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FÓRMULAS E TABELAS
BÁSICAS DE ESTATÍSTICA
Adriano Leal Bruni
Versão de 5/7/2002 10:59.
Fórmulas
Estatística descritiva e probabilidade
Tabulação de dados
Freqüência Simples :
% 100 %
1
∑
n
i
Fi
Fi Fi
Freqüência Acumulada :
% 100 %
1
∑
n
i
Fi
FAci FAci
Número de classes (K) Se n <= 25 : K = 5
Se n > 25 : K = n ou K = 1 + 3,22 log( n )
Amplitude da classe (h)
K
Maior Menor h
Medidas de posição central Média aritmética simples
n
x
n
i
∑ i μ = =^1 ou n
x x
n
i
∑ i = =^1
Média ponderada
∑
∑
=
n
i
i
n
i
i i w F
x F x
1
1
Média geométrica
n
n
i
xg n^ x x xn ∏ xn
1
Média harmônica
∑
= n
n i (^) i
h
x
n
x x x
n
x
1 2 1
Mediana Divide a série ordenada em duas partes iguais
Moda Valor com maior número de repetições
Medidas de dispersão Amplitude total (R) = Maior – Menor Desvio médio absoluto
n
x x DMA
n
i
∑ i
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Variância n
∑ Xi − X
2 σ 2 ( ) ou
2 (^ )^2
= ∑^ n
Xi X s
Desvio padrão n
∑ Xi − X
( )^2
σ ou
( )^2
∑ n
Xi X s
Populacional :
= (^) ∑ − ∑ n
x x n
2 σ^12
Amostral :
= (^) ∑ ∑ n
x x n
s
2 2 1
Coeficiente de variação
x
s CV ou μ
σ
Medidas de ordenamento e posição
QNúmero do
Quartil
= X Número do
Quartil x n 4
+
1
[ 2 ]
DNúmero doDecil =^ X^ Número doDecil x n
10
+
1
[ 2 ]
PNúmero do
Percentil
= X Número do
Percentil x n 100
+
1
[ 2 ]
Medidas de forma da distribuição Curtose
3 1
- P P
Q Q
K
¾ k=0,263 : mesocúrtica
¾ k > 0,263 : leptocúrtica ¾ k<0,263 : platicúrtica
Assimetria 1 o^ Coeficiente de Pearson :
σ
x Mo AS
2 o^ Coeficiente de Pearson :
3 1
Q Q
Q Q Q
AS
AS > 0 = assimetria à direita AS < 0 = assimetria à esquerda AS = 0 = simétrica
Probabilidade
Espaço Amostral
Eventos No.deeventospossíveis
No. deeventosfavoráveis P = =
Chance a favor = Número de eventos favoráveis / Número de eventos desfavoráveis
Teoremas P (E 1 U E 2 )= P(E 1 ) + P(E 2 ) − P ( E 1 I E 2 ) P ( E 1 I E 2 )= P(E 1 )* P ( E 2 | E 1 )
Permutações, arranjos e combinações
Pn = n!, An,r = ( )!
n r
n −
, r n r
n C (^) n r −
Probabilidade condicional
PB
P A B
P A B
I
Tabelas de probabilidade condicional
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n
x z (^) t σ
−μ 0 = ou
n
s
x t (^) t^0
− μ = ou
n
p q
p p Z ⋅
Duas amostras
2
2 2 1
2 1
1 2
n n
x x z (^) teste σ σ
= ou
2
2 2 1
2 1
1 2
n
s n
s
x x t (^) teste
ou
1 2 1 2
2 2 2
2 1 1
1 2 1 1 2
n n n n
n s n s
x x t (^) teste
Diferença alegada
2
2 2 1
2 1
n n
x x z (^) teste o σ σ
δ
= ou
2
2 2 1
2 1
n
s n
s
x x t (^) teste o
δ ou
1 2 1 2
2 2 2
2 1 1
1 2 1 1 2
n n n n
n s n s
x x t (^) teste o
δ
Proporções : igualdade
1 2
1 2 1 1 1 n n
p p
p p z (^) teste , onde
1
1 (^1) n
x p = , 2
2 (^2) n
x p = e 1 2
1 2 n n
x x p
Proporções : diferença
1 2
1 2 1 1 1 n n
p p
p p Z (^) teste
δ
Correlação e regressão
Y = a + b.X
a
y b x n
∑ − ∑
( ) ( )
( ) ( )
b
n xy x y
n x x
∑ ∑ ∑
∑ ∑
2 2
[ ( )] [ ( )] 2 2 2 2 .
∑ ∑ ∑ ∑
∑ ∑ ∑ − −
n x x n y y
n xy x y r
s
y a y b xy e (^) n
∑ ∑ ∑
2
2
( 1 ). x^2
e b n s
s s −
2
2
( 1 )
x
a e n s
x n
s s − ⋅
n
r s ρ
Números índices
Laspeyres ou Método do Ano Base
= × 100 % ∑
∑ poqo
pnqo PREÇO
= × 100 %
∑
∑ poqo
qnpo QUANTIDADE
Material extraído de http://www.infinitaweb.com.br
= × 100 %
∑
∑ poqo
pnqn VALOR
Paasche ou Método do Determinado ou Época Atual
= × 100 % ∑
∑ poqn
pnqn PREÇO
= × 100 %
∑
∑ o n
n n q p
pq QUANTIDADE
Fischer ou Método Ideal
IFP = LpxP p
IFQ = LqxP q
Marshall-Edgeworth ou Método Ideal
× 100 %
∑
∑ qopo pn
qnpo pn QUANTIDADE
× 100 %
∑
∑ poqo qn
pnqo qn PREÇO
Material extraído de http://www.infinitaweb.com.br
Tabela "t" de Student Tabela t de (^) Student Valores de t (supondo alfa bicaudal) Observação : valores gerados no MS Excel (função INVT) Por Adriano Leal Bruni α bicaudal α unicaudal
α
Material extraído de http://www.infinitaweb.com.br Tabela Qui-Quadrado Tabela de Qui-quadrado
- de 0,1000 0,0900 0,0800 0,0700 0,0600 0,0500 0,0400 0,0300 0,0200 0,0100 0,0050 0,0025 0,0010 0,0005 0, Graus
- (n -1) 0,0500 0,0450 0,0400 0,0350 0,0300 0,0250 0,0200 0,0150 0,0100 0,0050 0,0025 0,0013 0,0005 0,0003 0, liberdade - 1 6,3137 7,0264 7,9158 9,0579 10,5789 12,7062 15,8945 21,2051 31,8210 63,6559 127,3211 254,6608 636,5776 1.273,155 6.370, - 2 2,9200 3,1040 3,3198 3,5782 3,8964 4,3027 4,8487 5,6428 6,9645 9,9250 14,0892 19,9629 31,5998 44,7035 100, - 3 2,3534 2,4708 2,6054 2,7626 2,9505 3,1824 3,4819 3,8961 4,5407 5,8408 7,4532 9,4646 12,9244 16,3261 28, - 45 2,13182,0150 2,22612,0978 2,33292,1910 2,45592,2974 2,60082,4216 2,77652,5706 2,99852,7565 3,29763,0029 3,74693,3649 4,60414,0321 5,59754,7733 6,75825,6042 8,61016,8685 10,30517,9756 15,534511, - 6 1,9432 2,0192 2,1043 2,2011 2,3133 2,4469 2,6122 2,8289 3,1427 3,7074 4,3168 4,9807 5,9587 6,7882 9, - 7 1,8946 1,9662 2,0460 2,1365 2,2409 2,3646 2,5168 2,7146 2,9979 3,4995 4,0294 4,5946 5,4081 6,0815 7, - 8 1,8595 1,9280 2,0042 2,0902 2,1892 2,3060 2,4490 2,6338 2,8965 3,3554 3,8325 4,3336 5,0414 5,6170 7, - 9 1,8331 1,8992 1,9727 2,0554 2,1504 2,2622 2,3984 2,5738 2,8214 3,2498 3,6896 4,1458 4,7809 5,2911 6, - 10 1,8125 1,8768 1,9481 2,0283 2,1202 2,2281 2,3593 2,5275 2,7638 3,1693 3,5814 4,0045 4,5868 5,0489 6, - 11 1,7959 1,8588 1,9284 2,0067 2,0961 2,2010 2,3281 2,4907 2,7181 3,1058 3,4966 3,8945 4,4369 4,8633 5, - 12 1,7823 1,8440 1,9123 1,9889 2,0764 2,1788 2,3027 2,4607 2,6810 3,0545 3,4284 3,8065 4,3178 4,7166 5, - 13 1,7709 1,8317 1,8989 1,9742 2,0600 2,1604 2,2816 2,4358 2,6503 3,0123 3,3725 3,7345 4,2209 4,5972 5, - 14 1,7613 1,8213 1,8875 1,9617 2,0462 2,1448 2,2638 2,4149 2,6245 2,9768 3,3257 3,6746 4,1403 4,4995 5, - 15 1,7531 1,8123 1,8777 1,9509 2,0343 2,1315 2,2485 2,3970 2,6025 2,9467 3,2860 3,6239 4,0728 4,4168 5, - 16 1,7459 1,8046 1,8693 1,9417 2,0240 2,1199 2,2354 2,3815 2,5835 2,9208 3,2520 3,5805 4,0149 4,3464 5, - 17 1,7396 1,7978 1,8619 1,9335 2,0150 2,1098 2,2238 2,3681 2,5669 2,8982 3,2224 3,5430 3,9651 4,2858 5, - 18 1,7341 1,7918 1,8553 1,9264 2,0071 2,1009 2,2137 2,3562 2,5524 2,8784 3,1966 3,5101 3,9217 4,2332 4, - 19 1,7291 1,7864 1,8495 1,9200 2,0000 2,0930 2,2047 2,3457 2,5395 2,8609 3,1737 3,4811 3,8833 4,1869 4, - 20 1,7247 1,7816 1,8443 1,9143 1,9937 2,0860 2,1967 2,3362 2,5280 2,8453 3,1534 3,4554 3,8496 4,1461 4, - 21 1,7207 1,7773 1,8397 1,9092 1,9880 2,0796 2,1894 2,3278 2,5176 2,8314 3,1352 3,4324 3,8193 4,1095 4, - 2223 1,71711,7139 1,77341,7699 1,83541,8316 1,90451,9003 1,98291,9783 2,07392,0687 2,18292,1770 2,32022,3132 2,50832,4999 2,81882,8073 3,11883,1040 3,41183,3931 3,79223,7676 4,07694,0475 4,73584, - 24 1,7109 1,7667 1,8281 1,8965 1,9740 2,0639 2,1715 2,3069 2,4922 2,7970 3,0905 3,3761 3,7454 4,0207 4, - 25 1,7081 1,7637 1,8248 1,8929 1,9701 2,0595 2,1666 2,3011 2,4851 2,7874 3,0782 3,3606 3,7251 3,9965 4, - 26 1,7056 1,7610 1,8219 1,8897 1,9665 2,0555 2,1620 2,2958 2,4786 2,7787 3,0669 3,3464 3,7067 3,9744 4, - 27 1,7033 1,7585 1,8191 1,8867 1,9632 2,0518 2,1578 2,2909 2,4727 2,7707 3,0565 3,3334 3,6895 3,9540 4, - 2829 1,70111,6991 1,75611,7540 1,81661,8142 1,88391,8813 1,96011,9573 2,04842,0452 2,15392,1503 2,28642,2822 2,46712,4620 2,76332,7564 3,04703,0380 3,32133,3103 3,67393,6595 3,93483,9177 4,53094, - 30 1,6973 1,7520 1,8120 1,8789 1,9546 2,0423 2,1470 2,2783 2,4573 2,7500 3,0298 3,2999 3,6460 3,9017 4, - 40 1,6839 1,7375 1,7963 1,8617 1,9357 2,0211 2,1229 2,2503 2,4233 2,7045 2,9712 3,2266 3,5510 3,7884 4, - 50 1,6759 1,7289 1,7870 1,8516 1,9244 2,0086 2,1087 2,2338 2,4033 2,6778 2,9370 3,1840 3,4960 3,7230 4, - 60 1,6706 1,7232 1,7808 1,8448 1,9170 2,0003 2,0994 2,2229 2,3901 2,6603 2,9146 3,1562 3,4602 3,6808 4, - 7080 1,66691,6641 1,71921,7162 1,77651,7732 1,84011,8365 1,91181,9078 1,99441,9901 2,09272,0878 2,21522,2095 2,38082,3739 2,64792,6387 2,89872,8870 3,13663,1220 3,43503,4164 3,65083,6287 4,12694, - 90 1,6620 1,7138 1,7707 1,8337 1,9048 1,9867 2,0839 2,2050 2,3685 2,6316 2,8779 3,1108 3,4019 3,6118 4, - 100 1,6602 1,7120 1,7687 1,8315 1,9024 1,9840 2,0809 2,2015 2,3642 2,6259 2,8707 3,1018 3,3905 3,5984 4, - 200 1,6525 1,7036 1,7596 1,8217 1,8915 1,9719 2,0672 2,1857 2,3451 2,6006 2,8385 3,0621 3,3398 3,5387 3, - 250 1,6510 1,7020 1,7578 1,8197 1,8894 1,9695 2,0645 2,1826 2,3414 2,5956 2,8322 3,0543 3,3299 3,5268 3, - 500 1,6479 1,6987 1,7543 1,8158 1,8851 1,9647 2,0591 2,1763 2,3338 2,5857 2,8195 3,0387 3,3101 3,5038 3, - 750 1,6469 1,6976 1,7531 1,8145 1,8836 1,9631 2,0573 2,1742 2,3313 2,5824 2,8154 3,0336 3,3036 3,4960 3,
- 1000 1,6464 1,6970 1,7525 1,8139 1,8829 1,9623 2,0564 2,1732 2,3301 2,5807 2,8133 3,0310 3,3002 3,4922 3,
- 10000 1,6450 1,6956 1,7509 1,8121 1,8810 1,9602 2,0540 2,1704 2,3267 2,5763 2,8076 3,0241 3,2915 3,4820 3,
- Fi\Alfa 0,995 0,990 0,975 0,950 0,900 0,750 0,500 0,250 0,100 0,050 0,025 0,010 0, Valores gerados no Excel com a função =INV.QUI() - 1 0,00 0,00 0,00 0,00 0,02 0,10 0,45 1,32 2,71 3,84 5,02 6,63 7, - 2 0,01 0,02 0,05 0,10 0,21 0,58 1,39 2,77 4,61 5,99 7,38 9,21 10, - 3 0,07 0,11 0,22 0,35 0,58 1,21 2,37 4,11 6,25 7,81 9,35 11,34 12, - 4 0,21 0,30 0,48 0,71 1,06 1,92 3,36 5,39 7,78 9,49 11,14 13,28 14, - 5 0,41 0,55 0,83 1,15 1,61 2,67 4,35 6,63 9,24 11,07 12,83 15,09 16, - 6 0,68 0,87 1,24 1,64 2,20 3,45 5,35 7,84 10,64 12,59 14,45 16,81 18, - 7 0,99 1,24 1,69 2,17 2,83 4,25 6,35 9,04 12,02 14,07 16,01 18,48 20, - 8 1,34 1,65 2,18 2,73 3,49 5,07 7,34 10,22 13,36 15,51 17,53 20,09 21, - 9 1,73 2,09 2,70 3,33 4,17 5,90 8,34 11,39 14,68 16,92 19,02 21,67 23,
- 10 2,16 2,56 3,25 3,94 4,87 6,74 9,34 12,55 15,99 18,31 20,48 23,21 25,
- 11 2,60 3,05 3,82 4,57 5,58 7,58 10,34 13,70 17,28 19,68 21,92 24,73 26,
- 12 3,07 3,57 4,40 5,23 6,30 8,44 11,34 14,85 18,55 21,03 23,34 26,22 28,
- 13 3,57 4,11 5,01 5,89 7,04 9,30 12,34 15,98 19,81 22,36 24,74 27,69 29,
- 14 4,07 4,66 5,63 6,57 7,79 10,17 13,34 17,12 21,06 23,68 26,12 29,14 31,
- 15 4,60 5,23 6,26 7,26 8,55 11,04 14,34 18,25 22,31 25,00 27,49 30,58 32,
- 16 5,14 5,81 6,91 7,96 9,31 11,91 15,34 19,37 23,54 26,30 28,85 32,00 34,
- 17 5,70 6,41 7,56 8,67 10,09 12,79 16,34 20,49 24,77 27,59 30,19 33,41 35,
- 18 6,26 7,01 8,23 9,39 10,86 13,68 17,34 21,60 25,99 28,87 31,53 34,81 37,
- 19 6,84 7,63 8,91 10,12 11,65 14,56 18,34 22,72 27,20 30,14 32,85 36,19 38,
- 20 7,43 8,26 9,59 10,85 12,44 15,45 19,34 23,83 28,41 31,41 34,17 37,57 40,
- 21 8,03 8,90 10,28 11,59 13,24 16,34 20,34 24,93 29,62 32,67 35,48 38,93 41,
- 22 8,64 9,54 10,98 12,34 14,04 17,24 21,34 26,04 30,81 33,92 36,78 40,29 42,
- 23 9,26 10,20 11,69 13,09 14,85 18,14 22,34 27,14 32,01 35,17 38,08 41,64 44,
- 24 9,89 10,86 12,40 13,85 15,66 19,04 23,34 28,24 33,20 36,42 39,36 42,98 45,
- 25 10,52 11,52 13,12 14,61 16,47 19,94 24,34 29,34 34,38 37,65 40,65 44,31 46,
- 26 11,16 12,20 13,84 15,38 17,29 20,84 25,34 30,43 35,56 38,89 41,92 45,64 48,
- 27 11,81 12,88 14,57 16,15 18,11 21,75 26,34 31,53 36,74 40,11 43,19 46,96 49,
- 28 12,46 13,56 15,31 16,93 18,94 22,66 27,34 32,62 37,92 41,34 44,46 48,28 50,
- 29 13,12 14,26 16,05 17,71 19,77 23,57 28,34 33,71 39,09 42,56 45,72 49,59 52,
- 30 13,79 14,95 16,79 18,49 20,60 24,48 29,34 34,80 40,26 43,77 46,98 50,89 53,
- 31 14,46 15,66 17,54 19,28 21,43 25,39 30,34 35,89 41,42 44,99 48,23 52,19 55,
