Level 3 · Fit a line to spend and revenue

A line through the dots

A simple linear regression finds the straight line y = slope × x + intercept that sits closest to the points, in the sense that the squared vertical distances from the points to the line add up to as little as possible (“least squares”). In NumPy it is one call:

slope, intercept = np.polyfit(monthly["spend"], monthly["revenue"], 1)
# slope ≈ 5.50, intercept ≈ -40,288

The 1 is the degree: a straight line. The order of the arguments matters: x (the column you predict from) first, y (the one you predict) second. Swap them and you get a different line.

Reading the slope

The slope is the change in y per one unit of x. Here: months with $1,000 more marketing spend had about $5,500 more revenue. Read every word of that sentence. It compares months; it describes this data; it says nothing about what happens if you add $1,000 next month. Level 2 showed why: November and December have big budgets and holiday shoppers, so the slope carries both.

Reading the intercept

The intercept is the predicted y when x is 0: here, -$40,288 of revenue in a month with no marketing. Negative revenue is impossible. The intercept is just where the line crosses zero, far to the left of every real month (the lowest spend was $23,861). Treat it as a parameter of the line, not a forecast.

R²: how much of the variation the line follows

R² (R-squared) is the share of the variation in y around its mean that the line accounts for. For one x it is the correlation squared:

monthly["spend"].corr(monthly["revenue"]) ** 2   # 0.845

0.845 means the line follows revenue’s ups and downs closely. It doesn’t mean the line is causal, and it doesn’t mean predictions are accurate to 84.5%. A high R² with a confounder is still a confounded line.

Common mistakes

• Swapping x and y in np.polyfit. The slope of revenue on spend is not 1 / (slope of spend on revenue) unless the points sit exactly on a line.
• Calling the slope a return on investment. It is an association across months.
• Reading the intercept as a real scenario when x = 0 is far outside the data.
• Treating a high R² as proof. It measures fit, not cause.