Latest content was relocated to https://bintanvictor.wordpress.com. This old blog will be shutdown soon.

Sunday, October 19, 2014

quantile (+ quartile + percentile), briefly

http://en.wikipedia.org/wiki/Quantile_function is decent.

For a concrete example of quaNtile, i like the quaRtile concept. Wikipedia shows there are 3 quartile values q1, q2 and q3. On the pdf graph (usually bell-shaped, since both ends must show tails), these 3 quartile values are like 3 knifes cutting the probability mass "area-under-curve" into 4 equal slices consisting of 2 tails and 2 bodies.

Quantile function is related to inverse of the CDF function. Standard notation --

F(x) is the CDF function , strongly increasing from 0 to 1.
F -1() is the inverse function, whose support is (0,1)
F -1(0.25) = q1 , assuming one-to-one mapping

http://www.quora.com/What-is-an-intuitive-explanation-of-the-Probability-Integral-Transform-aka-Universality-of-the-Uniform explains in plain English that percentile function is a simplified, discrete version of our quantile function (or perhaps the inverse of it). The CDF is like a robot. You say your score, and he give you the percentage like "94% of test takers scored below you".

Conversely, the quantile function is another robot. You say a percentage like 25%, and she gives the score "25% of the test takers scored below 362 marks"

Obvious assumption -- one to one mapping, or equivalently, strongly increasing CDF.