A normal random variable is written as \(X \sim \mathcal{N}(\mu,\sigma^2)\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
This calculator finds probabilities like \(P(X<x)\), \(P(X>x)\), and \(P(a<X<b)\) by converting values to Z-scores and using the standard normal CDF \(\Phi(z)\).
Probability density function (PDF)
\[
f(x)=\frac{1}{\sigma\sqrt{2\pi}}
\exp\!\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)
\]
Z-score standardization
The Z-score measures how many standard deviations a value is from the mean.
\[
\begin{aligned}
z &= \frac{x-\mu}{\sigma}
\end{aligned}
\]
Using the standard normal CDF
The standard normal CDF is \(\Phi(z)=P(Z<z)\) for \(Z\sim\mathcal N(0,1)\). After converting to Z-scores:
\[
\begin{aligned}
P(X<x) &= \Phi\!\left(\frac{x-\mu}{\sigma}\right) \\
P(X>x) &= 1-\Phi\!\left(\frac{x-\mu}{\sigma}\right) \\
P(a<X<b) &= \Phi\!\left(\frac{b-\mu}{\sigma}\right)-\Phi\!\left(\frac{a-\mu}{\sigma}\right)
\end{aligned}
\]
Practical note: Printed Z-tables typically use Z rounded to 2 decimals; the calculator can mimic that with the “table approx” option.
