The correlation coefficient $r$ is a measure of the strength and direction of the linear relationship between two variables in a regression analysis. It is often found using the formula for the Pearson correlation coefficient.

Formula to find $r$:

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

Where:

• $n$ = number of data points
• $x$ and $y$ = values of the two variables
• $\sum{x}$ = sum of all $x$ values
• $\sum{y}$ = sum of all $y$ values
• $\sum{xy}$ = sum of the products of each corresponding pair of $x$ and $y$
• $\sum{x^2}$ = sum of the squares of the $x$-values
• $\sum{y^2}$ = sum of the squares of the $y$-values

Steps to Calculate $r$:

1. Sum the values for $x$, $y$, $x^2$, $y^2$, and $xy$.
2. Substitute the sums into the Pearson formula.
3. Calculate the value of $r$.

Example:

Consider the following data:

$x$	$y$
1	2
2	3
3	4
4	5
5	6

Let’s break it down:

1. Calculate the sums:

   • $\sum{x} = 1 + 2 + 3 + 4 + 5 = 15$
   • $\sum{y} = 2 + 3 + 4 + 5 + 6 = 20$
   • $\sum{xy} = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + (5 \times 6) = 2 + 6 + 12 + 20 + 30 = 70$
   • $\sum{x^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$
   • $\sum{y^2} = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 4 + 9 + 16 + 25 + 36 = 90$

2. Substitute into the Pearson formula:

[ r = \frac{5(70) - (15)(20)}{\sqrt{[5(55) - (15)^2][5(90) - (20)^2]}} ]

Simplify the numerator and denominator:

[ r = \frac{350 - 300}{\sqrt{[275 - 225][450 - 400]}} = \frac{50}{\sqrt{50 \times 50}} = \frac{50}{50} = 1 ]

Result:

The correlation coefficient $r = 1$, which indicates a perfect positive linear relationship between $x$ and $y$.

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Would you like further explanation on any step or have any specific questions? Here are some related topics you might find interesting:

1. How to interpret the value of $r$ (e.g., perfect correlation, no correlation).
2. How to find the regression line equation using the formula.
3. The significance of $r^2$ in regression analysis.
4. How outliers affect the value of $r$.
5. How to calculate $r$ using a statistical calculator or software.

Tip: When $r = 1$ or $r = -1$, the points lie perfectly on a straight line, indicating a perfect positive or negative linear relationship.