Question

Show that $$ R=\frac{\sum_{1}^{n}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right)}{\sqrt{\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} \sum_{1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}}}=\frac{\sum_{1}^{n} X_{i} Y_{i}-n \overline{X Y}}{\sqrt{\left(\sum_{1}^{n} X_{i}^{2}-n \bar{X}^{2}\right)\left(\sum_{1}^{n} Y_{i}^{2}-n \bar{Y}^{2}\right)}} $$

Short answer

The right side of the equation simplifies to represent the left side, thus proving the two representations of Pearson's correlation coefficient R are equivalent.

Step-by-step solution

Understand the given formula

The given formula is the Pearson correlation coefficient, R, which measures the linear relationship between two datasets. In this case, \(X\) and \(Y\) are the datasets and \(X_i\) and \(Y_i\) are individual points in these datasets. \(\bar{X}\) and \(\bar{Y}\) represent the mean of the datasets \(X\) and \(Y\) respectively.

Begin simplifying

We begin by focusing on the numerator of the right-hand side. We can express \(\sum_{1}^{n} X_{i} Y_{i} - n \overline{XY}\) as \(\sum_{1}^{n} X_{i} Y_{i} - \sum_{1}^{n} \bar{X} \bar{Y}\). This simplifies the expression.

Continue simplifying

Next we express \(\bar{X}\) and \(\bar{Y}\) as the sum of squares divided by the number of elements. So, \(\bar{X}\bar{Y}\) becomes \(\frac{\sum_{1}^{n} X_{i}}{n} \cdot \frac{\sum_{1}^{n} Y_{i}}{n}\), which simplifies to \(\frac{\sum_{1}^{n} X_{i} Y_{i}}{n^2}\). Substituting this into the previous equation we get \(\sum_{1}^{n} X_{i} Y_{i} - \sum_{1}^{n} \frac{\sum_{1}^{n} X_{i} Y_{i}}{n}\), which simplifies to \(\sum_{1}^{n}(X_{i} - \bar{X})(Y_{i} - \bar{Y})\), exactly the numerator of the left-hand side.

Simplify the denominator

The denominator can be simplified using similar steps. \(\sum_{1}^{n} X_{i}^{2} - n \bar{X}^{2}\) becomes \(\sum_{1}^{n}(X_{i}^{2} - \bar{X^2})\). Similarly, \(\sum_{1}^{n} Y_{i}^{2} - n \bar{Y}^{2}\) becomes \(\sum_{1}^{n}(Y_{i}^{2} - \bar{Y^2})\). When these two new expressions are multiplied together, they become \(\sum_{1}^{n}(X_{i} - \bar{X})^{2}\sum_{1}^{n}(Y_{i} - \bar{Y})^{2}\), which is exactly the denominator of the left-hand side.

Conclude equivalence

Having shown that both numerator and denominator of the right-hand side simplify to the numerator and denominator of the left-hand side, this proves the equivalence of the two representations of the Pearson correlation coefficient. This is the correlation coefficient represented in the covariance form and variance form.