sigmoid function

Also called: sigmoid, logistic function, logistic curve

The S-shaped function p = 1 ÷ (1 + e⁻ᶻ) that turns any score z into a probability between 0 and 1; z = 0 gives 0.5. It is the inverse of z = ln(odds).

A logistic model adds up a score, the log-odds z = b₀ + b₁x₁ + … The sigmoid turns that score into a probability:

$p = \frac{1}{1 + e^{-z}} \qquad z = \ln\frac{p}{1 - p}$

Whatever z is, p stays between 0 and 1: z = 0 gives 0.5, z = 1 gives 0.73, z = −1 gives 0.27. The curve is steep in the middle and flat at the ends, so the same step in z moves p most near 50%.

Each +1 in z multiplies the odds p ÷ (1 − p) by e ≈ 2.72, and a coefficient b multiplies them by eᵇ. That is why a logistic coefficient is reported as an odds multiplier, never as "+b to the probability".

Example

A model of buying after a trial: z = −1.5 + 0.5 × lessons in the first week.

1 lesson: z = −1, p = 1 ÷ (1 + e) ≈ 0.27.
3 lessons: z = 0, p = 0.5.
5 lessons: z = 1, p ≈ 0.73.
Each extra lesson multiplies the odds by e^0.5 ≈ 1.65.

Common mistakes

Reading b as a probability change. b = 0.7 multiplies the odds by about 2; it does not add 70 points.
Confusing odds and probability. Odds of 3 are p = 0.75, not 3%.
Same effect everywhere. The change in p depends on where you start on the curve.

Learn it in the course

e, ln, odds and the sigmoid · ln undoes a power, so it solves for time; odds = p ÷ (1 − p), log-odds and the sigmoid p = 1 ÷ (1 + e⁻ᶻ), eᵇ as an odds multiplier, and payback with churn.