2012_2 exerc Desvio Padrao - Moda e Mediana Solucao

Prévia do material em texto

2012_2 ex 1
4 6 8 11 13 4 4
4 7 8 12 13 15
6 7 9 12 14 20 Q3 = 12,25
6 8 10 13 15 11,00
4
4,0
2,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
10,5 2 ----| 6,0 5 5 4 20 -4,8 23,0400
6 6 ----| 10,0 7 12 8 56 -0,8 0,6400
5 10 ----| 14,0 7 19 12 84 3,2 10,2400
7 14 ----| 18,0 1 20 16 16 7,2 51,8400
 = 8,8571 20 ---- ---- 176 ---- ---- 
176,00 = 8,8 (3.n/4)+1/2 =
7 20 = 11,7143
10 s2 = s = 
0 s2 = s = 
6
0 . 4 = 10,00
0 + 6
Mo = 10 + 
3,4871
(xi - xm)
2.fi
115,2000
4,4800
Elemento Central = 
Li da classe Med = 
fantac = 
12,7999
51,8400
12,1599
243,2000
Cl Modal = 
71,6800
Li Cl Mo = 
d1 = 
d2 = 
n = 
total
Amp = máx - mín = 
K = raiz(n) = 4,4721 = 
xm =
pontos = mínimo = 
h = Amp/(k - 1) = 3,6667 = 
Limite Inferior = mínimo - h/2 = 
3,5777
máximo = 
fmed = 
(3.20/4 - 12).4
7
Média = 
Md = 6 + (20/2- 5).4
7
15,500
Q3 = 10 + 
2012_2 ex 2
100 130 140 150 180 100
104 134 146 170 184 188
116 138 150 176 186 20 Q3 = 176
120 140 150 176 188 88,00
4
29,0
85,5
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
10,5 85,5 ----| 114,5 2 2 100 200 -50,75 2575,5625
143,5 114,5 ----| 143,5 7 9 129 903 -21,75 473,0625
9 143,5 ----| 172,5 5 14 158 790 7,25 52,5625
5 172,5 ----| 201,5 6 20 187 1122 36,25 1314,0625
 = 149,3 20 ---- ---- 3015 ---- ---- 
3015,00 = 150,75 (3.n/4)+1/2 =
7 20 = 178,3
114,5 s2 = s = 
5 s2 = s = 
2
5 . 29 = 135,2143
5 + 2
Mo = 114,5 + 
(20/2- 9).29
5
Cl Modal = 
Li Cl Mo = 
d1 = 
d2 = 
29,5668
Elemento Central = 
Li da classe Med = 
fantac = 
fmed = 
Md = 143,5 + 
262,8125
mínimo = 
máximo = 
n = 
Amp = máx - mín = 
K = raiz(n) = 4,4721 = 
h = Amp/(k - 1) = 29,3333 = 
Limite Inferior = mínimo - h/2 = 
7884,3750
16609,7500
xm = Média = 
total
5151,1250
3311,4375
830,4887
874,1957
(xi - xm)
2.fi
15,500
Q3 = 172,5 + 
28,8182
(3.20/4 - 14).29
6
2012_2 ex 3
1 2 2 2 3 1 4
4 4 4 4 5 9
5 5 6 6 7 20 Q3 = 7
7 8 8 9 9 8,00 D4 =
4
3,0
-0,5
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
10,5 -0,5 ----| 2,5 4 4 1 4 -3,9 15,2100
2,5 2,5 ----| 5,5 8 12 4 32 -0,9 0,8100
4 5,5 ----| 8,5 6 18 7 42 2,1 4,4100
8 8,5 ----| 11,5 2 20 10 20 5,1 26,0100
 = 4,75 20 ---- ---- 98 ---- ---- 
98,00 = 4,9 (3.n/4)+1/2 =
8 20 = 7
2,5 s2 = s = 
4 s2 = s = 
2
4 . 3 = 4,5
4 + 2
d2 = 
Mo = 2,5 + 
2,7000
2,77017,6735
Q3 = 5,5 + 
52,0200
145,8000
fmed = 
(xi - xm)
2.fi
60,8400
6,4800
26,4600
15,500
8
Elemento Central = 
Li da classe Med = 
fantac = 
Md = 2,5 + total
Média = 
7,29
xm =
(20/2- 4).3
Cl Modal = 
Li Cl Mo = 
d1 = 
Limite Inferior = mínimo - h/2 = 
Amp = máx - mín = 
K = raiz(n) = 4,4721 = 
(3.20/4 - 12).3
6
mínimo = pontos = 
máximo = 
n = 
h = Amp/(k - 1) = 2,6667 = 
2012_2 ex 4
4 6 8 11 13 4 4
4 7 9 12 13 15
6 8 10 12 14 20 Q3 = 12,25
6 8 11 13 15 11,00
4
4,0
2,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
10,5 2 ----| 6,0 5 5 4 20 -5 25,0000
6 6 ----| 10,0 6 11 8 48 -1 1,0000
5 10 ----| 14,0 8 19 12 96 3 9,0000
6 14 ----| 18,0 1 20 16 16 7 49,0000
 = 9,3333 20 ---- ---- 180 ---- ---- 
180,00 = 9 (3.n/4)+1/2 =
8 20 = 12
10 s2 = s = 
2 s2 = s = 
7
2 . 4 = 10,89
2 + 7
mínimo = pontos = 
máximo = 
n = 
Amp = máx - mín = 
K = raiz(n) = 4,4721 = 
h = Amp/(k - 1) = 3,6667 = 
Limite Inferior = mínimo - h/2 = 
(xi - xm)
2.fi
Elemento Central = 125,0000
Li da classe Med = 6,0000
fantac = 72,0000
fmed = 49,0000
Md = 6 + (20/2- 5).4 total 252,0000
6
xm = Média = 15,500
Cl Modal = 
Li Cl Mo = 12,5997 3,5496
d1 = 13,2634 3,6419
d2 = 
Mo = 10 + 
Q3 = 10 + (3.20/4 - 11).4
8
2012_2 ex 5
100 128 146 156 170 180 100 4
104 130 150 160 176 180 188
116 134 150 160 176 184 37 Q3 = 176
120 138 152 162 176 186 88,00
122 140 156 162 176 186 6
126 140 156 164 178 188 18,0
128 91,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
19,0 91 ----| 109 2 2 100 200 -52,0541 2709,6293
145 109 ----| 127 4 6 118 472 -34,0541 1159,6817
13 127 ----| 145 7 13 136 952 -16,0541 257,7341
11 145 ----| 163 11 24 154 1694 1,9459 3,7865
 = 154 163 ----| 181 9 33 172 1548 19,9459 397,8389
181 ----| 199 4 37 190 760 37,9459 1439,8913
37 ---- ---- 5626 ---- ---- 
11
145 5626,00 = 152,0541 (3.n/4)+1/2 =
4 37 Q3 = 163 + 
2 s2 = s = 
4 . 18 = s2 = s = 
4 + 2
Cl Modal = 
Li Cl Mo = 
d1 = 
d2 = 
Mo = 145 + 157,0
Li da classe Med = 
fantac = 
fmed = 
Md = 145 + (4/2- 13).91
11
574,1583 23,9616
590,1061 24,2921
Elemento Central = 
21243,8919
(xi - xm)
2.fi
5419,2587
4638,7269
1804,1389
41,6518
3580,5503
5759,5653
Amp = máx - mín = 
Limite Inf = mínimo - h/2 = 
total
xm =
K = raiz(n) = 6,0828 = 
pontos = 
Média = 
h = Amp/(k - 1) = 17,6 = 
mínimo = 
máximo = 
n = 
 = 170,5
28,250
(3.37/4 - 24).18
9
2012_2 ex 6
4 6 8 10 11 13 4 4
4 7 8 10 12 13 15
4 7 8 10 12 13 36 Q3 = 12
5 7 9 10 12 14 11,00
6 7 9 11 12 14 6
6 8 10 11 13 15 2,0
3,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
18,5 3 ----| 5 4 4 4 16 -5 25
9 5 ----| 7 7 11 6 42 -3 9
17 7 ----| 9 6 17 8 48 -1 1
8 9 ----| 11 8 25 10 80 1 1
 = 9,25 11 ----| 13 8 33 12 96 3 9
13 ----| 15 3 36 14 42 5 25
36 ---- ---- 324 ---- ---- 
8
9 324,00 = 9 (3.n/4)+1/2 =
2 36 Q3 = 11 + 
0 s2 = s = 
2 . 2 = 11,0 s2 = s = 
2 + 0
8
11
0
5
0 . 2 = 11,0
0 + 5
Li da classe Med = 
fantac = 
fmed = 
Md = 9 + (3/2- 17).3
d2 = 
Mo = 11 + 
8
Cl Modal = 
Li Cl Mo = 
d1 = 
d2 = 
Mo = 9 + 
d1 = 
75
324
(xi - xm)
2.fi
100
63
6
8
72
3,0426
Elemento Central = 
n = 
Amp = máx - mín = 
K = raiz(n) = 6 = 
h = Amp/(k - 1) = 2,2 = 
pontos = 
9
9,2574
total
Limite Inferior = mínimo - h/2 = 
mínimo = 
máximo = 
3,0000
xm = Média = 27,500
(3.36/4 - 25).2 = 11,5
8
Cl Modal = 
Li Cl Mo = 
2012_2 ex 7
1 1 2 2 2 2 3 3 1 4
4 4 4 4 4 4 5 5 9
5 5 5 6 6 6 7 7 32 Q3 = 7
7 7 8 8 9 9 9 9 8,00
5
2,0
0,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
16,5 0 ----| 2 6 6 1 6 -3,625 13,1406
4 2 ----| 4 8 14 3 24 -1,625 2,6406
14 4 ----| 6 8 22 5 40 0,375 0,1406
8 6 ----| 8 6 28 7 42 2,375 5,6406
 = 4,5 8 ----| 10 4 32 9 36 4,375 19,1406
32 ---- ---- 148 ---- ---- 
8 148,00 = 4,625 (3.n/4)+1/2 =
4 32 Q3 = 6 + 
0 s2 = s = 
2 s2 = s = 
0 . 2 = 4,0000
0 + 2
8
2
2
0
2 . 2 = 4,0000
2 + 0
fantac = 
fmed = 
Md = 4 + (32/2- 14).2
8
Cl Modal = 
211,5000
2,5709
2,6120
(xi - xm)
2.fi
78,8438
21,1250
1,1250
33,8438
76,5625
mínimo = 
máximo = 
n = 
Amp = máx - mín = 
K = raiz(n) = 5,6569 = 
Média = 
Elemento Central = 
Li da classe Med = 
Li Cl Mo = 
d1 = 
d2 = 
Mo = 4 + 
Mo = 2 + 
d2 = 
pontos = 
h = Amp/(k - 1) = 2 = 
xm =
total
d1 = 
6,6095
6,8225
Limite Inferior = mínimo - h/2 = 
24,500
(3.32/4 - 22).2 = 6,6667
6
Cl Modal = 
Li Cl Mo = 
2012_2 ex 8
18 19 19 20 20 21 21 21 18 4
22 22 23 23 23 24 24 25 29
25 26 26 26 27 27 27 27 32 Q3 = 27
27 28 28 28 28 29 29 29 11,00
5
3,0
16,5
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
16,5 16,5 ----| 19,5 3 3 18 54 -6,4688 41,8454
22,5 19,5 ----| 22,5 7 10 21 147 -3,4688 12,0326
10 22,5 ----| 25,5 7 17 24 168 -0,4688 0,2198
7 25,5 ----| 28,5 12 29 27 324 2,5312 6,4070
 = 25,0714 28,5 ----| 31,5 3 32 30 90 5,5312 30,5942
32 ---- ---- 783 ---- ---- 
12 783,00 = 24,4688 (3.n/4)+1/2 =
25,5 32 Q3 = 25,5 + 
5 s2 = s = 
9 s2 = s = 
5 . 3 = 26,5714
5 + 9
Mo = 25,5 + 
Elemento Central = 
Li da classe Med = 
fantac = 
fmed = 
Md = 22,5 + 
125,5361
84,2280
1,5384
total
(32/2- 10).16,5
Cl Modal = 
76,8837
pontos = 
(xi - xm)
2.fi
mínimo = 
máximo = 
n = 
Amp = máx - mín = 
12,257
xm =
91,7825
379,9688
3,4459
3,5010
7
Li Cl Mo = 
d1 = 
d2 = 
 = 27,25
12
h = Amp/(k - 1) = 2,75 = 
K = raiz(n) = 5,6569 = 
11,8742
Média = 
Limite Inferior = mínimo - h/2 = 
24,500
(3.32/4 - 17).3
2012_2 ex 9
4 6 8 10 11 13 4 4
4 7 8 10 12 13 15
4 7 8 10 12 13 36 Q3 = 12
5 7 9 10 12 14 11,00
6 7 9 11 12 14 6
6 8 10 11 14 15 2,0
3,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
18,5 3 ----| 5 4 4 4 16 -5,055625,5591
9 5 ----| 7 7 11 6 42 -3,0556 9,3367
17 7 ----| 9 6 17 8 48 -1,0556 1,1143
8 9 ----| 11 8 25 10 80 0,9444 0,8919
 = 9,25 11 ----| 13 7 32 12 84 2,9444 8,6695
13 ----| 15 4 36 14 56 4,9444 24,4471
36 ---- ---- 326 ---- ---- 
8
9 326,00 = 9,0556 (3.n/4)+1/2 =
2 36 Q3 = 11 + 
1 s2 = s = 
2 . 2 = 10,33 s2 = s = 
2 + 1
d1 = 
d2 = 
Mo = 9 + 3,1163
Elemento Central = 
Li da classe Med = 
fantac = 
fmed = 
Md = 9 + (36/2- 17).2
8
Cl Modal = 
Li Cl Mo = 
mínimo = 
máximo = 
n = 
Amp = máx - mín = 
K = raiz(n) = 6 = 
h = Amp/(k - 1) = 2,2 = 
Limite Inferior = mínimo - h/2 = 
65,3568
6,6857
xm = Média = 
total
9,4415 3,0727
(3.36/4 - 25).2
9,7113
27,500
 = 11,5714
7
pontos = 
(xi - xm)
2.fi
102,2364
7,1351
60,6864
97,7884
339,8889
2012_2 ex 10
4 6 8 10 11 13 4 4
4 7 8 10 12 13 15
4 7 8 10 12 13 36 Q3 = 12
5 7 9 10 12 14 11,00
6 7 9 11 12 14 6
6 8 10 11 14 15 2,0
3,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
16,5 3 ----| 5 4 4 4 16 -4,4375 19,6914
7 5 ----| 7 7 11 6 42 -2,4375 5,9414
11 7 ----| 9 6 17 8 48 -0,4375 0,1914
6 9 ----| 11 8 25 10 80 1,5625 2,4414
 = 8,6667 11 ----| 13 7 32 12 84 3,5625 12,6914
32 ---- ---- 270 ---- ---- 
8 270,00 = 8,4375 (3.n/4)+1/2 =
9 32 Q3 = 9 + 
2 s2 = s = 
1 s2 = s = 
2 . 2 = 10,3333
2 + 1
Elemento Central = 
Li da classe Med = 
fantac = 
fmed = 
Md = 7 + (32/2- 11).2
6
Cl Modal = 
d1 = 
Li Cl Mo = 
d2 = 
Mo = 9 + 
2,6802
2,72317,4153
7,1835
78,7656
41,5898
1,1484
total 229,8750
n = 
Amp = máx - mín = 
K = raiz(n) = 6 = 
h = Amp/(k - 1) = 2,2 = 
pontos = 
xm = Média = 
mínimo = 
máximo = 
24,500
(3.32/4 - 17).2 = 10,75
8
Limite Inferior = mínimo - h/2 = 
19,5313
88,8398
(xi - xm)
2.fi
2012_2 ex 11
4 6 8 11 13 4 4
4 7 9 12 13 15
6 8 10 12 14 20 Q3 = 12,25
6 8 11 13 15 11,00
4
4,0
2,0
Freq 
Abs
Freq 
Acum
xi xi . fi xi - xm (xi - xm)2
10,5 2 ----| 6,0 5 5 4 20 -4,95 24,5025
6 6 ----| 10,0 6 11 8 48 -0,95 0,9025
5 10 ----| 14,0 8 19 12 96 3,05 9,3025
6 14 ----| 16,0 1 20 15 15 6,05 36,6025
 = 9,3333 20 ---- ---- 179 ---- ---- 
179,00 = 8,95 (3.n/4)+1/2 =
8 20 = 12
10 s2 = s = 
2 s2 = s = 
7
2 . 4 = 10,89
2 + 7
mínimo = pontos = 
máximo = 
n = 
Amp = máx - mín = 
K = raiz(n) = 4,4721 = 
h = Amp/(k - 1) = 3,6667 = 
Limite Inferior = mínimo - h/2 = 
(xi - xm)
2.fi
Elemento Central = 122,5125
Li da classe Med = 5,4150
fantac = 74,4200
fmed = 36,6025
Md = 6 + (20/2- 5).4 total 238,9500
6
xm = Média = 15,500
Cl Modal = Q3 = 10 + (3.20/4 - 11).4
Li Cl Mo = 11,9474 3,4565 8
d1 = 12,5762 3,5463
d2 = 
Mo = 10 +