Corso di STATISTICA. Prof. Giovanni LATORRE. e- mail: sito web: libri + slides + avvisi
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1 Corso di STATISTICA Prof. Giovanni LATORRE e- mail: sito web: libri + slides + avvisi
2 in Medicine Robert H. Riffenburgh Ed. Academic Press Price: 53,95 su Amazon
3 Steps in a scienffic process to increase knowdlege: a) observe; b) gather data; c) describe; d) explain; e) predict. Science is a set of facts and theories based upon informafon obtained with Scien&fic Method CharacterisFcs of the Scien&fic Method: a) the Method has to be objecfve or unbiased ; b) the Method should involve the control of variables; c ) the Method should be repeatable; d) the Method should allow the accumulafon of results.
4 Elementary Concepts StaFsFcal Unit or Unit PopulaFon Sample Sample Survey Experimental Design Variable: Discrete, ConFnuous (quanftafve), Categorical (qualitafve) Data: Discrete, ConFnuous, Categorical
5 Example In the Gynecological Department each woman was asked about the number of live births they had delivered so far. Answers: Raw Data This is an example of discrete variable and discrete data.
6 Frequency N of Live Births Frequency Total 50 Tabulated Data
7 Frequency Histogram Absolute Frequency N of Live Births N of Live Births 7
8 Volume of Prostate in 300 (VOL in ml) Raw Data 32,26 25,59 62,06 36,31 51,07 68,09 39,98 29,30 33,80 32,20 26,96 31,03 38,15 60,78 36,15 52,84 38,00 32,20 44,00 17,00 16,19 36,09 21,38 38,39 22,56 20,84 8,00 25,70 30,40 44,70 32,96 12,89 38,53 98,09 50,78 87,33 36,40 33,30 35,00 13,50 30,87 28,30 33,09 26,41 26,34 31,62 67,00 26,50 43,00 41,00 73,73 57,77 19,95 32,24 18,42 20,68 68,00 16,40 65,10 6,90 30,50 41,51 68,99 32,26 26,34 39,48 25,00 51,00 48,50 15,00 30,54 23,56 19,46 30,64 13,83 38,62 34,00 41,30 28,90 24,00 36,85 62,85 36,94 22,17 20,94 67,49 13,00 40,00 39,00 25,10 16,39 29,05 24,90 53,92 35,88 74,44 24,80 23,70 31,40 36,00 33,20 17,37 22,41 29,58 51,61 40,05 25,00 26,70 60,30 35,50 82,43 82,43 25,47 33,45 43,22 32,49 55,00 18,00 18,30 6,50 38,57 52,51 40,14 12,89 59,76 24,80 17,60 53,00 7,60 28,20 44,88 50,84 17,72 36,97 26,46 29,97 67,20 58,00 48,70 33,70 27,76 22,48 26,05 25,12 57,35 88,17 15,30 24,00 67,60 50,00 14,86 26,88 31,03 37,81 30,52 39,54 22,00 32,90 45,00 26,50 49,13 52,84 60,93 35,59 30,35 40,22 40,80 11,90 15,10 41,50 36,05 24,33 26,41 62,25 94,32 60,13 26,00 13,00 24,00 21,50 31,45 31,78 39,43 24,19 42,44 51,76 41,50 34,90 64,70 27,00 38,53 53,99 60,78 7,17 31,83 80,96 32,40 55,80 34,20 24,80 75,13 39,43 37,73 29,93 26,36 16,43 17,00 40,20 51,30 12,50 70,21 27,26 23,19 19,62 30,50 20,30 35,00 32,30 21,50 30,50 19,45 30,86 23,34 35,28 22,71 48,88 37,60 71,30 19,60 65,00 26,89 26,72 33,25 70,04 73,52 25,69 23,20 22,50 33,20 12,10 32,26 34,07 49,26 24,52 39,28 48,26 25,60 30,80 3,30 37,50 52,42 42,34 23,14 40,05 33,79 28,42 20,80 21,20 22,50 47,30 30,26 41,60 17,31 36,62 27,49 22,00 21,50 32,80 40,00 22,50 18,86 114,03 52,80 46,19 29,71 32,33 32,80 45,90 18,40 30,00 20,58 56,28 42,92 23,50 23,84 75,22 54,90 36,00 26,70 15,90 21,83 42,59 16,03 44,03 77,26 79,15 34,20 34,40 25,00 15,80 This is an example of confnuous variable and confnuous data.
9 Frequency Vol. Prostate (in ml.) Absolute Frequency Total 300 Tabulated Data
10 Volume of Prostate in Absolute Frequencies Volume of Prostate in ml
11 Frequency DRE Fr. negative 115 positive 185 Totale 300 DRE = digital rectal examina@on This is an example of categorical variable and categorical data.
12 nega@ve N of DRE 185 posi@ve
13 Frequency of the Volume of Prostate in 300 Vol. Prostate Absolute (in ml.) Frequency Frequency Frequency ,14 0, ,55 0, ,19 0, ,09 0, ,02 1, ,00 1,00 Total 300 1,
14 0,60 Volume of Prostate in 300 0,55 0,50 Frequencies 0,40 0,30 0,20 0,14 0,19 0,10 0,09 0,00 0,02 0, Volume of Prostate in ml
15 0,60 Volume of Prostate in 300 0,50 Frequencies 0,40 0,30 0,20 0,10 0, Volume of Prostate in ml
16 0,09 0,02 0,00 0,14 Volume of Prostate in 300 (in ml) , ,55 Frequencies
17 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
18 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
19 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
20 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
21 1,20 1,00 0,97 1,00 1,00 Frequencies x=0,8325 0,80 0,60 0,40 0,69 0,88 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
22 ! 0, = 0,88 0, ! 0,69 = (55 40)! = 0,69 + (55 40) 0,88 0, ,88 0, ! = 0, ,19 20 = 0,8325
23 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,50 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
24 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,50 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
25 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,50 0,40 0,69 0,20 0,14 0,00 0, Volume of Prostate in 300 Pa@ents (in ml.)
26 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,50 0,40 0,69 0,20 0,14 0,00 0,00 X=33, Volume of Prostate in 300 Pa@ents (in ml.)
27 ! 20 0,50 0,14 = ,69 0,14! 20 = (0,50 0,14)! = 20 + (0,50 0,14)! = , ,69 0, ,69 0, ,55 = 33,1
28 Symbols X = QuanFtaFve Variable (i.e.: N of births per woman, Volume of Prostate per pafent) x i = Value of X in the i- th out of n unit x 1, x 2,., x i,.,x n = Data (they can either be: PopulaFon or Sample Data)
29 Measures SummaFon Sign:!! +!! + +!! + +!! =!!!!!!! ProperFes: k + k k = k = n k n i=1 If k=1/b :!!!!! + +!!! + +!!! =!!!! =!!!!!!!!!!!!! + +!!! + +!!! =!!!! = 1!!!!!!!!!!!!
30 Mean M = 1 n (x x i x n ) = 1 n! n x i i=1 Proper@es:!! =!!!!! y 1 M y n!!!!!! =!!! =!"!" = 0!!!!!!!!!!!
31 Also:! =! 1!!!!! X Fr(X) = n i Fr.r(X) = f i x 1 n 1 f 1 x 2 n 2 f x i n i f i --- x k n k f k Total n 1!!!!!!! =!!!!!! Weighted Mean:! = 1! (!!!! + +!!!! + +!!!! ) = 1!! =!!!!!!!!!!!! ; with:!! =!;! = 1!!!!!!!!!!!!!
32 Examples: 1) N of Live Births per Woman: M = ( ) / 50 = 1,96; alternafvely: M = (0*5 + 1*12 + 2*19 + 3*9 + 4*4 + 5*1) / 50 = 1,96; alternafvely: M =(0*0,10+1*0,24+2*0,38+3*0,18+ 4*0,08+5*0,02)=1,96 2) Volume of Prostate per Pa@ent: M = (32,26+25, ,40+25,00+15,80)/300 = 36,30427; alternafvely: M=(10*42+30*166+50*56+70*28+90*7+110*1)/300=36,33333; alternafvely: M=(10*0,14+30*0,55+50*0,19+70*0,09+90*0,02+110*0,00)=35,5.
33 of the Mean for a Frequency Distribu@on of Grouped Data (1) (2) (3) Vol. Prostate Vol. Prostate Absolute (2) x (3) (in ml.) (central values) Frequency Total M=10.900/300=36,33
34 Median Let x 1, x 2,., x i,.,x n, n observafons on the variable X and y 1, y 2,., y i,., y n the same data arranged in increasing order, that is: y 1 y 2 y i y n Then, if n is an odd number: if n is an even number: Median = Me = y (n+1)/2 Median = Me = [ y (n/2) + y (n/2)+1 ]/ NoFce: y 1 Me y n.
35 Examples: 1)N of Live Births per Woman: n=50=even; (n/2)=25; (n/2)+1=26; data arranged in increasing order: ; Median = Me = (2 + 2)/2 = 2. 2) Volume of Prostate per Pa@ent: 32.26, 25.59, 62.06,.., n=300=even; (n/2)=150; (n/2)+1=151; data arranged in increasing order: 3.30, 6.50, 6.90, 7.17,.,98.09, ; Median = Me = (y 150 y 151 )/2 = ( ) / 2 = 32.36
36 of the Median for a Frequency Distribu@on of Grouped Data Vol. Prostate Absolute Rela@ve Cumua@ve (in ml.) Frequency Frequency Frequency ,14 0, ,55 0, ,19 0, ,09 0, ,02 1, ,00 1,00 Total 300 1, The Median Class
37 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,50 0,40 0,69 0,20 0,14 0,00 0, Me Volume of Prostate in 300 Pa@ents (in ml.)
38 1,20 1,00 0,97 1,00 1,00 0,88 Frequencie 0,80 0,60 0,50 0,40 0,69 0,20 0,14 0,00 0,00 Me=33, Volume of Prostate in 300 Pa@ents (in ml.)
39 !" =!" + (0,5!")!"!"!"!" Es=Upper Limit of the Median Class Ei=Lower Limit of the Median Class Fs=Cumulative Frequency of the Median Class Fi=Cumulative Frequency of the Class Preceding the Median Class!" = ,50 0, ,69 0,14 = 33,1
40 Altri indici di posizione: Moda (Md) Classe modale Distribuzione unimodale Distribuzione plurimodale Md Md 40
41 I percentili I Quartile = Q 1 è il valore tale che il 25% delle unità del collettivo hanno modalità Q 1 ; II Quartile = Q 2 =Me è il valore tale che il 50% delle unità del collettivo hanno modalità Q 2 ; III Quartile = Q 3 è il valore tale che il 75% delle unità del collettivo hanno modalità Q 3 ;
42 n = pari ed (n/2) = m = pari: Calcolo dei Quartili (1) Q 1 non è altro che la mediana delle prime m=(n/2) modalità ordinate, con m=n/2=pari, quindi: Q 1 =[y (m/2) +y (m/2)+1 ]/2 Q 3 non è altro che la mediana delle ultime m=(n/2) modalità ordinate, quindi: Esempio: n=12; m=6 Q 3 =[y m+(m/2) +y m+(m/2)+1 ]/2 y 1, y 2, y 3, y 4, y 5, y 6 y 7, y 8, y 9, y 10, y 11, y 12 Q 1 Q 3 42
43 Calcolo dei Quartili (2) n = pari ed (n/2) = m = dispari Q 1 non è altro che la mediana delle prime m=(n/2) modalità ordinate, con m=(n/2)=dispari, quindi: Q 1 =y (m+1)2 Q 3 non è altro che la mediana delle ultime m=(n/2) modalità ordinate, quindi: Esempio: n=10; m=5 Q 3 =y m+(m+1)/2 y 1, y 2, y 3, y 4, y 5 y 6, y 7, y 8, y 9, y 10 Q 1 Q 3
44 Calcolo dei Quartili (3) n = dispari e (n-1)/2 = m = pari Q 1 non è altro che la mediana delle prime m=(n-1)/2 modalità ordinate, con m=(n-1)/2=pari, quindi: Q 1 =[y (m/2) +y (m/2)+1 ]/2 Q 3 non è altro che la mediana delle ultime m=(n-1)/2 modalità ordinate, quindi: Q 3 =[y (m+1)+(m/2) +y (m+1)+(m/2)+1 ]/2 Esempio: n=13; m=6 y 1, y 2, y 3, y 4, y 5, y 6 y 7 y 8, y 9, y 10, y 11, y 12, y 13 Q 1 Q 3 44
45 Calcolo dei Quartili (4) n = dispari e (n-1)/2 = m = dispari Q 1 non è altro che la mediana delle prime m=(n-1)/2 modalità ordinate, con m=(n-1)/2 = dispari, quindi: Q 1 =y [(m+1)/2 Q 3 non è altro che la mediana delle ultime m=(n-1)/2 modalità ordinate, quindi: Esempio: n=11; m=5; Q 3 =y (m+1)+(m+1)/2 y 1, y 2, y 3, y 4, y 5 y 6 y 7, y 8, y 9, y 10, y 11 Q 1 Q 3 45
46 for the Prostate Volume Data n = 300, n/2 = m =150 = even: Q 1 = [ y m/2 + y (m/2)+1 ] / 2 = =[ y 75 + y 76 ] / 2 = = [24, ,19 ] /2 = 24,095 Q 3 = [ y m+(m/2) + y m+(m/2)+1 ] / 2 = =[ y y 226 ] / 2 = = [43, ,22 ] /2 = 43,11 (the indicies of the y s are called ranks)
47 1,20 1,00 0,97 1,00 1,00 0,88 Frequencies 0,80 0,75 0,60 0,40 0,69 0,25 0,20 0,14 0,00 0,00 Q Q Volume of Prostate in 300 Pa@ents (in ml.)
48 1,20 1,00 0,97 1,00 1,00 Frequencies 0,80 0,75 0,60 0,40 0,69 0,88 0,25 0,20 0,14 0,00 0,00 Q =24,00 Q 40 3 =46, Volume of Prostate in 300 Pa@ents (in ml.)
49 of the for a Frequency Distribu@on of Grouped Data Q 1 Q 3 Vol. Prostate Absolute Rela@ve Cumua@ve (in ml.) Frequency Frequency Frequency ,14 0, ,55 0, ,19 0, ,09 0, ,02 1, ,00 1,00 Total 300 1, The Quar@les Classes Q 1 Q 3
50 !! =!" + (0,25!")!"!"!"!"!! = ,25 0, ,69 0,14 = 24,00!"!"!! =!" + (0,75!")!"!" 60 40!! = ,75 0,69 0,88 0,69 = 46,32
51 Volume Prostata in 300 (ml.) (1) Frequenze Assolute (2) Valori Centrali (3)=(2)*(1) (4) Frequenze (5) Densità di Frequenza (6) Frequenze Cumulate ,5 2,5 0,003 0,0007 0, ,5 37,5 0,017 0,0033 0, ,5 125,0 0,033 0,0067 0, ,5 455,0 0,087 0,0173 0, ,5 900,0 0,133 0,0267 0, , ,0 0,133 0,0267 0, , ,5 0,163 0,0327 0, , ,5 0,117 0,0233 0, , ,0 0,080 0,0160 0, ,5 475,0 0,033 0,0067 0, ,5 892,5 0,057 0,0113 0, ,5 402,5 0,023 0,0047 0, ,5 562,5 0,030 0,0060 0, ,5 607,5 0,030 0,0060 0, ,5 435,0 0,020 0,0040 0, ,5 310,0 0,013 0,0027 0, ,5 247,5 0,010 0,0020 0, ,5 175,0 0,007 0,0013 0, ,5 92,5 0,003 0,0007 0, ,5 97,5 0,003 0,0007 0, ,5 0,0 0,000 0,0000 0, ,5 0,0 0,000 0,0000 0, ,5 112,5 0,003 0,0007 1,000 Totali ,0 1, Moda = ( )/2 = 32,5 Mediana = 30 + (0,5-0,407) * (35-30) / (0,570-0,407) = 32,8575 Media = ,0 /300 = 36,517
52 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0, Volume della Postata in ml.
53 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0, Moda=32,5 Volume della Postata in ml.
54 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0, Moda=32,5 Mediana=32,8575 Volume della Postata in ml.
55 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0,00 Md Quanto vale l AREA soio dell istogramma a sinistra della MEDIANA? Mediana=32,8575 Volume della Postata in ml.
56 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0,00 Md Quanto vale l AREA soio dell istogramma a sinistra della MEDIANA? 0,5 cioè il 50% dell area totale. Mediana=32,8575 Volume della Postata in ml.
57 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0,00 Md Me Media=36,517 Volume della Postata in ml.
58 d 1 =6m d 2 =3m d 3 =6m n 1 20kg A F n 2 10 Kg n 3 15 Kg L asta A, che può basculare intorno al fulcro F, è in equilibrio?
59 d 1 =6m d 2 =3m d 3 =6m n 1 20kg A F n 2 10 Kg n 3 15 Kg L asta A, che può basculare intorno al fulcro F, è in equilibrio? La risposta è SI perché: 20kg 6m = (10kg 3m) + (15kg 6m) = 120 kg m.
60 d 1 =6m d 2 =3m d 3 =6m n 1 20kg A F n 2 10 Kg n 3 15 Kg L asta A, che può basculare intorno al fulcro F, è in equilibrio? La risposta è SI perché: 20kg 6m = (10kg 3m) + (15kg 6m) = 120 kg m ed il punto F è detto BARICENTRO, cioè punto di equilibrio delle forze.
61 d 1 =6m d 3 =6m d 2 =3m n 1 20kg M n 2 10 Kg n 3 15 Kg x 1 A F x 2 x 3 L asta A, che può basculare intorno al fulcro F, è in equilibrio? La risposta è SI perché: 20kg 6m = (10kg 3m) + (15kg 6m) = 120 kg m ed il punto F è detto BARICENTRO, cioè punto di equilibrio delle forze. Se l asta A fosse graduata e x 1, x 2, x 3 ed M fossero le tacche che indicano il posizionamento dei tre pesi e del BARICENTRO, avremo che: n 1 (M-x 1 ) = n 2 (x 2 -M) + n 3 (x 3 -M), cioè, in generale, Σ(x i -M)n i =0 ovvero Baricentro=Media
62 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0, Qual è l interpretazione della Media? Media=36,517 Volume della Postata in ml.
63 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0, Qual è l interpretazione della Media? La Media è il Baricentro fisico. Media=36,517 Volume della Postata in ml.
64 0,04 Distribuzione di Frequenze del Volume della Prostata (in ml.) in 300 0,03 0,03 Densità di Frequenza 0,02 0,02 0,01 0,01 0,00 M Volume della Postata in ml.
65 M
66 M
67 M
68 Spread Variability - Dispersion It is of great iterest to measure the degree of variafon or scaier around the loca3on measure (mean). Example: In two groups of 15 pafents of the Urology Department it has been measure the: psa = prostate- specific an&gen level (in ng/ml, >0, rounded to one decimal place) Group1 PSA 7,80 4,10 5,90 9,00 6,80 8,00 7,70 4,40 6,10 7,90 5,30 6,60 7,60 4,80 5,70 M = 6,51 Group2 PSA 6,53 6,49 6,48 6,54 6,55 6,47 6,48 6,54 6,56 6,46 6,45 6,45 6,57 6,57 6,51 M = 6,51
69 Measures of: Spread Variability Dispersion 1) Range: R = y n y 1 ; PSA Min Max Range Group 1 4,10 9,00 4,90 Group 2 6,45 6,57 0,12 2) Interquar@le Range: IQR = Q 3 Q 1 ; PSA Q1 Q2 Q3 IQR Group 1 5,30 6,60 7,80 2,50 Group 2 6,47 6,51 6,55 0,08
70 ! 3) Variance:!! = 1! (!!!)!!!!!!! = 1! = 1!!!!!!!!!!!!! = 1! (!!! 2!!! +!! ) = 2! 1!!!!!!!!!!!!!!!! +!!
71 of th Variance for PSA values/group1 x i 7,80 4,10 5,90 9,00 6,80 8,00 7,70 4,40 6,10 7,90 5,30 6,60 7,60 4,80 5,70 97,70 (x i -M) 1,29-2,41-0,61 2,49 0,29 1,49 1,19-2,11-0,41 1,39-1,21 0,09 1,09-1,71-0,81 0,00 (x i -M) 2 1,66 5,82 0,38 6,18 0,08 2,21 1,41 4,47 0,17 1,92 1,47 0,01 1,18 2,94 0,66 30,56! AlternaFvely: M=97,70/15=6,51 ; V(x)=30,56/15=2,038 x i 7,80 4,10 5,90 9,00 6,80 8,00 7,70 4,40 6,10 7,90 5,30 6,60 7,60 4,80 5,70 97,70 2 x i 60,84 16,81 34,81 81,00 46,24 64,00 59,29 19,36 37,21 62,41 28,09 43,56 57,76 23,04 32,49 666,91 M=97,70/15=6,51 ; V(x)=666,91/15-M 2 =44,461-42,423=2, Computa@on of th Variance for PSA values/group2 x i 6,530 6,490 6,480 6,540 6,550 6,470 6,480 6,540 6,560 6,460 6,450 6,450 6,570 6,570 6,510 97,65 (x i -M) 0,020-0,020-0,030 0,030 0,040-0,040-0,030 0,030 0,050-0,050-0,060-0,060 0,060 0,060 0,000 0,00 (x i -M) 2 0,000 0,000 0,001 0,001 0,002 0,002 0,001 0,001 0,002 0,002 0,004 0,004 0,004 0,004 0,000 0,03! M=97,75/15=6,51 ; V(x)=0,03/15=0,002 AlternaFvely: x i 6,53 6,49 6,48 6,54 6,55 6,47 6,48 6,54 6,56 6,46 6,45 6,45 6,57 6,57 6,51 97,65 x i 2 42,64 42,12 41,99 42,77 42,90 41,86 41,99 42,77 43,03 41,73 41,60 41,60 43,16 43,16 42,38 635,73! M=97,75/15=6,51 ; V(x)=635,73/15-42,38=0,002
72 X Fr(X) = n i Fr.r(X) = f i x 1 n 1 f 1 x 2 n 2 f x i n i f i --- x k n k f k Total n 1 4) Variance for tabulated data: V(x) = 1 n k i=1 x i - M ( ) 2 n i
73 V(x) = 1 n = 1 n = 1 n = 1 n i i i i Computa@on method: ( x i - M ) 2 n i = ( x 2 i - 2Mx i +M 2 )n i = x i 2 n i x i 2 n i - 2M 1 n x i n i +M 2 1 n i = n - 2M 2 +M 2 i i k V = 1 2 x i n i - M 2 n i=1
74 (1) (2) N of Live Frequencies (3)=(1)x(2) (4)=(1)x(1) (5)=(4)X(2) Births Total M = 98/50 = 1,96 ; V(X) = 258/50 M 2 = 5,16 3,84 = 1,32
75 (1) (2) (3) Vol. Prostate Vol. Prostate Absolute (4)=(2) x (3) (5)=(2)x(2) (6)=(5)x(3) (in ml.) (central values) Frequency Total M = /300 = 36,33 V(X) = /300 M 2 = 1.665, ,87 = 345,46
76 Defects of the Variance and Measures of Dispersion a) Variance expressed in the square of the Unit Measure AlternaFve: Standard =!(!) b) Variance is an absolute measure of Variability AlternaFve: Coefficient of =!"!
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