$$ r = \frac { n \Sigma x y - \sum x \Sigma y } { \sqrt { [ n ( \Sigma x ^ { 2 } ) - ( \Sigma x ) ^ { 2 } ] [ n ( \Sigma y ^ { 2 } ) - ( \Sigma y ) ^ { 2 } ] } } $$

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**Step 1. Understand the Symbols**

- $$ n $$ is the number of paired data points.
- $$ \Sigma x $$ is the sum of all $$ x $$ values.
- $$ \Sigma y $$ is the sum of all $$ y $$ values.
- $$ \Sigma xy $$ is the sum of the products $$ x_i y_i $$ for each pair.
- $$ \Sigma x^2 $$ is the sum of the squares of the $$ x $$ values.
- $$ \Sigma y^2 $$ is the sum of the squares of the $$ y $$ values.

**Step 2. Express the Numerator**

The numerator of the formula is:
$$
n \Sigma xy - \Sigma x \Sigma y
$$
This term represents the covariance between the variables $$ x $$ and $$ y $$ multiplied by $$ n $$.

**Step 3. Express the Denominator**

The denominator consists of two parts under the square root:
$$
\sqrt{ \left[ n \Sigma x^2 - (\Sigma x)^2 \right] \left[ n \Sigma y^2 - (\Sigma y)^2 \right] }
$$
- The term $$ n \Sigma x^2 - (\Sigma x)^2 $$ computes a measure related to the variance of $$ x $$.
- Similarly, $$ n \Sigma y^2 - (\Sigma y)^2 $$ computes a measure related to the variance of $$ y $$.

Multiplying these two terms together and taking the square root standardizes the covariance by the standard deviations of $$ x $$ and $$ y $$.

**Step 4. Write the Full Correlation Coefficient Formula**

The formula for the Pearson correlation coefficient $$ r $$ is:
$$
r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{\left[ n \Sigma x^2 - (\Sigma x)^2 \right] \left[ n \Sigma y^2 - (\Sigma y)^2 \right]}}
$$
This formula measures the strength and direction of the linear relationship between $$ x $$ and $$ y $$.

**Step 5. Interpret the Formula**

- When $$ r $$ is close to $$ 1 $$, there is a strong positive linear relationship.
- When $$ r $$ is close to $$-1$$, there is a strong negative linear relationship.
- When $$ r $$ is around $$ 0 $$, there is little or no linear relationship between $$ x $$ and $$ y $$.

This step-by-step explanation clarifies how each component of the formula contributes to measuring the linear correlation between the two variables.
The Pearson correlation coefficient $$ r $$ is calculated using the formula: $$ r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{\left[ n \Sigma x^2 - (\Sigma x)^2 \right] \left[ n \Sigma y^2 - (\Sigma y)^2 \right]}} $$ This measures the strength and direction of the linear relationship between two variables $$ x $$ and $$ y $$.

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