Sigmoid and Logit Calculator

The logistic sigmoid is sigma(x) = 1 / (1 + e^-x), an S-shaped curve that maps any real number x to a probability between 0 and 1. You enter x, the log-odds, and the calculator returns sigma(x), its derivative, and the matching odds. It is the activation used in logistic regression and in many neural networks.

The derivative is sigma'(x) = sigma(x) times (1 - sigma(x)), so once you know sigma(x) you never differentiate from scratch. It peaks at 0.25 when x is 0 and shrinks toward 0 as x moves far from the center. This is the exact form backpropagation uses to push gradients through a sigmoid layer.

The logit is logit(p) = ln(p / (1 - p)), the natural log of the odds, and it is the exact inverse of the sigmoid. Switch to logit mode, enter a probability p, and you get the log-odds x such that sigma(x) = p. This is how logistic regression turns a probability back into the linear score it actually models.

Odds are p / (1 - p), so a probability of 0.8 is odds of 4 to 1. The log-odds (logit) is the natural log of those odds, and the sigmoid converts log-odds back into a probability. The calculator shows all three at once so you can move between them in either direction.

The logit divides by (1 - p) and then takes a logarithm, so p has to sit strictly inside the open interval from 0 to 1. At p = 0 the odds are 0 and the log is undefined, and at p = 1 the odds are infinite, so both ends return no result. Use values such as 0.01 to 0.99 to see finite log-odds.