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Lezione3stampa 10 [Modalità Compatibilità] Sommario • Indici statistici di posizione o locazione o di tendenza centrale Corso di Statistica • Medie Facoltà di Economia • Moda a.a. 20102010--20112011 • Mediana francesco mola • Quantile Lezione n° 3 lez 3 _2010-2011 statistica-francesco mola 2 Perché le medie? Medie secondo Chisini • Individuazione di una misura sintetica che tenda ad esprimere la tipicità = della distribuzione X carattere quantitati vo • Vengono definite misure di tendenza centrale perché “solitamente” si con posizionano al centro della distribuzione (ma non sempre rappresentano ≤ ≤ ⋯ ≤ ≤ ⋯ ≤ ≤ esattamente il centro) x1 x2 M xn−1 xn • Esistono molti indici di tendenza centrale alcuni che possono essere La quantità M è una media se… calcolati solo per fenomeni quantitativi altri sia per quantitativi che qualitativi . ⋯ = ⋯ • La scelta del tipo di media da utilizzare riguarda aspetti diversi, per lo più f (x1, x2 , , xn ) f (M , M , , M ) relativi alla conoscenza del fenomeno in studio (o la relazione che lega i dati o li ha generati), oppure ad una preliminare analisi grafica della Cioè: distribuzione. considerare la funzione f() sulle n modalità della X • Molti indici di tendenza centrale risentono infatti dei dati anomali (outliers ). o considerarla su n volte la costante M è equivalente. Questi a prima vista possono sembrare errori nei dati, ma non sempre è Seguendo questo approccio, al variare della funzione f() così. si ottengono differenti tipi medie. lez 3 _2010-2011 statistica-francesco mola 3 lez 3 _2010-2011 statistica-francesco mola 4 Un esempio su tredici studenti … Media aritmetica Voto in (semplice) Matematica Voto In Studenti (X) Statistica (Y) = Consideriamo come criterio Gregory House 28 30 n 13 la somma degli elementi Lisa Cuddy 30 18 = n Robert Chase 27 21 ∑xi 365 f (x , x ,⋯, x ) = x AllisonCameron 28 28 1 2 n ∑i=1 i ⇒ Eric Foreman 26 30 y =324 µ ∑ i Nella espressione del Chisini sostituisco a M il simbolo che indica la media aritmetica RemyHadley(13) 28 27 James Wilson 27 30 f (x , x ,⋯, x ) = f (µ, µ,⋯, µ) ⇔ ∑xi 365 1 2 n Amber Volakis(B.T.) 29 21 µ = = = x 28,08 n n n Chris Taub 28 22 n 13 = µ = µ Lawrence Kutner 29 26 ∑ xi ∑ ⇔ ∑ xi n ⇒ i=1 i=1 i=1 Edward Vogler 27 23 ∑yi 324 n StacyWarner 29 30 µ = = = 24,92 x y ∑i=1 i Michael Tritter 29 18 n 13 = µ 365 324 lez 3n _2010-2011 statistica-francesco mola 5 lez 3 _2010-2011 statistica-francesco mola 6 media aritmetica (ponderata) per… Un esempio su tredici studenti … distribuzioni di frequenze assolute con distribuzione di frequenza Voto in Matematica Frequenze (X) assolute (n) Consideriamo una variabile X nella forma: 26 1 27 3 (xi ,ni ) i=1,2,⋯,k 28 4 29 4 …la media aritmetica sarà: 30 1 13 nnn=+ + n + n + n = 13 k 1 2 3 4 5 = + + + += ∑ xini ∑ xi n i 26*1 27*3 28*4 29*4 30*1 365 µ = i=1 k ∑ xi n i 365 ∑ = ni Ogni x ha un’influenza µ = = = 28, 08 i 1 i x n 13 dettata da n i lez 3 _2010-2011 statistica-francesco mola 7 lez 3 _2010-2011 statistica-francesco mola 8 Un esempio su tredici studenti … media aritmetica (ponderata) per… con distribuzione di frequenza distribuzioni di frequenze relative Frequenze assolute Voto in Statistica (Y) (n) 18 2 Consideriamo una variabile X nella forma: 21 2 22 1 (x , f ) = ⋯ 23 1 i i i 1,2, ,k 26 1 …la media aritmetica sarà: 27 1 28 1 k 30 4 µ = x f 13 ∑i=1 i i =+++++++= nnnnnnnnn12345678 1 3 = + + ++ = ∑ yi n i 18*2 21*2 22*1 ... 30*4 324 Ogni x i ha un’influenza y n µ =∑ i i =3 2 4 = dettata da f y 24, 9 2 i lez 3 _2010-2011n 1 3 statistica-francesco mola 9 lez 3 _2010-2011 statistica-francesco mola 10 Un esempio su tredici studenti … Un esempio su tredici studenti … con distribuzione di frequenze relative con distribuzione di frequenze relative Voto in Statistica Frequenze Frequenze Voto in Frequenze Frequenze (Y) assolute (n) relative (f) Matematica (X) assolute (n) relative (f) 18 2 2/13 26 1 1/13 21 2 2/13 27 3 3/13 22 1 1/13 28 4 4/13 23 1 1/13 29 4 4/13 26 1 1/13 30 1 1/13 27 1 1/13 13 1 28 1 1/13 = + + + ∑ xi f i 26*(1/13) 27*(3/13) 28*(4/13) 30 4 4/13 13 1 +29*(4/13) + 30*(1/13) = =+ + + + = + + 2 6,230769 6,461538 8,615385 ∑ yi f i 18*(2/13) 21*(2/13) +8,923077 + 2,307692 = 28,08 +22*(1/13) ++ ... 30*(4/13) == 24,92 = µ µ = = y x∑ x i f i 28, 08 lez 3 _2010-2011 statistica-francesco mola 11 lez 3 _2010-2011 statistica-francesco mola 12 media aritmetica (semplice e ponderata) per… con alcuni esempi vediamo perché diciamo… distribuzioni in classi aritmetica Dati i valori 1, 2 e 3, la media è Si considerano i valori centrali di ogni classe + + Progressioni x + x µ = 1 2 3 = c = i−1 i 2 aritmetiche di i 2 3 ragione 1 con n …la media aritmetica sarà: dispari Dati i valori 1, 2, 3, 4 e 5, la media è k c k ∑i=1 i µ = c f La media in questo µ = ∑i=1 i i 1+ 2 + 3+ 4 + 5 n µ = = 3 caso è il VALORE 5 CENTRALE Si ipotizza uniformità nelle classi!!!! della distribuzione lez 3 _2010-2011 statistica-francesco mola 13 lez 3 _2010-2011 statistica-francesco mola 14 a con alcuni esempi vediamo perché diciamo… 1 Proprietà della media aritmetica aritmetica l’internalità o criterio di Cauchy Dati i valori 3, 8 e 13, la media è Progressioni ≤ µ ≤ x1 xn 3+ 8 +13 aritmetiche di µ = = 8 ragione 5 con n 3 la media è il baricentro della distribuzione dispari Dire che la media è il baricentro della distribuzione equivale a dire che: Dati i valori 3, 8, 13, 18 e 23, la media è La media in − µ = ∑ (xi ) 0 + + + + questo caso è il i 3 8 13 18 23 VALORE − µ µ = =13 dove (xi ) ∀i= ,2,1 ⋯,n 5 CENTRALE della Sono gli scarti dalla media lez 3 _2010-2011 statistica-francesco mola distribuzione15 lez 3 _2010-2011 statistica-francesco mola 16 Un esempio su tredici studenti … Voto in Voto In …graficamente abbiamo…. Matematica Statistica Studenti (X) (Y) X-mx Y-m Gregory House 28 30 -0,08 5,08 Lisa Cuddy 30 18 1,92 -6,92 Robert Chase 27 21 -1,08 -3,92 Allison Cameron 28 28 -0,08 3,08 Eric Foreman 26 30 -2,08 5,08 Remy Hadley(13) 28 27 -0,08 2,08 James Wilson 27 30 -1,08 5,08 Amber Volakis(B.T.) 29 21 0,92 -3,92 La media Chris Taub 28 22 -0,08 -2,92 aritmetica è Lawrence Kutner 29 26 0,92 1,08 il baricentro della Edward Vogler 27 23 -1,08 -1,92 Dove si distribuzione Stacy Warner 29 30 0,92 5,08 posizionerà la Michael Tritter 29 18 0,92 -6,92 media 365 324 0,00 0,00 lez 3 _2010-2011 statistica-francesco mola 17 lez 3 _2010-2011aritmetica? statistica-francesco mola 18 Un esempio su tredici studenti … a Voto in Voto In 2 proprietà: Matematica Statistica Studenti (X) (Y) X-mx (X-mx)^2 Y-my (Y-my)^2 la media è l’unico valore che rende minima la Gregory House 28 30 -0,08 0,01 5,08 25,78 somma degli scarti al quadrato Lisa Cuddy 30 18 1,92 3,70 -6,92 47,93 Robert Chase 27 21 -1,08 1,16 -3,92 15,39 AllisonCameron 28 28 -0,08 0,01 3,08 9,47 2 Eric Foreman 26 30 -2,08 4,31 5,08 25,78 − µ = RemyHadley(13) 28 27 -0,08 0,01 2,08 4,31 ∑(xi ) min James Wilson 27 30 -1,08 1,16 5,08 25,78 Amber Volakis(B.T.) 29 21 0,92 0,85 -3,92 15,39 Chris Taub 28 22 -0,08 0,01 -2,92 8,54 Lawrence Kutner 29 26 0,92 0,85 1,08 1,16 Edward Vogler 27 23 -1,08 1,16 -1,92 3,70 StacyWarner 29 30 0,92 0,85 5,08 25,78 Michael Tritter 29 18 0,92 0,85 -6,92 47,93 365 324 0,00 14,92 0,00 256,92 lez 3 _2010-2011 statistica-francesco mola 19 lez 3 _2010-2011 statistica-francesco mola 20 Minimo Es: H=4; n=22 3a Proprietà associativa x Consideriamo una variabile X ed H gruppi di numerosità non 2 x1 µ x x x7 necessariamente uguale e con ognuno una propria media 1 3 6 x x µ 8 x 9 2 Consideriamo quindi una variabile x4 5 x 10 x11 = ⋯ x18 X x , x , , x x x17 x 1 2 n 16 µ 19 x13 4 x x20 x 12 µ 21 x 3 22 raggruppata in H gruppi secondo l’esempio x15 x14 seguente: lez 3 _2010-2011 statistica-francesco mola 21 lez 3 _2010-2011 statistica-francesco mola 22 …allora abbiamo… …… µ n + µ n +⋯+ µ n H 1 1 2 2 H H = µ Media ponderata n + n +⋯+ n ∑ hnh delle medie dei 1 2 H = µ = h 1 singoli gruppi H n n n µ 1 + µ 2 +⋯+ µ H ∑ nh 1 2 H h=1 n n n + +⋯+ = cioè con n1 n2 nh n lez 3 _2010-2011 statistica-francesco mola 23 lez 3 _2010-2011 statistica-francesco mola 24 Aggiungiamo come informazione l’anno di corso ai nostri tredici studenti Cosa accade se vogliamo conoscere la media dei voti in funzione dell’anno di iscrizione? nh Voto in x n Studenti Matematica (X) Voto In Statistica (Y) Anno di corso Per ciascun gruppo ∑ i i per avere la media del gruppo h µ = i=1 Gregory House 28 30 Primo calcoliamo la media h n Lisa Cuddy 30 18 Secondo h Robert Chase 27 21 Secondo Primo AllisonCameron 28 28 H Eric Foreman 26 30 Secondo µ ∑ hn h RemyHadley(13) 28 27 Terzo La formula da utilizzare èµ = h=1 per avere la media generale James Wilson 27 30 Secondo H n Amber Volakis(B.T.) 29 21 Primo ∑ h h=1 Chris Taub 28 22 Secondo Lawrence Kutner 29 26 Secondo Edward Vogler 27 23 Terzo Stacy Warner 29 30 Secondo Numero di gruppi H = 3 Michael Tritter 29 18 Secondo 365 324 lez 3 _2010-2011 statistica-francesco mola 25 lez 3 _2010-2011 statistica-francesco mola 26 Abbiamo quindi per i voti in MATEMATICA : E’ come se avessimo costruito tre distribuzioni in funzione della provenienza 3 Voto in Matematica Frequenze ∑x n i i 28*2+ 29*1 Media del (X) assolute (n) Primo anno µ =i=1 = = 28 2 1 28,333 3 3 primo anno 29 1 3 8 ∑x n i i 26*1+ 27*2 + 28*1 + 29*3 + 30*1 Voto in Matematica Frequenze µ =i=1 = = (X) assolute (n) 2 28,125 Media del 8 8 secondo anno 27 1 Secondo anno 2 28 1 ∑ x n i i 27*1+ 28*1 2 µ =i=1 = = 27,5 Media del 3 2 2 terzo anno Voto in Matematica Frequenze (X) assolute (n) H µ n 26 1 ∑ h h 27 2 = 28,333*3+ 28,125*8 + 27,5*2 µ =h 1 = = 28,08 Media generale 28 1 Terzo anno n 13 29 3 30 1 8 lez 3 _2010-2011 statistica-francesco mola 27 lez 3 _2010-2011 statistica-francesco
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