# Sigmoid Function > The sigmoid takes any number, however big or small, and squeezes it into a value between 0 and 1 that can be read as a chance. Page: https://formulabro.yunusemrevurgun.com/ai-ml/sigmoid-function/ Category: AI & Machine Learning Published: 2026-10-22 Updated: 2026-10-22 ## Formula $$ \sigma(z) = \frac{1}{1 + e^{-z}} $$ Read it out loud: "Sigma of z equals one divided by the quantity one plus e to the power of negative z." ## What each symbol means - $\sigma(z)$: The sigmoid of $z$: the output. It is always strictly between 0 and 1. - $z$: The input number, a raw score. It can be any real number: positive, negative or zero. It has no unit in most models. - $e$: Euler's number, about 2.718. Here it is raised to the power $-z$. ## What the formula is doing Many models need an output they can read as a probability, and a probability has to sit between 0 and 1. A raw score, however, can be any number at all, such as $-40$ or $+300$. The sigmoid is the rule that bridges the two: whatever number goes in, a value between 0 and 1 comes out. Here is what happens step by step. The term $e^{-z}$ is the same as $1 / e^{z}$. When $z$ is a large positive number, $e^{-z}$ is almost 0, so the fraction is almost $1 / 1 = 1$. When $z$ is a large negative number, $e^{-z}$ is huge, so the fraction is almost 0. When $z$ is exactly 0, $e^{0} = 1$ and the answer is $1/2$. The curve has an S shape. It climbs smoothly from near 0 on the left to near 1 on the right, and it passes through 0.5 when $z = 0$. Inputs near the middle change the output the most. Very large inputs in either direction land close to 0 or 1 and barely move after that. The sigmoid is also a smooth curve with no sharp corners. Training methods rely on that smoothness, because they measure how a small change in $z$ changes the output. That is why the sigmoid appears in logistic regression and in early neural networks. ## An everyday picture Think of a dimmer switch with a stop at each end. Turn it all the way one way and the light is fully off; turn it all the way the other way and it is fully on. Around the middle, a small turn changes the brightness a lot, while near the ends you can turn much further without any visible change. The sigmoid turns a raw input into a brightness somewhere between off (0) and fully on (1). ## Worked example: Three inputs: positive, zero and negative Apply the sigmoid to three raw scores: $z = 2$, $z = 0$ and $z = -2$. All values are unitless here. 1. For $z = 2$: $\sigma(2) = \frac{1}{1 + e^{-2}} = \frac{1}{1 + 0.1353} \approx 0.881$. 2. For $z = 0$: $\sigma(0) = \frac{1}{1 + e^{0}} = \frac{1}{1 + 1} = 0.5$. 3. For $z = -2$: $\sigma(-2) = \frac{1}{1 + e^{2}} = \frac{1}{1 + 7.389} \approx 0.119$. **Answer:** A raw score of $+2$ becomes about **0.881**, a score of 0 lands exactly on **0.5**, and a score of $-2$ drops to about **0.119**. Positive scores lean toward 1, negative scores lean toward 0, and 0 sits in the middle. ## Common mistakes - Mixing up $e^{-z}$ with $e^{z}$. The minus sign decides whether big inputs push the output toward 1 or toward 0. With $z = 2$ the answer must be above 0.5. - Expecting the output to be exactly 0 or exactly 1. The sigmoid gets close to 0 or 1 but never reaches them for any finite input. - Treating every output between 0 and 1 as a well-calibrated probability. A model can output 0.9 for answers that turn out right only half the time. - Using the sigmoid for a whole list of scores that must add up to 1. The sigmoid works on one score at a time; for a list, use softmax. ## Where you will meet it - Logistic regression, the classic model for yes-or-no questions, which passes a weighted sum through a sigmoid. - The output layer of a binary neural network classifier, such as a spam-or-not-spam score. - The gates inside LSTM recurrent networks, which use sigmoids to decide how much information to keep or forget. - Early neural networks, before ReLU became the usual choice in deep networks. ## Questions people ask ### Why is it called sigmoid? Sigmoid means S-shaped. Plot the curve and it looks like a stretched letter S. ### How is the sigmoid different from softmax? The sigmoid works on one score and gives one number between 0 and 1. Softmax works on a whole list and makes the list add up to 1. See the softmax page for that rule. ### Can the sigmoid output be negative? No. Because $e^{-z}$ is always positive, the bottom of the fraction is always bigger than 1, so the result stays strictly between 0 and 1. ### Where is the sigmoid steepest? Right in the middle, at $z = 0$. That is where a small change in the input moves the output the most. Far out on either side, the curve is almost flat. ## Related formulas - [Artificial Neuron (Weighted Sum)](https://formulabro.yunusemrevurgun.com/ai-ml/artificial-neuron/) - [Softmax](https://formulabro.yunusemrevurgun.com/ai-ml/softmax/) - [Cross-Entropy Loss](https://formulabro.yunusemrevurgun.com/ai-ml/cross-entropy-loss/) ## Sources - [Sigmoid function, Wikipedia](https://en.wikipedia.org/wiki/Sigmoid_function) - [Logistic function, Wikipedia](https://en.wikipedia.org/wiki/Logistic_function) From Formula Bro (https://formulabro.yunusemrevurgun.com/), a side project developed by yunusemrevurgun.com.