30
Statistique appliquée aux sciences de la vie
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−3 −2 −1 0 1 2 3
−2
−1
0
1
2
quantiles normale standardisée
observations standardisées
(a) normale
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−3 −2 −1 0 1 2 3
−5
0
5
quantiles normale standardisée
observations standardisées
(b) valeurs extrêmes
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−3 −2 −1 0 1 2 3
−1
0
1
quantiles normale standardisée
observations standardisées
(c) uniforme
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−3 −2 −1 0 1 2 3
−2
−1
0
1
2
quantiles normale standardisée
observations standardisées
(d) bimodale
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−3 −2 −1 0 1 2 3
−5 −4 −3 −2 −1
0
1
quantiles normale standardisée
observations standardisées
(e) asymétrique à gauche
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−3 −2 −1 0 1 2 3
0
2
4
6
8
1 0
quantiles normale standardisée
observations standardisées
(f) asymétrique à droite
Figure 2.12 – Exemples de qq-plots pour différentes distributions continues.
5 %
15 % 25 % 35 % 45 % 55 % 65 % 75 % 85 % 95 %
d’une variable normale standardisée, c’est-à-dire avec (selon le tableau A.1) :
−1.645 −1.035 −0.675 −0.385 −0.135 0.135 0.385 0.675 1.035 1.645
Plus généralement, n observations standardisées (et rangées par ordre croissant)
d’une variable normale devraient à peu près coïncider avec les quantiles (i −
0.5)/n d’une distribution normale standardisée (pour i = 1, · · · , n). Un qqplot est un graphique nous montrant ces observations standardisées en fonction
de ces quantiles d’une distribution normale standardisée. Si une variable est
proche de la normalité, les points sur ce graphique seront à peu près alignés
sur la droite identité.
La figure 2.12 nous montre des qq-plots obtenus à partir de n = 500 observations (standardisées) provenant des densités de probabilité montrées dans la
figure 2.4. Seul le graphique (a), où la distribution est effectivement normale,
nous montre un bel alignement. Les autres graphiques nous montrent à quoi
ressemble un qq-plot pour différentes distributions non normales.