Meaning of uniform percentile rank distribution? I'm new to statistics and I've been following Think Stats 2. I've just gotten to Cumulative Distribution Functions. I have some questions regarding uniform distribution:

*

*What does it mean for the CDF of the percentile ranks of a random sample to be distributed uniformly? Does it have any special meaning?

*Is having a uniform percentile rank distribution the same as having a uniform distribution of data?

*Do QQ plots also determine the same thing above?

Thanks for any help you could provide!
 A: To answer (1)+(2).
The answers to these are something more general actually. You don't need to talk about the percentile ranks only. In fact, if you have any distribution $X$ with cdf $F_X$, then $F_X(X)$ follows the uniform distribution
\begin{equation*}
F_X(X) \sim U(0,1). 
\end{equation*}
For a proof, observe that if $Y:=F_X(X)$, then
\begin{equation*}
F_Y(y)=P(Y\leq y)=P(F_X(X)\leq y)=P(X\leq F^{-1}_X(y))=F_X(F^{-1}_X(y))=y
\end{equation*}
and so $F_Y(y)=y$, which is the cumulative density function of the uniform distribution.  Hence
\begin{equation*}
Y=F_X(X) \sim U(0,1). 
\end{equation*}
In other words, for every random sample
\begin{equation*}
\{x_1,...,x_n\}
\end{equation*}
from $X$,
\begin{equation*}
\{F_X(x_1),...,F_X(x_n)\}
\end{equation*}
is a random sample from $U(0,1)$. This provides a way to generate random values from the uniform distribution, assuming that you have a  random sample from $X$ and that you know the formula of the cdf $F_x$.
To answer (3).
Yes. If you have a random sample $\{x_1,...,x_n\}$ from $X$ and plot the theoritical quantiles of the uniform distribution $U(0,1)$ vs the estimated quantiles of $\{F_X(x_1),...,F_X(x_n)\}$, then the points should fall approximately along the line $y=x$. The approximation improves as the sample size $n$ increases.
