How is the slope $b_1$ computed from the two column sums in the least-squares regression example?

Conditions

• The data set consists of paired observations.
• The sample means $\bar{x}$ and $\bar{y}$ have been computed.
• The denominator $\sum (x_i-\bar{x})^2$ is nonzero.

Reasoning, step by step

1. Identify the column of squared x-deviations $(x_i-\bar{x})^2$ and sum its values to get the denominator.
2. Identify the column of cross-products $(x_i-\bar{x})(y_i-\bar{y})$ and sum its values to get the numerator.
3. Divide the numerator sum by the denominator sum to find $b_1$.
4. Simplify the resulting fraction to a decimal value.

Example

The video shows the squared x-deviations summing to 10 and the cross-products summing to 6. The calculation is explicitly written as $b_1 = 6/10 = .6$.

Common misconceptions

• Assuming the slope is the ratio of the sum of y-deviations to the sum of x-deviations.
• Forgetting to square the x-deviations before summing them for the denominator.
• Believing that rows with a zero deviation in either x or y contribute to the numerator sum.

Watch the explanation

Explore next

• Why does the video square the $x - \bar{x}$ column?
• What does the column ($x - \bar{x}$)($y - \bar{y}$) represent?
• How is b0 calculated once b1 is known?