Meaning of an inverse normal distribution calculator
An inverse normal distribution calculator returns a quantile: the value on the horizontal axis whose cumulative probability matches a specified area. For the standard normal variable \(Z\sim N(0,1)\), this means finding a number \(z\) such that \[ P(Z \le z)=p. \] The inverse function \(z=\Phi^{-1}(p)\) is called the inverse standard normal (normal quantile).
Step 1: Identify the probability form
Calculators usually accept one of these probability inputs:
| Target probability statement | Convert to left-tail area \(p\) |
|---|---|
| \(P(X \le x)=p\) | Already a left-tail area: \(p\). |
| \(P(X \ge x)=r\) | Convert using complement: \(p=1-r\). |
| \(P(a \le X \le b)=c\) with symmetry about the center | If centered, split tails: \(p_{\text{left}}=(1-c)/2\) and \(p_{\text{right}}=1-(1-c)/2\). |
Key conversion rule. Most inverse normal outputs are based on a left-tail probability \(p\). If a right-tail probability is given, use \(p=1-r\).
Step 2: Standardize to connect x-values and z-values
For a general normal variable \(X\sim N(\mu,\sigma^2)\), standardizing gives \[ Z=\frac{X-\mu}{\sigma}\sim N(0,1). \] Therefore, the x-value corresponding to a left-tail probability \(p\) is \[ x=\mu+z\sigma \quad \text{where } z=\Phi^{-1}(p). \]
Worked example (percentile as an x-value)
A service time in minutes is modeled as \(X\sim N(18,3^2)\). Find the 90th percentile \(x_{0.90}\), meaning \(P(X\le x_{0.90})=0.90\).
- Set the left-tail probability. The statement already has a left-tail form, so \(p=0.90\).
- Use inverse standard normal. \[ z=\Phi^{-1}(0.90)\approx 1.2816. \]
- Convert back to x-units. \[ x_{0.90}=\mu+z\sigma=18+(1.2816)\cdot 3=18+3.8448=21.8448\approx 21.84. \]
Result. The inverse normal distribution calculator returns an x-value of about \(21.84\) minutes for the 90th percentile when \(\mu=18\) and \(\sigma=3\).
Quick variation (right-tail probability)
If the requirement is \(P(X \ge x)=0.10\) for the same \(X\sim N(18,3^2)\), then the left-tail area is \(p=1-0.10=0.90\), so the same quantile is obtained: \[ x\approx 21.84. \]
Visualization: left-tail area and the quantile
Checklist to avoid common input mistakes
- Use a probability in \((0,1)\). Percent values must be converted (e.g., 90% becomes \(0.90\)).
- Match tail direction. For right-tail inputs \(P(X\ge x)=r\), convert to \(p=1-r\) before applying \(\Phi^{-1}\).
- Use the correct parameters. A general normal quantile needs \(\mu\) and \(\sigma\); the standard normal quantile uses \(\mu=0\) and \(\sigma=1\).