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Statistic Distributions

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iΟι Κατανομές χ2, t και F

Οι Κατανομές χ2, t και F
Σε αυτή την ενότητα παρουσιάζουμε συνοπτικά τρεις συνεχείς κατανομές οι οποίες, όπως και η κανονική κατανομή, είναι πολύ χρήσιμες στη Στατιστική Συμπερασματολογία. Είναι αξιοσημείωτο, ότι και οι τρεις έχουν ως αφετηρία την κανονική κατανομή1. Πρόκειται για την κατανομή χ 2 (διαβάζεται χι τετράγωνο), την κατανομή t και την κατανομή F. Γιατί οι τρεις αυτές κατανομές, είναι χρήσιμες στη Στατιστική Συμπερασματολογία, θα γίνει εύκολα κατανοητό όταν στα επόμενα μιλήσουμε για τις στατιστικές συναρτήσεις (δειγματοσυναρτήσεις) και το ρόλο τους στις συμπερασματικές στατιστικές διαδικασίες. Παρατηρείστε στον ορισμό αυτών των κατανομών, που δίνουμε στη συνέχεια, ότι πρόκειται για κατανομές συναρτήσεων ανεξάρτητων τυχαίων μεταβλητών και θυμηθείτε ότι όταν λέμε τυχαίο δείγμα μεγέθους n από έναν πληθυσμό εννοούμε n ανεξάρτητες και ισόνομες τυχαίες μεταβλητές X 1 , X 2 , ..., X n . Την κατανομή χ 2 εισήγαγε ο F. R. Helmert το 1876, την κατανομή t το 1908 ο W. L. Gosset και την κατανομή F ο G.W Snedecor (το 1934) ο οποίος της έδωσε το όνομα F προς τιμήν του διακεκριμένου στατιστικού (και γενετιστή) R. A. Fisher, γι’ αυτό στη βιβλιογραφία συναντάται και ως κατανομή Fisher, ως κατανομή Snedecor ή ως κατανομή Snedecor-Fisher. Σημειώνουμε, επίσης, ότι η κατανομή t είναι γνωστή και ως κατανομή Student, ψευδώνυμο του Gosset με το οποίο δημοσίευε τα άρθρα του. Για την προέλευση (γέννηση) της κατανομής t, θα δώσουμε σε επόμενη ενότητα και κάποιες επιπλέον χρήσιμες πληροφορίες. Στη συνέχεια, για κάθε μια από αυτές τις τρεις κατανομές, δίνουμε μόνο τον ορισμό της, τη γραφική παράσταση της συνάρτησης πυκνότητάς της, τη μέση τιμή της, τη διασπορά της και τα άνω α -ποσοστιαία σημεία της, δηλαδή, μόνο τις ελάχιστες πληροφορίες που μας είναι απαραίτητες για να κατανοήσουμε και να μπορούμε να εφαρμόζουμε σωστά βασικές στατιστικές μεθόδους της…...

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