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?

Thanks for any help you could provide!


1 Answer 1


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.


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