2.4.4. The quantile computation
There are several ways of estimating the quantiles implemented in Uranie. This part describes the most commonly used and starts with a definition of quantile.
A quantile \(x_p\), as discussed in the following parts, for \(p\) a probability going from 0 to 1, is the lowest value of the random variable \(X\) leading to \(P\left\{X \leq x_p \right\} = p\). This definition holds equally if one is dealing with a given probability distribution (leading to a theoretical quantile), or a sample, drawn from a known probability distribution or not (leading to an empirical quantile). In the latter case, the sample is split into two sub-samples: one containing \(pN\) points, the other one containing \((1-p)N\) points.
It can be easily pictured by looking at Figure 2.44 which represents the cumulative distribution function (CDF) of the attribute \(x_2\). The quantile at 50 percent for \(x_2\) can be seen by drawing an horizontal line at 0.5, the value of interest being the one on the abscissa where this line crosses the CDF curve.