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Monte

Simple regression

  1. Read it, not yet
  2. Practice it, not yet
  3. Master it, not yet

Regression fits the line with the smallest squared misses. Slope: how much y moves per unit of x. R²: the share of y's variation explained. t-test: does x help?

Why it matters

Regression draws the best straight line through a cloud of points, so x can predict y.

It turns “these move together” into a number: each extra unit of x is worth this much y.

  • PricingHow much demand drops for every $1 of price.
  • MarketingSales per $1,000 of ad spend, with care about cause and effect.
  • Real estatePrice per extra square foot.

Everyday example

Does advertising move sales? Six weeks give Sales = 1.43 + 1.14 Adv: each extra unit of ads adds about 1.14 units of sales. R² = 87.1%: the line explains most of the scatter. t = 5.19 (p = 0.007): not luck.

Least squares: six weeks of data plus one point you control

0369120816Advertising (x)

Base line (dashed): Sales = 1.43 + 1.14 Adv, R² = 87.1%. Tick the box to add a point you control.

Check yourself

  1. Σx = 24, Σy = 36, Σxy = 176, Σx² = 124, n = 6. Slope?

  2. Intercept?

  3. Predict sales at 8 units of advertising?

The trap

The mistake people make most on this one.

Correlation is symmetric; regression is not. Swap x and y and you get a different line. Never predict outside the x range you have.

Formulas

7 formulas and a way to remember them.
Line
ŷ = b₀ + b₁x
Slope
b₁ = [nΣxy − ΣxΣy] / [nΣx² − (Σx)²]
Intercept
b₀ = ȳ − b₁x bar
Fit
SST = SSR + SSE, R² = SSR / SST = 1 − SSE / SST
Residual SD
s = √(SSE / (n − 2))
Slope test
t = b₁ / SE(b₁), df = n − 2
Correlation
r = ±√R², with the sign of the slope
Remember it
Slope is rise over run. R² is how much scatter the line explains. The t-test asks: is the slope really nonzero?

Watch and go deeper

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