The sigmoid squashes any real number into the range 0 to 1, so it is used for probabilities in logistic regression, as an activation function in neural networks and in growth models that level off.
How to use the Sigmoid Calculator
- Finite real input x
- Probability for the logit (0 to 1)
Results update as you type, and everything is calculated in your browser.
How it works
σ(0) = 0.5; large positive x gives values near 1, large negative x values near 0. The inverse, the logit, turns a probability back into log-odds.
Inputs explained
| Field | What to enter |
|---|---|
| Finite real input x | A number |
| Probability for the logit (0 to 1) | A number |
What each result means
- σ(x) = 1÷(1 + exp(-[Finite real input x]))
- Derivative σ′(x) = 1÷(1 + exp(-[Finite real input x]))×(1 – 1÷(1 + exp(-[Finite real input x])))
- logit(p) = ln(p/(1 − p)) = ln([Probability for the logit (0 to 1)]÷(1 – [Probability for the logit (0 to 1)]))
- tanh(x/2) = 2σ(x) − 1 = 2÷(1 + exp(-[Finite real input x])) – 1
Example values
How the main result changes when Finite real input x changes and everything else stays at the example values above:
| Finite real input x | σ(x) |
|---|---|
| 0.75 | 0.6792 |
| 1.13 | 0.7549 |
| 1.5 | 0.8176 |
| 1.88 | 0.867 |
| 2.25 | 0.9047 |
| 3 | 0.9526 |
Formula
σ(x) = 1 / (1 + e^(−x)), σ′(x) = σ(x)(1 − σ(x)), logit(p) = ln(p / (1 − p))
Worked example
Inputs — Finite real input x: 1.5; Probability for the logit (0 to 1): 0.8.
Results — σ(x) = 0.81757448; Derivative σ′(x) = 0.14914645; logit(p) = ln(p/(1 − p)) = 1.38629436; tanh(x/2) = 2σ(x) − 1 = 0.63514895.
Key terms
- Logistic function
- σ(x) = 1/(1 + e⁻ˣ).
- Logit
- ln(p/(1 − p)).
Frequently asked questions
What is σ(0)?
0.5.
Why is the sigmoid used in neural networks?
It is smooth and differentiable and maps outputs to (0, 1); modern networks often prefer ReLU in hidden layers.
How is it related to tanh?
tanh(x) = 2σ(2x) − 1.
What is the logit?
The inverse of the sigmoid: the log-odds of a probability.