Proof
By the definition of percentile with \(y = P^s\text{,}\)
\begin{equation*}
y = (1-r) y_m + r y_{m+1}.
\end{equation*}
Solving for r and equating to the formula for \(r\) in the definition of percentile yields
\begin{equation*}
r = \frac{y - y_m}{y_{m+1} -y_m} = (n+1)s - m.
\end{equation*}
Solving this for s given
\begin{equation*}
s = \frac{r+m}{n+1}.
\end{equation*}
Hence,
\begin{equation*}
y = P^s.
\end{equation*}