How this calculator works
The logistic sigmoid maps the real line smoothly into the interval from zero to one. Its inverse converts an interior probability to log odds. The implementation evaluates exponentials with a nonpositive argument to avoid overflow for large negative x. Very large magnitudes can round the probability to zero or one in floating-point arithmetic, even though the exact function never reaches either boundary at a finite x.
Formula
sigma(x) = 1/(1+exp(-x)); sigma prime = sigma(x)(1-sigma(x)); logit(p) = ln(p) - ln(1-p).
Worked example
At x = 0, the sigmoid is 0.5 and its derivative is 0.25. Probability 0.75 has inverse ln(3), about 1.098612289.
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Frequently asked questions
Why reject probabilities zero and one for the inverse?
They correspond to negative and positive infinity. This calculator returns finite inputs only.
Is every S-shaped curve this sigmoid?
No. Sigmoid describes a broad family; this tool uses the standard logistic function without shift or scale parameters.
Does the output automatically give a calibrated probability?
No. That interpretation depends on the model and data. Applying the function alone does not establish calibration.