Probit

In statistics, the probit function converts a probability (a number between 0 and 1) into a score. This score indicates how many standard deviations a value from a standard normal distribution (or "bell curve") is from the mean. For example, a probability of 0.5 (50%) represents the exact middle of the distribution, so its probit score is 0. A smaller probability like 0.025 (2.5%) is far to the left on the curve, corresponding to a probit score of approximately −1.96.

The function is widely used in probit models, a type of regression analysis for binary outcomes (e.g., success/failure or pass/fail). It was first developed in toxicology to analyze dose-response relationships, such as how the percentage of pests killed by a pesticide changes with its concentration. The probit function is also used to create Q–Q plots, a graphical tool for assessing whether a dataset is normally distributed.

Mathematically, the probit function is the quantile function (the inverse of the cumulative distribution function (CDF)) associated with the standard normal distribution. If the CDF is denoted by ${\displaystyle \Phi (z)}$, then the probit function is defined as:

${\displaystyle \operatorname {probit} (p)=\Phi ^{-1}(p)\quad {\text{for}}\quad p\in (0,1)}$.

This means that for any probability ${\displaystyle p}$, the probit function finds the value ${\displaystyle z}$ such that the area under the standard normal curve to the left of ${\displaystyle z}$ is equal to ${\displaystyle p}$.