Probit transformation is widely used to transform a probability, percentage, or proportion to a value in the unconstrained interval (−∞,∞), which is usually referred to as a quantile in probability theory. Strictly speaking, probit transformation is the inverse of the cumulative distribution function of the standard normal distribution. For any observed value x ∈ (−∞,∞), the cumulative distribution function of the standard normal distribution, denoted by Φ(x), is defined as follows:

$$\Phi \left(x\right)=\underset{-\infty }{\overset{x}{{\displaystyle \int }}}\frac{1}{\sqrt{2\text{π}}}{e}^{-\frac{t^2}{2}}dt,$$

with t being a value that the standard normal distributed variable could take. It converts a value in the interval (−∞,∞) to a value p in the interval (0,1) such that p = Φ(x). For a probability p, or more generally any value between 0 and 1, Φ−1(p) is its probit transformation to transform p ...