43  Sample Correlation Distribution

When \(n\) bivariate normal observations are drawn under zero population correlation (\(\rho = 0\)), the sample correlation coefficient \(r\) follows a specific symmetric distribution on \((-1, 1)\). This sampling distribution provides the exact basis for testing whether an observed correlation is statistically distinguishable from zero.

Formally, when \((X_i, Y_i)\) are i.i.d. bivariate Normal with correlation \(\rho = 0\), the sample Pearson correlation \(R\) defined on \((-1, 1)\) follows the distribution \(R \sim \text{CorDist}(n)\) with sample size \(n \geq 3\).

Note: This chapter describes the distribution of the sample correlation under \(H_0: \rho = 0\). For the full hypothesis test of \(H_0: \rho = 0\), including software output and interpretation, see Chapter 71.

WarningScope and Applicability

This distribution applies to Pearson correlation computed from i.i.d. paired observations under a bivariate Normal model and the null hypothesis \(H_0:\rho = 0\). It should not be used for sample autocorrelations of a single time series, nor for correlations between serially dependent time series. In those settings, inference is usually based on large-sample approximations such as white-noise confidence bands, Bartlett-type formulas, or large-lag variance results rather than the exact finite-sample distribution described here.

43.1 Probability Density Function

\[ f(r) = \frac{(1 - r^2)^{(n-4)/2}}{\text{B}\!\left(\tfrac{1}{2},\, \tfrac{n-2}{2}\right)}, \quad -1 < r < 1 \]

The figure below shows examples of the sample correlation distribution for different sample sizes.

Code 
dcorr <- function(r, n) {
  ifelse(abs(r) < 1,
         (1 - r^2)^((n - 4)/2) / beta(0.5, (n - 2)/2),
         0)
}

par(mfrow = c(2, 2))
r <- seq(-1, 1, length = 500)

plot(r, dcorr(r, 5), type = "l", lwd = 2, col = "blue",
     xlab = "r", ylab = "f(r)", main = "n = 5")

plot(r, dcorr(r, 10), type = "l", lwd = 2, col = "blue",
     xlab = "r", ylab = "f(r)", main = "n = 10")

plot(r, dcorr(r, 30), type = "l", lwd = 2, col = "blue",
     xlab = "r", ylab = "f(r)", main = "n = 30")

plot(r, dcorr(r, 100), type = "l", lwd = 2, col = "blue",
     xlab = "r", ylab = "f(r)", main = "n = 100")

par(mfrow = c(1, 1))
Figure 43.1: Sampling distribution of Pearson r under H0: rho = 0 for various sample sizes

43.2 Purpose

The sample correlation distribution serves as the null reference distribution for testing independence between two continuous variables in a bivariate normal population. It enables exact inference for any sample size \(n \geq 3\) without requiring large-sample approximations. Common applications include:

  • Testing whether two continuous variables are linearly uncorrelated in a bivariate normal population
  • Determining critical values for the Pearson correlation test statistic
  • Computing exact p-values for observed correlations under \(H_0: \rho = 0\)
  • Understanding the sampling variability of \(r\) when the true correlation is zero
  • Teaching the foundations of correlation hypothesis testing

Relation to the discrete setting. This is an inferential tool, not a data-generating model. Its closest discrete analog is the distribution of a sample proportion under \(H_0: p = 0.5\), which is also symmetric and used for hypothesis testing of association.

43.3 Distribution Function

The CDF is obtained via the regularized incomplete beta function. In R:

# CDF of sample correlation under H0: rho = 0
pcorr <- function(r, n) {
  pbeta((1 + r) / 2, shape1 = (n - 2) / 2, shape2 = (n - 2) / 2)
}

The figure below shows the sample correlation CDF for \(n = 30\).

Code 
pcorr <- function(r, n) {
  pbeta((1 + r) / 2, shape1 = (n - 2) / 2, shape2 = (n - 2) / 2)
}

r <- seq(-1, 1, length = 500)
plot(r, pcorr(r, 30), type = "l", lwd = 2, col = "blue",
     xlab = "r", ylab = "F(r)", main = "Sample Correlation CDF (n = 30)",
     sub = expression(H[0] ~ ":" ~ rho == 0))
Figure 43.2: CDF of sample correlation under H0: rho = 0 (n = 30)

43.4 Moment Generating Function

The MGF does not have a simple closed form. Moments are derived from the Beta distribution representation.

43.5 1st Uncentered Moment

\[ \mu_1' = \text{E}(R) = 0 \]

By symmetry of the distribution about zero under \(H_0: \rho = 0\).

43.6 2nd Uncentered Moment

\[ \mu_2' = \text{E}(R^2) = \frac{1}{n-1} \]

43.7 3rd Uncentered Moment

\[ \mu_3' = 0 \]

By symmetry.

43.8 4th Uncentered Moment

\[ \mu_4' = \frac{3}{(n-1)(n+1)} \]

43.9 2nd Centered Moment

\[ \mu_2 = \text{Var}(R) = \frac{1}{n-1} \]

43.10 3rd Centered Moment

\[ \mu_3 = 0 \]

43.11 4th Centered Moment

\[ \mu_4 = \frac{3}{(n-1)(n+1)} \]

43.12 Expected Value

\[ \text{E}(R) = 0 \]

43.13 Variance

\[ \text{Var}(R) = \frac{1}{n-1} \]

As \(n\) increases, the sampling distribution of \(r\) concentrates around zero, reflecting increased precision of the correlation estimate.

43.14 Median

\[ \text{Med}(R) = 0 \]

43.15 Mode

\[ \text{Mo}(R) = 0 \]

Both the median and mode equal zero, reflecting the perfect symmetry of the null distribution.

43.16 Coefficient of Skewness

\[ g_1 = 0 \]

The distribution is symmetric about zero under \(H_0: \rho = 0\).

43.17 Coefficient of Kurtosis

\[ g_2 = \frac{\mu_4}{\mu_2^2} = \frac{3/(n-1)(n+1)}{1/(n-1)^2} = \frac{3(n-1)}{n+1} \]

For large \(n\), \(g_2 \to 3\) (approaching the Normal), consistent with the asymptotic normality of \(r\).

43.18 Key Result: t-Transform

The test statistic

\[ T = \frac{R\sqrt{n-2}}{\sqrt{1-R^2}} \]

follows a \(t\)-distribution with \(n-2\) degrees of freedom under \(H_0: \rho = 0\) (see Chapter 25). This is the most commonly used result for testing correlation.

43.19 Parameter Estimation

The sample correlation coefficient \(r = \sum(x_i-\bar x)(y_i - \bar y) / \sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}\) is computed directly from data. The parameter \(n\) is the sample size.

43.20 R Module

43.20.1 RFC

The sample correlation distribution is covered in the context of the correlation test: see the “Hypothesis Testing / Pearson Correlation Test” module in RFC.

43.20.3 R Code

The following code demonstrates the t-transform test for a correlation coefficient:

r <- 0.42; n <- 30

# t-test statistic for H0: rho = 0
t_stat <- r * sqrt(n - 2) / sqrt(1 - r^2)

# Two-sided p-value
p_val <- 2 * pt(-abs(t_stat), df = n - 2)

cat("r =", r, "\n")
cat("n =", n, "\n")
cat("t =", round(t_stat, 4), "\n")
cat("df =", n - 2, "\n")
cat("p-value (two-sided) =", round(p_val, 4), "\n")
r = 0.42 
n = 30 
t = 2.4489 
df = 28 
p-value (two-sided) = 0.0208

43.21 Example

With \(n = 30\) observations, we observe a sample correlation of \(r = 0.42\). We test \(H_0: \rho = 0\) against \(H_1: \rho \neq 0\).

r <- 0.42; n <- 30

# t-test statistic
t_stat <- r * sqrt(n - 2) / sqrt(1 - r^2)
p_val  <- 2 * pt(-abs(t_stat), df = n - 2)

cat("t =", round(t_stat, 3), "  p =", round(p_val, 4), "\n")

if (p_val < 0.05) {
  cat("Conclusion: reject H0 at 5% level\n")
} else {
  cat("Conclusion: fail to reject H0 at 5% level\n")
}

# 95% critical value
t_crit <- qt(0.975, df = n - 2)
r_crit <- t_crit / sqrt(t_crit^2 + n - 2)
cat("Critical r (two-sided, alpha = 0.05):", round(r_crit, 4), "\n")
t = 2.449   p = 0.0208 
Conclusion: reject H0 at 5% level
Critical r (two-sided, alpha = 0.05): 0.361
Interactive Shiny app (click to load).
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43.22 Random Number Generator

Sample correlations under \(H_0: \rho = 0\) can be simulated by generating bivariate normal data with zero correlation and computing the sample correlation:

set.seed(123)
n <- 30; N_sim <- 1000

# Simulate N_sim sample correlations under H0: rho = 0
r_sims <- replicate(N_sim, {
  x <- rnorm(n); y <- rnorm(n)
  cor(x, y)
})

cat("Simulated mean of r:", round(mean(r_sims), 4), "\n")
cat("Theoretical mean:", 0, "\n")
cat("Simulated var of r:", round(var(r_sims), 4), "\n")
cat("Theoretical var:", round(1/(n-1), 4), "\n")
Simulated mean of r: 0.006 
Theoretical mean: 0 
Simulated var of r: 0.0357 
Theoretical var: 0.0345
Interactive Shiny app (click to load).
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43.23 Property 1: t-Transform

Under \(H_0: \rho = 0\):

\[ T = \frac{R\sqrt{n-2}}{\sqrt{1-R^2}} \sim t(n-2) \]

This is the fundamental result that makes correlation testing tractable with standard t-tables. See Chapter 25.

43.24 Property 2: Asymptotic Normality

As \(n \to \infty\):

\[ \sqrt{n-1}\, R \xrightarrow{d} N(0, 1) \]

The variance \(1/(n-1)\) decreases to zero, so the sample correlation concentrates around the true population correlation.

43.25 Property 3: Beta Representation

Under \(H_0: \rho = 0\), the shifted correlation \((1+R)/2\) follows:

\[ \frac{1+R}{2} \sim \text{Beta}\!\left(\frac{n-2}{2},\, \frac{n-2}{2}\right) \]

This connects the correlation distribution directly to the Beta distribution family. See Chapter 30.