Correlation and Regression

Introduction

Correlation measures how two variables move together; regression uses that relationship to make predictions. These tools are everywhere in business: ad spend vs revenue, temperature vs energy use, engagement vs churn. This lesson shows how to measure the relationship, read it correctly, and fit a prediction line.

The Pearson Correlation Coefficient r

The Pearson correlation r measures the strength and direction of a linear relationship, always between -1 and 1:

• r = 1: perfect positive linear relationship.
• r = -1: perfect negative linear relationship.
• r = 0: no linear relationship.

import numpy as np
from scipy import stats
rng = np.random.default_rng(3)
x = rng.normal(0, 1, 120)
y = 0.9  x + np.sqrt(1 - 0.92)  rng.normal(0, 1, 120)
r, p = stats.pearsonr(x, y)
print(f"r = {r:.3f}   p = {p:.4f}")

Output:

r = 0.899   p = 0.0000

Interpreting r

Correlation Is Not Causation

Three reasons two variables can correlate without one causing the other:

1. Confounding: a third variable drives both (ice cream and drowning both rise with summer heat). 2. Reverse causation: maybe Y causes X, not X causes Y. 3. Spurious correlation: pure coincidence with enough data.

Correlation justifies investigating a relationship, not acting on it.

Simple Linear Regression

Regression fits a line that predicts Y from X:

y = b0 + b1 * x

• b0 (intercept): predicted Y when X = 0.
• b1 (slope): expected change in Y per 1-unit change in X.

ad_spend = rng.uniform(0, 10, 80)
revenue = 1.8 + 0.7 * ad_spend + rng.normal(0, 0.9, 80)
slope, intercept, r_value, p_value, std_err = stats.linregress(ad_spend, revenue)
print(f"intercept: {intercept:.2f}")
print(f"slope:     {slope:.3f}")
print(f"r:         {r_value:.3f}")
print(f"p-value:   {p_value:.4f}")

Output:

intercept: 1.79
slope:     0.700
r:         0.921
p-value:   0.0000

Interpretation: each extra dollar of ad spend adds about 0.70 dollars of revenue (within the observed range).

Making Predictions

pred = intercept + slope * 8.0   # ad spend = 8
print(f"predicted revenue at spend 8: {pred:.2f}")

Output:

predicted revenue at spend 8: 7.39

Caveats:

• Only interpolate inside the observed X range; extrapolation is dangerous.
• The prediction is a mean estimate, not a guarantee.
• Regression assumes a roughly linear relationship and normally distributed errors.

R-Squared: How Good Is the Fit?

R-squared is the square of r (for simple regression). It tells you the share of variance in Y explained by X.

print(f"R-squared: {r_value2:.3f}")

Output:

R-squared: 0.848

84.8% of the variation in revenue is explained by ad spend; the rest comes from other factors and noise.

Correlation vs Regression: Quick Comparison

Common Pitfalls

• Concluding causation from correlation.
• Trusting r without plotting the data (Anscombe's quartet!).
• Extrapolating the regression line far beyond the data.
• Ignoring outliers that drag the regression line.

Summary

• r measures linear association: sign = direction, magnitude = strength.
• Correlation is NOT causation; watch for confounders.
• Regression fits y = b0 + b1*x and enables prediction.
• R-squared tells you the share of variance explained.
• Always plot before you conclude.

Next Lesson

Congratulations - you finished Statistics for Data Analysts! Review the quizzes, and if you want to go deeper, check out the Python Programming and Data Analysis courses on the platform.