Taken directly, from my lecture notes, we are given that
$$ \hat{\sigma}_X^2 = \frac{1}{k-1} \sum_{i=1}^k (X_i - \hat{\mu}_X)^2 \\[5mm] \hat{\rho} = \frac{1}{k-1} \sum_{i=1}^k \frac{(X_i - \hat{\mu}_X)(Y_i - \hat{\mu}_Y)}{\hat{\sigma}_X^2 \cdot \hat{\sigma}_Y^2} $$
However, from the formula for $\hat{\sigma}_X^2$, we can deduce that $$ \hat{\sigma}_X = \sum_{i=1}^k \frac{X_i - \hat{\mu}_X}{\sqrt{k-1}} $$ and hence $$ \hat{\sigma}_X \cdot \hat{\sigma}_Y = \sum_{i=1}^k \frac{X_i - \hat{\mu}_X}{\sqrt{k-1}} \cdot \sum_{i=1}^k \frac{Y_i - \hat{\mu}_Y}{\sqrt{k-1}} \\[5mm] = \frac{1}{k-1}\sum_{i=1}^k (X_i - \hat{\mu}_X) (Y_i - \hat{\mu}_Y) $$ and, thus we have $$ \hat{\rho} = \frac{1}{k-1} \sum_{i=1}^k \frac{(X_i - \hat{\mu}_X)(Y_i - \hat{\mu}_Y)}{\hat{\sigma}_X^2 \cdot \hat{\sigma}_Y^2} \\[5mm] = \hat{\rho} = \sum_{i=1}^k \frac{\hat{\sigma}_X^2 \cdot \hat{\sigma}_Y^2}{\hat{\sigma}_X^2 \cdot \hat{\sigma}_Y^2} \\[5mm] = 1 $$ Clearly the correlation coefficient of 2 variables is not always 1, so how have I derieved this result incorrectly?