what is statistics – Quantile Function – Invert ECDF

What is Percentiles and Quartiles: Definition
They describe e.g. which position (of a value) separate the Results of a Test in such a way, e.g. so the 50% of the Test Results to be before this position ( the Middle one ) and another 50% of the values to be after this position ( e.g. 50th position ). There are only 99 Percentiles ( 99th Positions ), from the 1st one to the 99th one, but they produce 100 pieces. When you cut a Cake into 99th positions, you produce 100 Cake Pieces. Correspondingly, there are only three ( 3 ) Quartiles, the 1st ( 25th position ), and the 2nd one ( 50th position which equals the position of the Median ), and the 3rd one ( 75th position ), BUT they produce four ( 4 ) pieces. When you cut a Cake in three ( 3 ) positions, you produce four ( 4 ) Cake pieces.

What is Quantiles: Etymology Quantile
I could not find a reliable Internet source about the Etymology / Origins of Quartile word. However, I found that it may related to the Latin word “Qua” (From Where) + tile “Covering”, by braking the word into two halves: Qua -n- tile, see here.

What is Quantiles: Definition
In Statistics, it has two distinct meanings / definitions that can confuse someone. The SAME word is given to describe BOTH:

i) The values of a Dataset, the numbers of a Variable as well

ii) the results that are obtained from the Quantile Function and they can be interpreted as “percentages”.

Quantile Function - Numbers and Percentages

Quantile Function and Cumulative Distribution Function: General Definition
The Quantile Function Q of a Probability Distribution ( pdf ), is the Inverse Function of the Cumulative Distribution Function ( cdf ) of this Distristribution ( pdf ). If we hypothesize that we have a Random Variable X which includes Real Values, then ( each ) Cumulative Distribution Function can be defined as:

F\left ( x \right )=\mathbb{P}(X\leq x) when  x \in \mathbb{R}

Quantile Function: Statistical Definition
As it was described, the Quantile Function Q of a Probability Distribution is the Inverse Function of the Cumulative Distribution ( cdf ) of that Distribution. Therefore, Statistically, it is defined as:

Q(\left t \right)=F^{-1}(\left t \right)= inf \left \{ x:F(\left x\right) \geq  t \right \},  x,t \in \mathbb{R}

Quantile Function: Symbol Explanation
\mathbb{R} shows all the Real Numbers.
inf shows that the function has an Open Lower Bound, that is, its Lower bound is restricted by -\infty, which is called infimum, but in some cases, its Minimum value is not included as a possible result.

Quantile Function - Numbers and Percentages_Real Numbers

Quantile Function: Using the Complementary Rule of the Probabilities
In order to calculate the Inverse cdf^{-1} of the Cumulative Distribution ( cdf ), we must use the Complementary Rule of the Probabilities which it states that:

—Two Mutually Exclusive and exhaustive events A and {A}'
—which they happen under the same framework of reference
—Then, their probability to happen equal to 1, in such way that:
\mathbb{P}\left ( A \right ) + \mathbb{P}\left ( {A}' \right )=1 and thereore
\mathbb{P}\left ( {A}' \right )=1-\mathbb{P}\left ( A \right ) and because

—This statistical manipulation F^{-1} produces the Inverse Function, then, it can be said that:
Q\left ( t \right )=F^{-1}\left ( t \right )=1-F\left ( x \right )

Quantile Function - Numbers and Percentages_Inverse

Quantile Function: An example of Inverse Function of the Empirical Cumulative Distribution
According to the Example and Results that we presented on the Article about the Empirical Cumulative Distribution Function, and in relation to the below Table:
—The final values of ECDF are placed in the column before the last one
—The final values of the Invert Function of the Empirical Cumulative Distribution
—That is, the Results of the Quantile Function are placed in the Last Column of the Table, and thus:

Therefore, the Results of the Quantile Function are also presented below. They are placed on the Y axis on the related Graph Figure:

Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 0.125 = 0.875
Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 0.125 = 0.875
Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 0.375 = 0.625
Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 0.375 = 0.625
Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 0.500 = 0.500
Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 0.875 = 0.125
Q\left ( t \right )=F_{n}^{-1}(t) = 1 - F_{n}(t) = 1- 1.000 = 0.000

These results represent the X data as Quantiles which are expressed from 0 to 1. Note that every result of the Quantile Function is expressed from 0 to 1, therefore, it can be interpreted easily as Percentages.

X value placed on Χ=tt step for the axis X=t Indicator: how many X for each tΣ ( Ι ) explanationΣ ( Ι ) for t step 0 --> 6( 1*Σ ( Ι ) ) /n for n=8
ECDF value for Y=Fn ( t )

Fn^-1(t) = 1 - Fn(t)
0011 =>I ( Xi < 0 ) = 10.13
1-0.13=0.88
---101+0 =>I ( Xi < 1 ) = 10.13
1-0.13=0.88
2, 2221+0+2 =>I ( Xi < 2 ) = 30.381-0.38=0.63
---301+0+2+0 =>I ( Xi < 3 ) = 30.381-0.38=0.63
4411+0+2+0+1 =>I ( Xi < 4 ) = 40.501-0.50=0.50
5, 5 ,5531+0+2+0+1+3 =>I ( Xi < 5 ) = 70.881-0.88=0.13
6611+0+2+0+1+3+1=>I ( Xi < 6 ) = 81.001-1.00=0.00

Quantile Function: Conclusion
Therefore, What these Results can tell us about the Real values of the Random Variable X ? Because of ECDF has been Inverted, therefore, their Inequal sign also has been Inverted, from \leq to >. Therefore, these Results can tell us that:

—The 88\% of the values are Higher than X > 0 with upper ( Right ) limit to be 6
—The 63\% of the values are Higher than X > 2 with upper ( Right ) limit to be 6
—The 50\% of the values are Higher than X > 4 with upper ( Right ) limit to be 6
—The 13\% of the values are Higher than X > 5 with upper ( Right ) limit to be 6
—The 0.0\% of the values are Higher than X > 6 with upper ( Right ) limit to be 6

Quantile Function: The Graph Figure
—Note that there are only 5 “Steps”, because so many unique X values exist.
—The points that do not exist in the values of the X variable, like the value 1 and 3, they are created with the method of the Linear Interpolation: That is:
—Each “Step” can cover so many t steps, until the next “Step” ‘s appearance, that is, until the next Unique X value.
—Therefore, value 1 is Linearly Interpolated in X axis by extending the step / point of 0 until the next Real Existed value of X variable, that is, until 2, BUT value 4 is not covered / by that step / point.
—In this case, value 1 has the same value as the X axis value of 0, on Y axis.

Quantile Function - Inverse Empirical Cumulative Distribution function_ECDF_figure_graph

Sources
Quantile Etymology / Quantile Origins II
Infimum: What is it?
Quantile Function