# 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:

1. 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?
2. Is having a uniform percentile rank distribution the same as having a uniform distribution of data?
3. Do QQ plots also determine the same thing above?

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 $$F_x$$.
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.