For 0<p<1, define the p-quantile by q_p=F-1(p), so P(X<q_p)=p. Calculator inverse-normal commands return this boundary value. Left-tail quantile Right-tail quantile **Given boundary** | **Monotonic comparison** P(X<q_p)=p | P(X<x)<p x<q_p; P(X<x)>p x>q_p. P(X>q_1-p)=p | P(X>x)<p x>q_1-p; P(X>x)>p x<q_1-p. A right-tail probability p uses the left-tail cumulative probability 1-p: P(X>q_1-p)=p.

For \(0<p<1\), define the \(p\)-quantile by \(q_p=F^{-1}(p)\), so \(P(X<q_p)=p\). Calculator inverse-normal commands return this boundary value.

A right-tail probability \(p\) uses the left-tail cumulative probability \(1-p\): \(P(X>q_{1-p})=p\).

Given boundary	Monotonic comparison
\(P(X<q_p)=p\)	\(P(X<x)<p\Rightarrow x<q_p\); \(P(X<x)>p\Rightarrow x>q_p\).
\(P(X>q_{1-p})=p\)	\(P(X>x)<p\Rightarrow x>q_{1-p}\); \(P(X>x)>p\Rightarrow x<q_{1-p}\).

Cumulative Probabilities and Inverse Normal

Cumulative Probabilities and Inverse Normal