Rough approximations to move between logit and probability scales

The logit and expit functions are commonly used to move between the probability scale [0, 1] and the whole real number line, especially when working with supervised learning models with binary outcomes. These functions are very easy to define, here is an implementation in R:

logit <- function(p) {
  log(p / (1 - p))
}

expit <- function(x) {
  exp(x) / (1 + exp(x))
}

When working in this space, it’s helpful to have some approximations to move between the logit and probability scales. (As an analogy, it is helpful to know that for a normal distribution, the interval standard deviations around the mean covers about 95% of the distribution, while standard deviations covers about 99%.)

Here are some rough approximations to help you estimate probabilities from logit scores and vice versa:

Logit	Probability
-7	0.001
-5	0.01
-3	0.05
-2	0.1
-1	0.25
0	0.5
1	0.75
2	0.9
3	0.95
5	0.99
7	0.999