QQ-plot¶

The Quantile - Quantile - Plot (QQ Plot) enables to validate whether two given samples of data are drawn from the same continuous distribution of dimension 1.

We denote by $\left\{ x_1,\ldots,x_{\sampleSize} \right\}$ and $\left\{ y_1,\ldots,y_{\sampleSize} \right\}$ two given samples of dimension 1.

A QQ-Plot is based on the comparison of some empirical quantiles. Let $q_{X}(\alpha)$ be the quantile of order $\alpha$ of the distribution $F$ , with $\alpha \in (0, 1)$ . It is defined by:

$\begin{aligned} q_{X}(\alpha) = \inf \{ x \in \Rset \, |\, F(x) \geq \alpha \} \end{aligned}$

The empirical quantile of order $\alpha$ built on the sample is defined by:

$\begin{aligned} \widehat{q}_{X}(\alpha) = x_{([\sampleSize \alpha]+1)} \end{aligned}$

where $[\sampleSize\alpha]$ denotes the integral part of $\sampleSize \alpha$ and $\left\{ x_{(1)},\ldots,x_{(\sampleSize)} \right\}$ is the sample sorted in ascended order:

$x_{(1)} \leq \dots \leq x_{(\sampleSize)}$

Thus, the $j^\textrm{th}$ smallest value of the sample $x_{(j)}$ is an estimate $\widehat{q}_{X}(\alpha)$ of the $\alpha$ -quantile where $\alpha = (j-1)/\sampleSize$ , for $1 < j \leq \sampleSize$ .

The QQ-plot draws the couples of empirical quantiles of the same order from both samples: $(x_{(j)}, y_{(j)})_{1 < j \leq \sampleSize}$ . If both samples follow the same distribution, then the points should be close to the diagonal.

The following figure illustrates a QQ-plot with two samples of size $\sampleSize=50$ . In this example, the points remain close to the diagonal and the hypothesis “Both samples are drawn from the same distribution” does not seem false, even if a more quantitative analysis should be carried out to confirm this.

(Source code, png)