What is the normal distribution?

The normal (Gaussian) distribution is a continuous probability distribution completely determined by its mean (μ) and standard deviation (σ). It is symmetric and bell-shaped.

How do I find the probability between two values?

Compute the cumulative probability at the upper bound, subtract the cumulative probability at the lower bound: P(a < X < b) = Φ((b−μ)/σ) − Φ((a−μ)/σ). This tool does it for you.

Can I get inverse CDF / quantile?

Yes. Enter a probability p and the calculator returns x such that P(X ≤ x) = p for your specified μ and σ.

Normal Distribution Calculator

Enter mean (μ) and standard deviation (σ) of your normal distribution and compute PDF, CDF, tail probability, the probability between two values, and the inverse CDF (quantile). The shaded bell curve updates to show the selected area.

Mean (μ)

Std. deviation (σ)

σ > 0

Illustrative only: area under the curve refers to current calculation.

z-score

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PDF f(x)

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CDF P(X ≤ x)

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Area / Result

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Formulas used

PDF:

\( f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( - \frac{(x-\mu)^2}{2 \sigma^2} \right) \)

CDF:

\( F(x) = \frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma \sqrt{2}} \right) \right] \)

Standardize: \( z = \frac{x-\mu}{\sigma} \)

Typical tasks

Find P(X ≤ x) for a normal variable.
Find P(a < X < b) by subtracting two CDF values.
Find a critical value x_α given a probability (inverse CDF).