Regression

Linear regression fits a straight line y = slope × x + intercept through a set of points, minimizing the total squared vertical distance between the line and every actual point — that's what "least squares" means.

def slope_intercept(xs, ys):
    n = len(xs)
    x_bar = sum(xs) / n
    y_bar = sum(ys) / n
    numerator = sum((x - x_bar) * (y - y_bar) for x, y in zip(xs, ys))
    denominator = sum((x - x_bar) ** 2 for x in xs)
    slope = numerator / denominator
    intercept = y_bar - slope * x_bar
    return slope, intercept

The slope tells you the direction and steepness of the relationship — R² tells you how well the line actually fits. R² of 1.0 means every point sits exactly on the line; R² of 0 means the line explains none of the variation in y (you'd do just as well predicting the average every time).

import math

def r_squared(xs, ys):
    x_bar = sum(xs) / len(xs)
    y_bar = sum(ys) / len(ys)
    numerator = sum((x - x_bar) * (y - y_bar) for x, y in zip(xs, ys))
    denom_x = sum((x - x_bar) ** 2 for x in xs)
    denom_y = sum((y - y_bar) ** 2 for y in ys)
    correlation = numerator / math.sqrt(denom_x * denom_y)
    return correlation ** 2

A high R² doesn't prove x causes y — it only measures how tightly the two move together, which is why "correlation isn't causation" comes up constantly in any discussion of regression results.