Normal Distribution Calculator: Solve the Bell Curve Free 2026

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Normal Distribution Calculator Gaussian

Find probabilities and the density for a normal (Gaussian) distribution.

P(X < x)
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P(X > x)
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z-score
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Density f(x)
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Show working (LaTeX)

This free normal distribution calculator finds probabilities and the density for any normal (Gaussian) distribution — P(X < x), P(X > x) and the area between two values.

How to use the normal distribution calculator

Enter a value x, the mean and the standard deviation, then press Calculate. The normal distribution calculator standardizes to a z-score and returns the left-tail, right-tail and between probabilities, plus the density. As a gaussian distribution calculator and normal probability calculator, it covers the everyday cases instantly.

normal distribution calculator bell curve showing the probability area under the Gaussian
Probabilities are areas under the bell-shaped normal curve.

What is the normal distribution?

The normal, or Gaussian, distribution is the symmetric bell curve that describes much natural variation. See the normal distribution reference for the details.

Definition. The normal distribution with mean $\mu$ and standard deviation $\sigma$ has density $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$.

Normal distribution formula

$$z=\frac{x-\mu}{\sigma},\qquad P(Xwhere $\Phi$ is the standard normal cumulative distribution function.

How to find a normal probability step by step

  1. Standardize: convert x to a z-score with the z-score formula.
  2. Look up the area to the left, $\Phi(z)$.
  3. Subtract or combine areas for right-tail and between probabilities.
✅ The 68-95-99.7 ruleAbout 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Worked example

For a standard normal (mean 0, SD 1), $P(X<1.5)=\Phi(1.5)\approx 0.9332$, so about 93.3% of values fall below 1.5.

Why the normal distribution matters in machine learning

The normal distribution is everywhere in machine learning for beginners — from Gaussian Naive Bayes to weight initialization and the assumptions behind many tests. It is the basis of the z-score and the standard deviation.

🤖 ML insight

Neural network weights are often initialized from a normal distribution, and the central limit theorem is why averaged quantities tend to look Gaussian.

Frequently asked questions

What does this normal distribution calculator compute?
Left-tail, right-tail and between probabilities, the z-score and the density f(x).
What is the standard normal distribution?
A normal distribution with mean 0 and standard deviation 1.
How do I find the area between two values?
Enter x and an upper x; the calculator returns the area between them.
What is the 68-95-99.7 rule?
The share of values within 1, 2 and 3 standard deviations of the mean.
Is the normal distribution calculator free?
Yes, completely free and browser-based.

Standard normal and the empirical rule

Any normal curve can be converted to the standard normal — mean 0, standard deviation 1 — by subtracting the mean and dividing by the standard deviation. That single transformation lets one table or one function answer every probability question.

The empirical rule is the quick mental version: roughly 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. It is why a value beyond three standard deviations is treated as a rare outlier, and why so much of statistics leans on the bell curve.

Normal distribution calculator: summary

This normal distribution calculator turns any value into a probability under the bell curve. Pair it with the z-score calculator and the standard deviation calculator.

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