To calculate the slope of the line of best fit, commonly referred to as the regression line, you can use the following formula:

The slope represents how much the variable $y$ changes for each unit change in the variable $x$. To compute this slope, follow these steps:

1. Gather Data Points: Collect your data points.
2. Calculate Means: Compute the mean of the $x$-values, denoted as $\bar{x}$, and the mean of the $y$-values, denoted as $\bar{y}$.
3. Compute the Products: For each data point, multiply the corresponding $x$-value by the $y$-value. Sum these products to find $\Sigma(xy)$.
4. Square the $x$-Values: Square each $x$-value and sum these squares to obtain $\Sigma(x^2)$.

Next, determine the number of data points, denoted as $n$. With all the required values at hand, you can substitute them into the slope formula:

This formula effectively measures the covariance of $x$ and $y$ (indicating how they vary together) divided by the variance of $x$ (showing how much $x$ varies).

Example Calculation

Consider the data points $(1, 2)$, $(2, 3)$, and $(3, 5)$. We will compute the slope step by step:

1. Calculate Means:

   • For $x$: $\bar{x} = \frac{1 + 2 + 3}{3} = 2$
   • For $y$: $\bar{y} = \frac{2 + 3 + 5}{3} = 3$

2. Compute $\Sigma(xy)$:

   • $\Sigma(xy) = (1 \cdot 2) + (2 \cdot 3) + (3 \cdot 5) = 2 + 6 + 15 = 23$

3. Compute $\Sigma(x^2)$:

   • $\Sigma(x^2) = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$

4. Count Data Points:

   • Here, $n = 3$.

Now, substitute these values into the slope formula:

Thus, the slope of the line of best fit for the given data points is $2.5$.