Probable Error

The first Quartile of a standard Normal Distribution occurs when

$$\int_0^t \Phi(z)\,dz={\textstyle{1\over 4}}.$$

The solution is $t=0.6745\dots$. The value of $t$ giving $1/4$ is known as the probable error of a Normally Distributed variate. However, the number $\delta$ corresponding to the 50% Confidence Interval,

$$P(\delta)\equiv 1-2\int^{\vert\delta\vert}_0 \phi(t)\,dt={\textstyle{1\over 2}},$$

is sometimes also called the probable error.

See also Significance