm(100, 0, 10)

plot(x, y)

cor(x, y)

## [1] 0.4547479

greater.than.10 <- x > 10
x.sub <- x[greater.than.10]
y.sub <- y[greater.than.10]

plot(x.sub, y.sub)

cor(x.sub, y.sub)

## [1] 0.4229429

# correlations are not equal
# cor(x, y) != cor(x.sub, y.sub)

Does z-scoring this restricted-range subset leads to a smaller correlation?

#### how does z-score change? ####
z.score = function(g) {
  return((g - mean(g)) / sd(g))
}

x.z <- z.score(x)
y.z <- z.score(y)
cor(x.z, y.z)

## [1] 0.4547479

# correlation unchanged

x.sub.z <- z.score(x.sub)
y.sub.z <- z.score(y.sub)
cor(x.sub.z, y.sub.z)

## [1] 0.4229429

# correlation unchanged