curs 6-2009.doc
TRANSCRIPT
BIOSTATISTICS
81
CURS 6
Biostatistic
1. Noiuni de baz; definiii
Statistica este o ramur a matematicii aplicate cu rdcini n teoria probabilitilor i este fundamentat pe toate tiinele experimentale: fizica, biologia, sociologia. Statistica este tiina care se ocup cu descrierea i analizarea numeric a fenomenelor de mas. Ea studiaz latura cantitativ a fenomenelor, legile statistice manifestndu-se sub form de tendine.
Biostatistica intereseaz aplicarea statisticii fenomenelor biologice, incluznd biologia uman, medicina i sntatea public.
Concepte specifice:
1. Studiul reprezint o organizare tiinific a sarcinilor cu definirea unei mulimi de obiective.
2. Rezumatul este un studiu al crui scop este s evalueze condiiile care exist n natur i modificrile orict de nensemnate (mici).
3. Experimentul este un studiu care modific condiiile existente ntr-o manier definit pentru a evalua efectele unuia sau mai multor tratamente.
4. Unitatea este cel mai mic obiect sau individ care poate fi investigat, sursa informaiei de baz.
5. Populaia este un foarte mare grup de uniti avnd aceleai caracteristici cuantificabile, cu privire la care sunt fcute inferene tiinifice.
6. Eantionul de uniti este o submulime finit de uniti din populaia de uniti.
7. Parametrul este o caracteristic a populaiei.
8. Variabila este o caracteristic observabil pe uniti.9. Populaia de observaii este un grup care const n valori numerice ale unei caracteristici cuantificabile determinate pentru fiecare membru al populaiei de uniti.10. Eantionul de observaii este o submulime finit din populaia de observaii.11. Statistica este o caracteristic a irului, utilizat pentru verificarea inferenelor.
Observarea statistic a diverselor boli sau fenomene s-a fcut din antichitate; primele statistici adevrate s-au fcut n sec. XVII, de ctre John Grownt, care a fcut primele recensminte de bolnavi n timpul unor epidemii de cium din Londra, construind tabele de natalitate i mortalitate. Termenul de biostatistic a fost introdus n 1772, de ctre Achenwald, matematician german. Pearson, matematician i statistician englez, a condus revista de biostatistic Biometrika; n 1900 a introdus testul (2. Un alt nume important este cel al lui Francisc Gatton, expert n probleme de ereditate. 2. Elemente de statistic descriptiv. Descrierea unei serii statistice
O serie statistic este constituit dintr-un ansamblu (mulime) de valori numerice rezultat al unei observaii.
De exemplu, se studiaz numrul de biei ntr-un grup de familii, nlimea, greutatea ntr-un grup de recrui, procentul urinar sau sanguin al metabolismului ntr-un grup de bolnavi.
Prima etap a activitii statistice const n clasificarea rezultatelor obinute, prezentarea sub o form facil, accesibil i care ne d o descriere fidel pe ct posibil.
Ordonarea datelor. Distribuia de frecven
Fie un grup de valori numerice experimentale ale tipului pe care l citm. Rezultatele sunt obinute ntr-o ordine oarecare. Este logic s ncepem cu o ordonare cresctoare sau descresctoare, etap care poart numele de punerea n ordine a datelor. Se nscrie pentru fiecare valoare observat x, numrul F de apariii, care se va numi frecvena (efectivul) valorii.
Mulimea valorilor efective ale frecvenelor respective constituie distribuia frecvenei, care se poate prezenta printr-un tabel.
Numrul total n de cazuri se definete ca fiind suma efectivelor fiecrei valori,
Exemplul 1: Se studiaz numrul de biei n 1877 familii de 7 copii. Rezultatele sunt prezentate n funcie de numrul x de biei, cuprins ntre 0 i 7. Se observ fiecare dintre aceste valori, care indic efectivul F = numrul de apariii ntlnite, adic numrul de familii avnd acest numr de copii.
Numr de biei XNumr de familii FFrecvena
f=F/nProcentajul
100*f
0220.011
11120.066
22850.1515
34700.2525
45130.2727
53200.1717
61320.077
7230.011
Total18771100
Tabelul 1: Distribuia numrului de biei n 1877 familii cu cte 7 copii
Gruparea datelor n clase
Cnd valorile studiate variaz continuu, cum este cazul greutii i al nlimii etc., adic atunci cnd variabila poate lua toate valorile posibile ntr-un interval dat, diferitele valori observate pot fi foarte numeroase. Distribuia de frecven este atunci foarte dispersat i nu ofer o interpretare semnificativ a mulimii de valori. De aceea, se recurge la reducerea numrului de valori posibile, grupnd valorile vecine. Mai precis, se mparte domeniul de variaii posibile ntr-un numr de intervale sau clase n interiorul crora se grupeaz toate valorile care cad n intervalul corespunztor.
Exemplul 2: Se studiaz distribuia greutii ntr-un grup de 100 aduli normali de sex feminin. Greutatea variaz ntre 41 74 kg. Se mparte domeniul de variaie n intervale de 5 kg (40 44, 45 49, 50 54, ), care constituie clase n interiorul crora se grupeaz toi subiecii a cror greutate este cuprins ntre limitele intervalului; mulimea acestor subieci constituie efectivul clasei.
Clase
xEfectiv
FFrecvena
f=F/nProcentaj
100*f
40-4450.055
45-49120.1212
50-54310.3131
55-59310.3131
60-64160.1616
65-6930.033
70-7420.022
TotalN=1001100
Tabelul 2: Distribuia greutii a 100 aduli de sex feminin
Intervalul de clasn aceast activitate este important modul de precizare a domeniului claselor alese, numit interval de clas, care trebuie s fie n principiu acelai pentru toate clasele; intervalele de clas trebuie de asemeni s fie alturate i fr suprapuneri.
Intervalele de clas se pot preciza n trei moduri (figura 1, tabel 3):
Figura 1: Alegerea intervalelor de clas
Indicnd limitele reale ale fiecrei clase; n acest caz limita superioar a unei clase i limita inferioar a clasei urmtoare coincid, ca de exemplu valoarea 44.5 pentru clasele (39.5 .. 44.5) i (44.5 .. 49.5). Indicnd msurile limit, adic cea mai mic i cea mai mare msur corespunztoare apartenenei la clas, innd cont de precizia msurtorii. Indicnd valoarea care se gsete exact n centrul clasei, numit punct median al clasei, i care este dat de semisuma limitelor unei clase, ca de exemplu 42, 47, 52 etc.
Msuri
limitLimitele
realePuncte medianeEfective
40-4439.5 44.5425
45-4944.5 49.54712
50-5449.5 54.55231
55-5954.5 59.55731
60-6459.5 64.56216
65-6964.5 69.5673
70-7469.5 74.5722
100
Tabelul 3: Distribuia greutii a 100 aduli de sex feminin
Grupnd datele n clase, se remarc asimilarea tuturor valorilor unei clase la o valoare unic, aceea a punctului median (n acest mod, se face reducerea la cazul unei variabile discontinue).
Observaie: Experiena arat c n practic numrul intervalelor de clas este bine s fie cuprins ntre 10 i 20.
Frecvena relativEfectivul F al unei valori (sau al unei clase) reprezint frecvena sa absolut, adic numrul de apariii ale acestei valori (sau al acestei clase) n ansamblul distribuit.
Dac dorim s comparm serii statistice comportnd un numr diferit de cazuri, este interesant s raportm aceast frecven absolut la numrul n de cazuri, numit efectivul total, care conine seria studiat. Se definete la fel frecvena relativ, sau frecvena propriu-zis a valorilor considerate,
Se poate atunci completa tabloul distribuiei de frecvene printr-o coloan suplimentar indicnd valoarea frecvenei relative, care poate fi de asemeni exprimat sub form de procente (coloana 4 din tabelele 1, 2).
O variant a frecvenei relative o constituie frecvena procentual, obinut nmulind frecvena relativ cu 100,
fp = fr ( 100
Suma frecvenelor relative va fi egal cu 1. Analog, suma frecvenelor procentuale va fi egal cu 100.
Frecvena cumulat
Plecnd de la valoarea cea mai mic (prima din tabelulul ordonat) se adun succesiv frecvenele fiecrei valori (sau clase); prin urmare, pentru fiecare valoare se consider nu numai frecvena sa proprie, ci suma acestei frecvene cu a tuturor valorilor inferioare mrime numit frecven cumulat. Distribuia de frecvene corespunztoare se va numi distribuia frecvenelor cumulate (tabelul 4).
Numr de biei xNumr de
familii FNumr cumulat
de biei xcNumr cumulat
de familii Fc.
021021
11110 1 132
22870 2 419
34800 3 899
45290 41428
53040 51732
61260 61858
7190 71877
Total: 1877
Tabelul 4: Frecvene cumulate pentru numrul de biei n familii cu 7 copii
Diagrama frecvenelor
Este interesant s nlocuim tabloul cifrelor unei distribuii de frecvene cu o reprezentare grafic, care d distribuiei de frecven o imagine mai gritoare, permind a face s apar cu uurin alura general a caracteristicilor eseniale, adic diagrama frecvenelor.
Modul de reprezentare cel mai des utilizat este histograma: fiecare valoare (clas) este figurat printr-un dreptunghi a crui baz corespunde valorii (sau intervalului de clas) reprezentat pe axa absciselor i a crui nlime este proporional cu efectivul (numrul, procentajul) acestei valori (clase), fiind reprezentat pe ordonat.
Figura 2: Histograma numrului de biei n familii cu 7 copii
Se obine astfel o mulime de dreptunghiuri cu aceeai lime i a cror nlime i suprafa sunt respectiv proporionale cu efectivele fiecrei valori (sau clase).
Se poate construi identic diagrama frecvenelor, notnd pe ordonate nu frecvenele absolute, ci pe cele relative, . n acest caz suprafaa histogramei va fi egal cu suma frecvenelor relative, deci va fi egal cu 1. Acest mod de lucru nu modific aspectul histogramei, cu condiia de a alege pentru scara frecvenelor relative uniti mai mari. Axa ordonatelor este atunci gradat n procente n raport cu numrul total de cazuri.
Similar, reprezentarea grafic a frecvenelor cumulate duce la obinerea unei diagrame numit diagrama cumulat a datelor , cu un aspect grafic caracteristic.
Figura 3. Exemplu de diagram cumulat
Principalele tipuri de diagram de frecvene
Diagramele de frecvene folosite pentru observaiile din biologie i medicin pot acoperi aspecte foarte variate. Totodat, se pot reduce adesea la tipuri bine-definite, evocnd cu certitudine distribuiile teoretice ale calculului probabilitilor. Exist urmtoarele tipuri generale de diagrame:
1. Diagrame simetrice: frecvenele diferitelor clase se grupeaz simetric descresctor de o parte i de alta a unei frecvene centrale maximale (figura 4).
Figura 4. Distribuia a 8879 proteine marcate cu iod
Acest tip de diagram corespunde cazului unei bine-cunoscute distribuii teoretice din calculul probabilitilor, numit distribuia normal (gaussian), care joac un rol fundamental n statistic. Ea se ntlnete efectiv foarte adesea n biologie, cnd eantionul studiat aparine unei populaii normal distribuite.
2. Diagrame asimetrice: corespund cazului unei frecvene maximale n jurul creia se grupeaz i frecvenele diferitelor clase. Frecvenele descresc mai rapid fa de cea maxim ntr-o parte a diagramei, comparativ cu cealalt parte (figura 5).
Figura 5. Distribuia diametrelor a 100 cochilii de cepaea memoralis3. Diagrame hiperbolice: sunt un caz particular al distribuiei asimetrice, unde frecvena maximal se situeaz la una dintre extremitile distribuiei (figura 6).
Figura 6. Distribuia deceselor prin scarlatin (Anglia 1933)
Uneori, distribuia asimetric este mascat prin adoptarea unui interval de clas mai mare.
4. Distribuia bimodal: prezint dou frecvene maximale corespunztoare diferitelor valori ale variabilei (figura 7). Acest aspect sugereaz existena, ntr-un eantion studiat, a dou populaii distincte.
Figura 7. Vrsta de apariie a gimcomastiei la 98 subieci
Poligoane de frecvenSe unesc mijloacele marginilor superioare ale fiecrui dreptunghi al histogramei reprezentative a unei serii de frecvene. Se obine o linie frnt, numit poligon de frecvene al seriei corespunztoare, care indic cum variaz frecvena de-a lungul mulimii valorilor seriei (figura 8). Dup construirea poligonului de frecvene, se vede c fiecare dintre colurile amputate sunt compensate cu triunghiuri adiacente, astfel nct suprafaa nglobat n poligonul de frecven rmne aceeai (fiind echivalent cu suprafaa histogramei).
Observaie: Poligonul de frecven are aceeai semnificaie cu marginea superioar a histogramei.
Figura 8. Poligon de frecvene
Limite reale
Msuri limit
Puncte mediane: 42 47 52 57
39.5 44.5 49.5 54.5 59.5
40 44 45 49 50 54 55 59
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Valorile diametrului
Frecvena absolut
Diagrama asimetric
Sheet1
Boys number
x
F
f=F/n
100*fFamilies numberFrequencyPercentage
0220.011
11120.066
22850.1515
34700.2525
45130.2727
53200.1717
61320.077
7230.011
Total18771100
220.01
1120.06
2850.15
4700.25
5130.27
3200.17
1320.07
230.01
40-4450.055
45-49120.1212
50-54310.3131
55-59310.3131
60-64160.1616
65-6930.033
70-7420.022
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3542
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4522
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5523
6021
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7034
7537
8032
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3512
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5480
6230
7230
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10230
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18105
19105
20105
2195
2295
2395
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103
2027
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4010
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6014
7017
806
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1010
2121
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r/n0.3
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0.31.67193339170.83596669590.32.79070530431.39535265210.6976763261
0.41.73148717230.86574358610.42.10916969621.05458484810.527292424
0.51.81118382130.90559191060.51.47151776470.73575888230.3678794412
0.61.91358926780.95679463390.60.94771103470.47385551740.2369277587
0.72.04210405611.02105202810.70.56343368370.28171684180.1408584209
0.82.20115159661.10057579830.80.30921896180.15460948090.0773047404
0.92.39643130961.19821565480.90.15665558040.07832779020.0391638951
12.63525691031.317628455110.07326255560.03663127780.0183156389
Sheet1
00
00
00
00
00
00
00
00
Boys Number
Absolute Frequency
Percentage Frequency
Sheet2
00
00
00
00
00
00
00
Classes of Weight
Absolute Frequency
Percentage Frequency
Sheet3
0
0
0
0
0
0
0
0
0
0
0
0
0
Absolute Frequency
Symmetrical Diagrame
0
0
0
0
0
0
0
0
0
Diameter value
Absolute frequency
Asymmetrical Diagrame
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
The scarlet fever death (England 1933)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
0
0
0
0
0
0
0
0
Age
F - absolute frequency
The age of "gimcomastite" appearance
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequency
0
0
0
0
0
0
0
0
0
0
0
0
0
Cumulate Diagram
0
0
0
0
0
0
Probability
00
00
00
00
00
00
00
00
00
&A
Page &P
Probability
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequencies
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
0
0
0
0
Cumulative probabilities
p1+p2+p3+p4
p1+p2+p3
p1+p2
p1
0
0
0
0
probabilities
p4
p3
p2
p1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
y=f(x)
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
x2
Y2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
x
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Success number
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0
0
0
0
0
0
0
P
Boys number
The Percent Distribution
0
0
0
0
0
0
0
Number of families
Boys number
The Percent Distribution
0
0
0
0
0
0
0
0
0
0
0
0
P
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
Pr y
Pr(y)
r(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
0
x=x-m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
t
-1
+1
-s
+s
x
0
0
m-s
m+s
m
X
Y
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
s=0.5
s=0.25
s=1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
P
y
x
m
0
X-m
-
+
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
y=f(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
-
+
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
Y2
x2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Yt
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
-
+
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t1
-
+
F(t1)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
2F(t1)
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
P(t1)
1-P(t1)
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t2
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
t
x
-2.6 -2 -1 0 1 2 2.6
m-2.6s m -2s m-1s m m-1s m-2s m-2.6s
68.30%
95.50%
99%
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-s
+s
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-2.6s
+2.6s
Y
_983091052.xlsChart7
3
27
14
10
10
14
17
6
Varsta
Frecventa absoluta
Diagrama bimodala
Sheet1
Boys number
x
F
f=F/n
100*fFamilies numberFrequencyPercentage
0220.011
11120.066
22850.1515
34700.2525
45130.2727
53200.1717
61320.077
7230.011
Total18771100
220.01
1120.06
2850.15
4700.25
5130.27
3200.17
1320.07
230.01
40-4450.055
45-49120.1212
50-54310.3131
55-59310.3131
60-64160.1616
65-6930.033
70-7420.022
00
12
27
310
414
523
627
722
813
911
107
113
122
194
206
2122
2226
233411
2411
255
263
271
5330
10302
15286
20195
2540
3039
3542
4023
4522
5024
5523
6021
6520
7035
6536
7034
7537
8032
190
2450
3512
4490
5480
6230
7230
8230
9230
10230
11180
12180
13180
14180
15180
16105
17105
18105
19105
20105
2195
2295
2395
2495
2595
103
2027
3014
4010
5010
6014
7017
806
00
55
1010
2121
1919
1717
1414
1212
00
000
122
279
31019
41433
52356
62783
722105
813118
911129
107136
113139
122141
0.3
0.7
0.5
0
0
0.4
0.070.07
0.180.18
0.30.3
0.40.4
0.50.5
0.60.6
0.680.68
0.750.75
0.820.82
0.070.07
0.180.18
0.30.3
0.40.4
0.50.5
0.60.6
0.680.68
0.750.75
0.820.82
0.70.7
0.620.62
0.520.52
0.450.45
0.380.38
0.30.3
0.20.2
0.020.02
0.070.0630.07
0.180.1620.18
0.30.270.3
0.40.360.4
0.50.450.5
0.60.540.6
0.680.6120.68
0.750.6750.75
0.820.750.82
0.70.630.7
0.620.5580.62
0.520.4680.52
0.450.4050.45
0.380.3420.38
0.30.270.3
0.20.180.2
0.020.0180.02
p10.2x1
p1+p20.3x2
p1+p2+p30.5x3
p1+p2+p3+p40.8x4
x10.2
x20.1
x30.2
x40.3
-10.01831563890.01831563890.01831563890.0183156389
-0.90.03916389510.03916389510.0574795340.057479534
-0.80.07730474040.07730474040.13478427440.1347842744
-0.70.14085842090.14085842090.27564269540.2756426954
-0.60.23692775870.23692775870.5125704540.512570454
-0.50.36787944120.36787944120.88044989520.8804498952
-0.40.5272924240.5272924241.40774231921.4077423192
-0.30.69767632610.69767632612.10541864532.1054186453
-0.20.8521437890.8521437892.95756243432.9575624343
-0.10.96078943920.96078943923.91835187343.9183518734
-0114.91835187344.9183518734
0.10.96078943920.96078943925.87914131265.8791413126
0.20.8521437890.8521437896.73128510166.7312851016
0.30.69767632610.69767632617.42896142767.4289614276
0.40.5272924240.5272924247.95625385177.9562538517
0.50.36787944120.36787944128.32413329288.3241332928
0.60.23692775870.23692775878.56106105158.5610610515
0.70.14085842090.14085842098.70191947248.7019194724
0.80.07730474040.07730474048.77922421298.7792242129
0.90.03916389510.03916389518.8183881088.818388108
10.01831563890.01831563898.83670374698.8367037469
-10.01831563890.0183156389
-0.90.0574795340.057479534
-0.80.13478427440.1347842744
-0.70.27564269540.2756426954
-0.60.5125704540.512570454
-0.50.88044989520.8804498952
-0.41.40774231921.4077423192
-0.32.10541864532.1054186453
-0.22.95756243432.9575624343
-0.13.91835187343.9183518734
-04.91835187344.9183518734
0.15.87914131265.8791413126
0.26.73128510166.7312851016
0.37.42896142767.4289614276
0.47.95625385177.9562538517
0.58.32413329288.3241332928
0.68.56106105158.5610610515
0.78.70191947248.7019194724
0.88.77922421298.7792242129
0.98.8183881088.818388108
18.83670374698.8367037469
-10.0183156389
-0.90.057479534
-0.80.1347842744
-0.70.2756426954
-0.60.512570454
-0.50.8804498952
-0.41.4077423192
-0.32.1054186453
-0.22.9575624343
-0.13.9183518734
04.9183518734
0.15.8791413126
0.26.7312851016
0.37.4289614276
0.47.9562538517
0.58.3241332928
0.68.5610610515
-10.01831563890.0183156389
-0.90.03916389510.0391638951
-0.80.07730474040.0773047404
-0.70.14085842090.1408584209
-0.60.23692775870.2369277587
-0.50.36787944120.3678794412
-0.40.5272924240.527292424
-0.30.69767632610.6976763261
-0.20.8521437890.852143789
-0.10.96078943920.9607894392
011
0.10.96078943920.9607894392
0.20.8521437890.852143789
0.30.69767632610.6976763261
0.40.5272924240
0.50.36787944120
0.60.23692775870
0.70.14085842090
0.80.07730474040
0.90.03916389510
10.01831563890
0.852143789
0.9607894392
1
0.9607894392
0.852143789
0.6976763261
00.2
10.4
20.7
30.6
40.4
50.5
.
.
.
.
.
r0.4
.
.
n0.1
-10.0183156389
-0.90.0391638951
-0.80.0773047404
-0.70.1408584209
-0.60.2369277587
-0.50.3678794412
-0.40.527292424
-0.30.6976763261
-0.20.852143789
-0.10.9607894392
01
0.30.6976763261
0.40.527292424
0.80.0773047404
0.90.0391638951
10.0183156389
-0.80.0773047404
-0.40.527292424
-0.20.852143789
01
0.10.9607894392
0.20.852143789
0.30.6976763261
0.40.527292424
0.50.3678794412
0.60.2369277587
0.70.1408584209
0.80.0773047404
0.90.0391638951
10.0183156389
0.8521437890.5272924240.36787944120.0183156389
0.96078943920.69767632610.5272924240.0391638951
10.8521437890.69767632610.0773047404
0.96078943920.96078943920.8521437890.1408584209
0.85214378910.96078943920.2369277587
0.69767632610.960789439210.3678794412
0.23692775870.5272924240.96078943920.527292424
0.03916389510.36787944120.69767632610.6976763261
0.01831563890.23692775870.5272924240.852143789
0.03916389510.36787944120.9607894392
0.01831563890.23692775871
0.07730474040.9607894392
40.03916389510.852143789
0.01831563890.6976763261
70.527292424
120.3678794412
0.2369277587
0.1408584209
0.0773047404
0.0391638951
200.0183156389
00.016
10.094
20.234
30.312
40.234
50.094
60.016
32
188
468
624
468
188
32
0/n0.1
1/n0.2
2/n0.3
3/n0.4
.
.
.
r/n0.3
.
.
.
n/n0.2
-12.63525691031.3176284551-10.07326255560.03663127780.0183156389
-0.92.39643130961.1982156548-0.90.15665558040.07832779020.0391638951
-0.82.20115159661.1005757983-0.80.30921896180.15460948090.0773047404
-0.72.04210405611.0210520281-0.70.56343368370.28171684180.1408584209
-0.61.91358926780.9567946339-0.60.94771103470.47385551740.2369277587
-0.51.81118382130.9055919106-0.51.47151776470.73575888230.3678794412
-0.41.73148717230.8657435861-0.42.10916969621.05458484810.527292424
-0.31.67193339170.8359666959-0.32.79070530431.39535265210.6976763261
-0.21.63065320920.8153266046-0.23.40857515591.70428757790.852143789
-0.11.60637594570.8031879729-0.13.84315775661.92157887830.9607894392
01.59836411230.79918205610421
0.11.60637594570.80318797290.13.84315775661.92157887830.9607894392
0.21.63065320920.81532660460.23.40857515591.70428757790.852143789
0.31.67193339170.83596669590.32.79070530431.39535265210.6976763261
0.41.73148717230.86574358610.42.10916969621.05458484810.527292424
0.51.81118382130.90559191060.51.47151776470.73575888230.3678794412
0.61.91358926780.95679463390.60.94771103470.47385551740.2369277587
0.72.04210405611.02105202810.70.56343368370.28171684180.1408584209
0.82.20115159661.10057579830.80.30921896180.15460948090.0773047404
0.92.39643130961.19821565480.90.15665558040.07832779020.0391638951
12.63525691031.317628455110.07326255560.03663127780.0183156389
Sheet1
00
00
00
00
00
00
00
00
Boys Number
Absolute Frequency
Percentage Frequency
Sheet2
00
00
00
00
00
00
00
Classes of Weight
Absolute Frequency
Percentage Frequency
Sheet3
0
0
0
0
0
0
0
0
0
0
0
0
0
Absolute Frequency
Symmetrical Diagrame
0
0
0
0
0
0
0
0
0
Diameter value
Absolute frequency
Asymmetrical Diagrame
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
Hyperbolic Diagrame
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
0
0
0
0
0
0
0
0
Age
F - absolute frequency
Bimodal Diagrame
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequency
0
0
0
0
0
0
0
0
0
0
0
0
0
Cumulate Diagram
0
0
0
0
0
0
Probability
00
00
00
00
00
00
00
00
00
&A
Page &P
Probability
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequencies
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
0
0
0
0
Cumulative probabilities
p1+p2+p3+p4
p1+p2+p3
p1+p2
p1
0
0
0
0
probabilities
p4
p3
p2
p1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
y=f(x)
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
x2
Y2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
x
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Success number
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0
0
0
0
0
0
0
P
Boys number
The Percent Distribution
0
0
0
0
0
0
0
Number of families
Boys number
The Percent Distribution
0
0
0
0
0
0
0
0
0
0
0
0
P
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
Pr y
Pr(y)
r(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
0
x=x-m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
t
-1
+1
-s
+s
x
0
0
m-s
m+s
m
X
Y
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
s=0.5
s=0.25
s=1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
P
y
x
m
0
X-m
-
+
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
y=f(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
-
+
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
Y2
x2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Yt
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
-
+
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t1
-
+
F(t1)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
2F(t1)
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
P(t1)
1-P(t1)
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t2
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
t
x
-2.6 -2 -1 0 1 2 2.6
m-2.6s m -2s m-1s m m-1s m-2s m-2.6s
68.30%
95.50%
99%
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-s
+s
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-2.6s
+2.6s
Y
_983177235.xlsChart9
0
2
9
19
33
56
83
105
118
129
136
139
141
Diagrama cumulat
Sheet1
Boys number
x
F
f=F/n
100*fFamilies numberFrequencyPercentage
0220.011
11120.066
22850.1515
34700.2525
45130.2727
53200.1717
61320.077
7230.011
Total18771100
220.01
1120.06
2850.15
4700.25
5130.27
3200.17
1320.07
230.01
40-4450.055
45-49120.1212
50-54310.3131
55-59310.3131
60-64160.1616
65-6930.033
70-7420.022
00
12
27
310
414
523
627
722
813
911
107
113
122
194
206
2122
2226
233411
2411
255
263
271
5330
10302
15286
20195
2540
3039
3542
4023
4522
5024
5523
6021
6520
7035
6536
7034
7537
8032
190
2450
3512
4490
5480
6230
7230
8230
9230
10230
11180
12180
13180
14180
15180
16105
17105
18105
19105
20105
2195
2295
2395
2495
2595
103
2027
3014
4010
5010
6014
7017
806
00
55
1010
2121
1919
1717
1414
1212
00
000
122
279
31019
41433
52356
62783
722105
813118
911129
107136
113139
122141
0.3
0.7
0.5
0
0
0.4
0.070.07
0.180.18
0.30.3
0.40.4
0.50.5
0.60.6
0.680.68
0.750.75
0.820.82
0.070.07
0.180.18
0.30.3
0.40.4
0.50.5
0.60.6
0.680.68
0.750.75
0.820.82
0.70.7
0.620.62
0.520.52
0.450.45
0.380.38
0.30.3
0.20.2
0.020.02
0.070.0630.07
0.180.1620.18
0.30.270.3
0.40.360.4
0.50.450.5
0.60.540.6
0.680.6120.68
0.750.6750.75
0.820.750.82
0.70.630.7
0.620.5580.62
0.520.4680.52
0.450.4050.45
0.380.3420.38
0.30.270.3
0.20.180.2
0.020.0180.02
p10.2x1
p1+p20.3x2
p1+p2+p30.5x3
p1+p2+p3+p40.8x4
x10.2
x20.1
x30.2
x40.3
-10.01831563890.01831563890.01831563890.0183156389
-0.90.03916389510.03916389510.0574795340.057479534
-0.80.07730474040.07730474040.13478427440.1347842744
-0.70.14085842090.14085842090.27564269540.2756426954
-0.60.23692775870.23692775870.5125704540.512570454
-0.50.36787944120.36787944120.88044989520.8804498952
-0.40.5272924240.5272924241.40774231921.4077423192
-0.30.69767632610.69767632612.10541864532.1054186453
-0.20.8521437890.8521437892.95756243432.9575624343
-0.10.96078943920.96078943923.91835187343.9183518734
-0114.91835187344.9183518734
0.10.96078943920.96078943925.87914131265.8791413126
0.20.8521437890.8521437896.73128510166.7312851016
0.30.69767632610.69767632617.42896142767.4289614276
0.40.5272924240.5272924247.95625385177.9562538517
0.50.36787944120.36787944128.32413329288.3241332928
0.60.23692775870.23692775878.56106105158.5610610515
0.70.14085842090.14085842098.70191947248.7019194724
0.80.07730474040.07730474048.77922421298.7792242129
0.90.03916389510.03916389518.8183881088.818388108
10.01831563890.01831563898.83670374698.8367037469
-10.01831563890.0183156389
-0.90.0574795340.057479534
-0.80.13478427440.1347842744
-0.70.27564269540.2756426954
-0.60.5125704540.512570454
-0.50.88044989520.8804498952
-0.41.40774231921.4077423192
-0.32.10541864532.1054186453
-0.22.95756243432.9575624343
-0.13.91835187343.9183518734
-04.91835187344.9183518734
0.15.87914131265.8791413126
0.26.73128510166.7312851016
0.37.42896142767.4289614276
0.47.95625385177.9562538517
0.58.32413329288.3241332928
0.68.56106105158.5610610515
0.78.70191947248.7019194724
0.88.77922421298.7792242129
0.98.8183881088.818388108
18.83670374698.8367037469
-10.0183156389
-0.90.057479534
-0.80.1347842744
-0.70.2756426954
-0.60.512570454
-0.50.8804498952
-0.41.4077423192
-0.32.1054186453
-0.22.9575624343
-0.13.9183518734
04.9183518734
0.15.8791413126
0.26.7312851016
0.37.4289614276
0.47.9562538517
0.58.3241332928
0.68.5610610515
-10.01831563890.0183156389
-0.90.03916389510.0391638951
-0.80.07730474040.0773047404
-0.70.14085842090.1408584209
-0.60.23692775870.2369277587
-0.50.36787944120.3678794412
-0.40.5272924240.527292424
-0.30.69767632610.6976763261
-0.20.8521437890.852143789
-0.10.96078943920.9607894392
011
0.10.96078943920.9607894392
0.20.8521437890.852143789
0.30.69767632610.6976763261
0.40.5272924240
0.50.36787944120
0.60.23692775870
0.70.14085842090
0.80.07730474040
0.90.03916389510
10.01831563890
0.852143789
0.9607894392
1
0.9607894392
0.852143789
0.6976763261
00.2
10.4
20.7
30.6
40.4
50.5
.
.
.
.
.
r0.4
.
.
n0.1
-10.0183156389
-0.90.0391638951
-0.80.0773047404
-0.70.1408584209
-0.60.2369277587
-0.50.3678794412
-0.40.527292424
-0.30.6976763261
-0.20.852143789
-0.10.9607894392
01
0.30.6976763261
0.40.527292424
0.80.0773047404
0.90.0391638951
10.0183156389
-0.80.0773047404
-0.40.527292424
-0.20.852143789
01
0.10.9607894392
0.20.852143789
0.30.6976763261
0.40.527292424
0.50.3678794412
0.60.2369277587
0.70.1408584209
0.80.0773047404
0.90.0391638951
10.0183156389
0.8521437890.5272924240.36787944120.0183156389
0.96078943920.69767632610.5272924240.0391638951
10.8521437890.69767632610.0773047404
0.96078943920.96078943920.8521437890.1408584209
0.85214378910.96078943920.2369277587
0.69767632610.960789439210.3678794412
0.23692775870.5272924240.96078943920.527292424
0.03916389510.36787944120.69767632610.6976763261
0.01831563890.23692775870.5272924240.852143789
0.03916389510.36787944120.9607894392
0.01831563890.23692775871
0.07730474040.9607894392
40.03916389510.852143789
0.01831563890.6976763261
70.527292424
120.3678794412
0.2369277587
0.1408584209
0.0773047404
0.0391638951
200.0183156389
00.016
10.094
20.234
30.312
40.234
50.094
60.016
32
188
468
624
468
188
32
0/n0.1
1/n0.2
2/n0.3
3/n0.4
.
.
.
r/n0.3
.
.
.
n/n0.2
-12.63525691031.3176284551-10.07326255560.03663127780.0183156389
-0.92.39643130961.1982156548-0.90.15665558040.07832779020.0391638951
-0.82.20115159661.1005757983-0.80.30921896180.15460948090.0773047404
-0.72.04210405611.0210520281-0.70.56343368370.28171684180.1408584209
-0.61.91358926780.9567946339-0.60.94771103470.47385551740.2369277587
-0.51.81118382130.9055919106-0.51.47151776470.73575888230.3678794412
-0.41.73148717230.8657435861-0.42.10916969621.05458484810.527292424
-0.31.67193339170.8359666959-0.32.79070530431.39535265210.6976763261
-0.21.63065320920.8153266046-0.23.40857515591.70428757790.852143789
-0.11.60637594570.8031879729-0.13.84315775661.92157887830.9607894392
01.59836411230.79918205610421
0.11.60637594570.80318797290.13.84315775661.92157887830.9607894392
0.21.63065320920.81532660460.23.40857515591.70428757790.852143789
0.31.67193339170.83596669590.32.79070530431.39535265210.6976763261
0.41.73148717230.86574358610.42.10916969621.05458484810.527292424
0.51.81118382130.90559191060.51.47151776470.73575888230.3678794412
0.61.91358926780.95679463390.60.94771103470.47385551740.2369277587
0.72.04210405611.02105202810.70.56343368370.28171684180.1408584209
0.82.20115159661.10057579830.80.30921896180.15460948090.0773047404
0.92.39643130961.19821565480.90.15665558040.07832779020.0391638951
12.63525691031.317628455110.07326255560.03663127780.0183156389
Sheet1
00
00
00
00
00
00
00
00
Boys Number
Absolute Frequency
Percentage Frequency
Sheet2
00
00
00
00
00
00
00
Classes of Weight
Absolute Frequency
Percentage Frequency
Sheet3
0
0
0
0
0
0
0
0
0
0
0
0
0
Absolute Frequency
Symmetrical Diagrame
0
0
0
0
0
0
0
0
0
Diameter value
Absolute frequency
Asymmetrical Diagrame
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
Hyperbolic Diagrame
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
0
0
0
0
0
0
0
0
Age
F - absolute frequency
Bimodal Diagrame
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequency
0
0
0
0
0
0
0
0
0
0
0
0
0
Cumulate Diagrame
0
0
0
0
0
0
Probability
00
00
00
00
00
00
00
00
00
&A
Page &P
Probability
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequencies
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
0
0
0
0
Cumulative probabilities
p1+p2+p3+p4
p1+p2+p3
p1+p2
p1
0
0
0
0
probabilities
p4
p3
p2
p1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
y=f(x)
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
x2
Y2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
x
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Success number
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0
0
0
0
0
0
0
P
Boys number
The Percent Distribution
0
0
0
0
0
0
0
Number of families
Boys number
The Percent Distribution
0
0
0
0
0
0
0
0
0
0
0
0
P
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
Pr y
Pr(y)
r(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
0
x=x-m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
t
-1
+1
-s
+s
x
0
0
m-s
m+s
m
X
Y
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
s=0.5
s=0.25
s=1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
P
y
x
m
0
X-m
-
+
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
y=f(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
-
+
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
Y2
x2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Yt
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
-
+
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t1
-
+
F(t1)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
2F(t1)
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
P(t1)
1-P(t1)
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t2
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
t
x
-2.6 -2 -1 0 1 2 2.6
m-2.6s m -2s m-1s m m-1s m-2s m-2.6s
68.30%
95.50%
99%
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-s
+s
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-2.6s
+2.6s
Y
_1225696267.xlsChart1
220.01
1120.06
2850.15
4700.25
5130.27
3200.17
1320.07
230.01
Numr de biei
Frecven absolut
Frecven procentual
Sheet1
Boys number
x
F
f=F/n
100*fFamilies numberFrequencyPercentage
0220.011
11120.066
22850.1515
34700.2525
45130.2727
53200.1717
61320.077
7230.011
Total18771100
220.01
1120.06
2850.15
4700.25
5130.27
3200.17
1320.07
230.01
40-4450.055
45-49120.1212
50-54310.3131
55-59310.3131
60-64160.1616
65-6930.033
70-7420.022
00
12
27
310
414
523
627
722
813
911
107
113
122
194
206
2122
2226
233411
2411
255
263
271
5330
10302
15286
20195
2540
3039
3542
4023
4522
5024
5523
6021
6520
7035
6536
7034
7537
8032
190
2450
3512
4490
5480
6230
7230
8230
9230
10230
11180
12180
13180
14180
15180
16105
17105
18105
19105
20105
2195
2295
2395
2495
2595
103
2027
3014
4010
5010
6014
7017
806
00
55
1010
2121
1919
1717
1414
1212
00
000
122
279
31019
41433
52356
62783
722105
813118
911129
107136
113139
122141
0.3
0.7
0.5
0
0
0.4
0.070.07
0.180.18
0.30.3
0.40.4
0.50.5
0.60.6
0.680.68
0.750.75
0.820.82
0.070.07
0.180.18
0.30.3
0.40.4
0.50.5
0.60.6
0.680.68
0.750.75
0.820.82
0.70.7
0.620.62
0.520.52
0.450.45
0.380.38
0.30.3
0.20.2
0.020.02
0.070.0630.07
0.180.1620.18
0.30.270.3
0.40.360.4
0.50.450.5
0.60.540.6
0.680.6120.68
0.750.6750.75
0.820.750.82
0.70.630.7
0.620.5580.62
0.520.4680.52
0.450.4050.45
0.380.3420.38
0.30.270.3
0.20.180.2
0.020.0180.02
p10.2x1
p1+p20.3x2
p1+p2+p30.5x3
p1+p2+p3+p40.8x4
x10.2
x20.1
x30.2
x40.3
-10.01831563890.01831563890.01831563890.0183156389
-0.90.03916389510.03916389510.0574795340.057479534
-0.80.07730474040.07730474040.13478427440.1347842744
-0.70.14085842090.14085842090.27564269540.2756426954
-0.60.23692775870.23692775870.5125704540.512570454
-0.50.36787944120.36787944120.88044989520.8804498952
-0.40.5272924240.5272924241.40774231921.4077423192
-0.30.69767632610.69767632612.10541864532.1054186453
-0.20.8521437890.8521437892.95756243432.9575624343
-0.10.96078943920.96078943923.91835187343.9183518734
-0114.91835187344.9183518734
0.10.96078943920.96078943925.87914131265.8791413126
0.20.8521437890.8521437896.73128510166.7312851016
0.30.69767632610.69767632617.42896142767.4289614276
0.40.5272924240.5272924247.95625385177.9562538517
0.50.36787944120.36787944128.32413329288.3241332928
0.60.23692775870.23692775878.56106105158.5610610515
0.70.14085842090.14085842098.70191947248.7019194724
0.80.07730474040.07730474048.77922421298.7792242129
0.90.03916389510.03916389518.8183881088.818388108
10.01831563890.01831563898.83670374698.8367037469
-10.01831563890.0183156389
-0.90.0574795340.057479534
-0.80.13478427440.1347842744
-0.70.27564269540.2756426954
-0.60.5125704540.512570454
-0.50.88044989520.8804498952
-0.41.40774231921.4077423192
-0.32.10541864532.1054186453
-0.22.95756243432.9575624343
-0.13.91835187343.9183518734
-04.91835187344.9183518734
0.15.87914131265.8791413126
0.26.73128510166.7312851016
0.37.42896142767.4289614276
0.47.95625385177.9562538517
0.58.32413329288.3241332928
0.68.56106105158.5610610515
0.78.70191947248.7019194724
0.88.77922421298.7792242129
0.98.8183881088.818388108
18.83670374698.8367037469
-10.0183156389
-0.90.057479534
-0.80.1347842744
-0.70.2756426954
-0.60.512570454
-0.50.8804498952
-0.41.4077423192
-0.32.1054186453
-0.22.9575624343
-0.13.9183518734
04.9183518734
0.15.8791413126
0.26.7312851016
0.37.4289614276
0.47.9562538517
0.58.3241332928
0.68.5610610515
-10.01831563890.0183156389
-0.90.03916389510.0391638951
-0.80.07730474040.0773047404
-0.70.14085842090.1408584209
-0.60.23692775870.2369277587
-0.50.36787944120.3678794412
-0.40.5272924240.527292424
-0.30.69767632610.6976763261
-0.20.8521437890.852143789
-0.10.96078943920.9607894392
011
0.10.96078943920.9607894392
0.20.8521437890.852143789
0.30.69767632610.6976763261
0.40.5272924240
0.50.36787944120
0.60.23692775870
0.70.14085842090
0.80.07730474040
0.90.03916389510
10.01831563890
0.852143789
0.9607894392
1
0.9607894392
0.852143789
0.6976763261
00.2
10.4
20.7
30.6
40.4
50.5
.
.
.
.
.
r0.4
.
.
n0.1
-10.0183156389
-0.90.0391638951
-0.80.0773047404
-0.70.1408584209
-0.60.2369277587
-0.50.3678794412
-0.40.527292424
-0.30.6976763261
-0.20.852143789
-0.10.9607894392
01
0.30.6976763261
0.40.527292424
0.80.0773047404
0.90.0391638951
10.0183156389
-0.80.0773047404
-0.40.527292424
-0.20.852143789
01
0.10.9607894392
0.20.852143789
0.30.6976763261
0.40.527292424
0.50.3678794412
0.60.2369277587
0.70.1408584209
0.80.0773047404
0.90.0391638951
10.0183156389
0.8521437890.5272924240.36787944120.0183156389
0.96078943920.69767632610.5272924240.0391638951
10.8521437890.69767632610.0773047404
0.96078943920.96078943920.8521437890.1408584209
0.85214378910.96078943920.2369277587
0.69767632610.960789439210.3678794412
0.23692775870.5272924240.96078943920.527292424
0.03916389510.36787944120.69767632610.6976763261
0.01831563890.23692775870.5272924240.852143789
0.03916389510.36787944120.9607894392
0.01831563890.23692775871
0.07730474040.9607894392
40.03916389510.852143789
0.01831563890.6976763261
70.527292424
120.3678794412
0.2369277587
0.1408584209
0.0773047404
0.0391638951
200.0183156389
00.016
10.094
20.234
30.312
40.234
50.094
60.016
32
188
468
624
468
188
32
0/n0.1
1/n0.2
2/n0.3
3/n0.4
.
.
.
r/n0.3
.
.
.
n/n0.2
-12.63525691031.3176284551-10.07326255560.03663127780.0183156389
-0.92.39643130961.1982156548-0.90.15665558040.07832779020.0391638951
-0.82.20115159661.1005757983-0.80.30921896180.15460948090.0773047404
-0.72.04210405611.0210520281-0.70.56343368370.28171684180.1408584209
-0.61.91358926780.9567946339-0.60.94771103470.47385551740.2369277587
-0.51.81118382130.9055919106-0.51.47151776470.73575888230.3678794412
-0.41.73148717230.8657435861-0.42.10916969621.05458484810.527292424
-0.31.67193339170.8359666959-0.32.79070530431.39535265210.6976763261
-0.21.63065320920.8153266046-0.23.40857515591.70428757790.852143789
-0.11.60637594570.8031879729-0.13.84315775661.92157887830.9607894392
01.59836411230.79918205610421
0.11.60637594570.80318797290.13.84315775661.92157887830.9607894392
0.21.63065320920.81532660460.23.40857515591.70428757790.852143789
0.31.67193339170.83596669590.32.79070530431.39535265210.6976763261
0.41.73148717230.86574358610.42.10916969621.05458484810.527292424
0.51.81118382130.90559191060.51.47151776470.73575888230.3678794412
0.61.91358926780.95679463390.60.94771103470.47385551740.2369277587
0.72.04210405611.02105202810.70.56343368370.28171684180.1408584209
0.82.20115159661.10057579830.80.30921896180.15460948090.0773047404
0.92.39643130961.19821565480.90.15665558040.07832779020.0391638951
12.63525691031.317628455110.07326255560.03663127780.0183156389
Sheet1
00
00
00
00
00
00
00
00
Boys Number
Absolute Frequency
Percentage Frequency
Sheet2
00
00
00
00
00
00
00
Classes of Weight
Absolute Frequency
Percentage Frequency
Sheet3
0
0
0
0
0
0
0
0
0
0
0
0
0
Absolute Frequency
Symmetric Diagrame
0
0
0
0
0
0
0
0
0
Diameter value
Absolute frequency
The diameter distribution of "memoralis cepaea"
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
The scarlet fever death (England 1933)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Age
Absolute Frequency
0
0
0
0
0
0
0
0
Age
F - absolute frequency
The age of "gimcomastite" appearance
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequency
0
0
0
0
0
0
0
0
0
0
0
0
0
Cumulate Diagram
0
0
0
0
0
0
Probability
00
00
00
00
00
00
00
00
00
&A
Page &P
Probability
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
Frequencies
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
0
0
0
0
Cumulative probabilities
p1+p2+p3+p4
p1+p2+p3
p1+p2
p1
0
0
0
0
probabilities
p4
p3
p2
p1
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
y=f(x)
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
&A
Page &P
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
x2
Y2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
x
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Success number
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0
0
0
0
0
0
0
P
Boys number
The Percent Distribution
0
0
0
0
0
0
0
Number of families
Boys number
The Percent Distribution
0
0
0
0
0
0
0
0
0
0
0
0
P
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
Pr y
Pr(y)
r(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
y
y
x
0
x=x-m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
t
-1
+1
-s
+s
x
0
0
m-s
m+s
m
X
Y
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
s=0.5
s=0.25
s=1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x
P
y
x
m
0
X-m
-
+
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
x
y=f(x)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
X
Y
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
y=f(x)
X1
-
+
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Y=F(x)
Y1
x1
-
Y2
x2
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Yt
-
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
X1
X2
-
+
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t1
-
+
F(t1)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
2F(t1)
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
P(t1)
1-P(t1)
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t1
t2
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
t2
t1
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y=f(x)
+
-
t
x
-2.6 -2 -1 0 1 2 2.6
m-2.6s m -2s m-1s m m-1s m-2s m-2.6s
68.30%
95.50%
99%
X
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-s
+s
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
t
-2.6s
+2.6s
Y
_983175432.xlsChart8
00
55
1010
2121
1919
1717
1414
1212
00
Frecvena
Sheet1
Boys number
x
F
f=F/n
100*fFamilies numberFrequencyPercentage
0220.011
11120.066
22850.1515
34700.2525
45130.2727
53200.1717
61320.077
7230.011
Total18771100
220.01
1120.06
2850.15
4700.25
5130.27
3200.17
1320.07
230.01
40-4450.055
45-49120.1212
50-54310.3131
55-59310.3131
60-64160.1616
65-6930.033
70-7420.022
00
12
27
310
414
523
627
722
813
911
107
113
122
194
206
2122
2226
233411
2411
255
263
271
5330
10302
15286
20195
2540
3039
3542
4023
4522
5024
5523
6021
6520
7035
6536
7034
7537
8032
190
2450
3512
4490
5480
6230
7230
8230
9230
10230
11180
12180
13180
14180
15180
16105
17105
18105
19105
20105
2195
2295
2395
2495
2595
103
2027
3014
4010
5010
6014
7017
806