Fundamentos da Estatística
Autor: Equipe de Estatística Aplicada | Publicado: 2011-08-16T21:00:00-07:00


Estatística
Refere-se ao conjunto de técnicas usadas na coleção, organização e interpretação de dados e para a utilização dos mesmos na tomada de decisões. Estes dados podem ser quantitativos, com valores expressos numericamente, ou qualitativos, representados por características tais quais as preferências dos consumidores obtidas em uma pesquisa, com calculo nas médias. &nbsp;&nbsp;

&nbsp;. Média&nbsp;Aritmética é definida como a soma dos valores no grupo de dados dividida pelo número de valores.
Formula:&nbsp;
. Média Mediana:&nbsp;é o valor que divide um conjunto de dados ordenados ao meio( 50%) ou seja, ela nos fornece o elemento central desse conjunto de dados ela e dividida em amostra par e impar.Formula da Amostra impar:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Formula da Amostra par:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  e &nbsp;

. Média Moda: é o valor que ocorre com mais&nbsp;frequência em um conjunto de valores. Podendo ser &nbsp;distribuição&nbsp;como unimodal, bimodal,&nbsp;multimodal ou sem moda.&nbsp;
Exemplo(1):&nbsp;O número de acidentes que ocorreram durante um determinado mês em 13 departamentos de manufaturas de uma planta industrial foi ; 2,0.0,3,3,12,1,0,8,1,0,5,1,calcule.
Média&nbsp;Aritmética:&nbsp;&nbsp; &nbsp;logo&nbsp;
 
X= &nbsp;2+0+0+3+3+12+1+0+8+1+0+5+1 &nbsp;= &nbsp;&nbsp; 36 &nbsp; = X 2,7692&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;13 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 13&nbsp;
Média Mediana:&nbsp;&nbsp; X= { 2,0,0,3,3,12,1,0,8,1,0,5,1}&nbsp;formula impar.
X=&nbsp;&nbsp; logo&nbsp;
X= 13+1 &nbsp;= &nbsp;X =&nbsp;14 &nbsp; &nbsp;X= 7 &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2&nbsp;. Media Moda&nbsp;
X = { 2,0,0,3,3,12,1,0,8,1,8,5,1}&nbsp; X = { &nbsp;0 &nbsp;} unimodal
Exemplo ( 2)&nbsp;As seguintes notas de provas organizadas em ordem crescente, foram&nbsp;obtidas por 20 estudantes inscritos em um curso de análise de decisão: 39,46,57,65,70,72,72,75,77,79 ,81,81,84,84,84 ,87,93,94, 97 97, determine:&nbsp;
Média aritmética.&nbsp;:&nbsp;, logoX= 1.534,00&nbsp;= &nbsp; X= 76,70&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;20   Média Mediana: &nbsp; &nbsp; &nbsp; &nbsp;X= &nbsp; &nbsp;  e &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; X= &nbsp;{&nbsp;39,46,57,65,70,72,72,75,77,79 ,81,81,84,84,84 ,87,93,94, 97 97}&nbsp;X= ( &nbsp;n &nbsp;&nbsp;) + (&nbsp; n &nbsp;+1) = &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;X = (20&nbsp;)+(20&nbsp;+1)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; 2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2&nbsp; &nbsp; &nbsp; &nbsp;2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2

X= 10 +11&nbsp; &nbsp;= &nbsp; X = 79 + 81&nbsp; &nbsp; &nbsp;X= 160 &nbsp; &nbsp; &nbsp;X = 80 &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2

Média Moda :
moda { &nbsp;84 }unimodal


Exemplo ( 3) &nbsp;O numero de carros vendidos por cada um dos 10 representantes de venda em uma agência de&nbsp;automóveis, durante um determinado mês, grupando em ordem crescente; 2,4,7,10,10,10,12,12, 14,15.&nbsp;Determine&nbsp;&nbsp;

Média&nbsp;aritmética:&nbsp;&nbsp;&nbsp; &nbsp;, logo

X= &nbsp;96 &nbsp;&nbsp;&nbsp;&nbsp; &nbsp;X= &nbsp; 9,60&nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;10&nbsp;
Média Mediana:
X={ 2,4,7,10,10,10,12,12,14,15}


X=&nbsp;(&nbsp;&nbsp;n &nbsp;&nbsp;) + (&nbsp; n&nbsp;&nbsp;+1) = &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;X =&nbsp;(10&nbsp;)+(10&nbsp;+1)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp;2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2&nbsp; &nbsp; &nbsp; &nbsp;2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2

&nbsp; &nbsp; &nbsp;X= 5+6 &nbsp; &nbsp;X = 10+10 &nbsp; &nbsp; &nbsp;X= 10&nbsp; &nbsp; &nbsp; &nbsp; 2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 2
Média Moda :&nbsp;
X = &nbsp;{ 10 } unimodal

Média ageométrica: Realiza a multiplicação entre números, logo em seguida extrair a raiz com índice igual ao número.
Formula:



&nbsp;Exemplo ( &nbsp;1 )Qual o cálculo dos seguintes números {4,6,8,10}
XG=&nbsp;4√4. 6.8.1 , logo&nbsp;
XG = 4√ 1.920.00&nbsp;
&nbsp;XG= 6,6195&nbsp;
Na HP=&nbsp;1.920,00 enter 4 1/X &nbsp;Yx

Exemplo (2 )&nbsp;São dados os valores 4,7,9,10. calcule a média geométrica&nbsp;
XG = 4√4.7.9.10&nbsp; &nbsp; &nbsp;XG= &nbsp;4√2.520,00
&nbsp;XG= &nbsp;7,0852 &nbsp;
Na HP = 2.520,00 enter 4 &nbsp;1/x Yx

Exemplo (3)calcule a média geométrica dos seguintes dados: 8,10,12,14,16
&nbsp;XG=&nbsp;5√8.10.12.14.16
&nbsp;&nbsp;XG=&nbsp; &nbsp;5√215.040,00
XG= &nbsp; 11,6548

Na HP = &nbsp;215,040,00 enter &nbsp;5 &nbsp;1/x &nbsp;Yx

Média Harmônia: é definida como sendo o número de membros dividido pela soma do inverso dos membros .&nbsp;Formula&nbsp;






Exemplo (1)&nbsp;calcule o valor da média harmônia dos seguinte números &nbsp;12 e 16
XH=&nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; XH = &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;XH =&nbsp; &nbsp; &nbsp; 2 &nbsp; &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1 &nbsp;+ &nbsp;1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0,0833 + 0,065 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;0,1483&nbsp; &nbsp; &nbsp; &nbsp; 12 &nbsp; &nbsp; 16

XH= &nbsp;13,4862
Exemplo (2) qual a média harmônica entre 2,6,8,10
XH =&nbsp; &nbsp; &nbsp; &nbsp; 4 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; XH= &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 4 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1 &nbsp;+ 1 + 1 + 1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0,5 +0,1667 +0,1250 +0,10&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2 &nbsp; &nbsp; &nbsp; 6 &nbsp; &nbsp;8 &nbsp; &nbsp; 10
XH= &nbsp; &nbsp;4 &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; XH= 4,4858&nbsp; &nbsp; &nbsp; &nbsp; 0,8917

Exemplo (3)&nbsp;&nbsp;calcule os valores 8,10,15,17,19
XH= &nbsp; &nbsp; &nbsp; &nbsp;5 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; XH= &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 5 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1 +1+ 1+ 1+ 1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0,1250+0,10+0,066+0,588+0,0526&nbsp; &nbsp; &nbsp; &nbsp; 8 &nbsp; 10 &nbsp;15 &nbsp;17 &nbsp;19
XH=&nbsp; &nbsp; &nbsp; &nbsp;5 &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp;XH= &nbsp;12,4039 &nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0,4031

&nbsp; &nbsp; &nbsp; Média Ponderada&nbsp;
Consideremos uma coleção formada por n números: , de forma que cada um esteja sujeito a um peso [Nota: "peso" é sinônimo de "ponderação"], respectivamente, indicado pela&nbsp;&nbsp; média aritmética ponderada desses n números é a soma dos produtos de cada um multiplicados por seus respectivos pesos, dividida pela soma dos pesos:
&nbsp; &nbsp; &nbsp;Formula:&nbsp;



Exemplo ( 1 )

1) João deseja calcular a média das notas que tirou em  inglês. Calcule a média ponderada de suas notas, sendo que as duas primeiras provas valem 2 pontos e as outras duas valem 3 pontos:
Inglês

   1º prova   6,5  
   2º prova   7,8  
   3º prova   8,0  
   4º prova   7,1  


6,5*2 + 7,8*2 + 8,0*3 + 7,1*3 =­­­­­­­­­­­­­­­­­­­­­_________________________&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2 + 2 + 3 + 3

(13 + 15,6 + 24 + 21,3) =___________________&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 10
&nbsp; &nbsp; &nbsp; 73,9 &nbsp; =&nbsp;&nbsp; 7,39&nbsp; &nbsp; &nbsp; 10

&nbsp;Exemplo (2)&nbsp;Qual é a média ponderada dos números 10, 14, 18 e 30 sabendo-se que os seus pesos são respectivamente 1,2,3,5?

10*1 + 14*2 + 18*3 + 30*5&nbsp; =&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1 + 2 + 3 + 5

10 + 28 + 54 + 150 =&nbsp;&nbsp;&nbsp;&nbsp; 242&nbsp; =&nbsp;&nbsp; 22&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 11&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;11



&nbsp;Exemplo (3)&nbsp;
&nbsp;Calcule a média aritmética abaixo:

   notas   fi  
   0 - 2   5  
   2 – 4   8  
   4 – 6   14  
   6 – 8   10  
   8 – 10   7  
   
   ∑ = 44  




1*5 + 3*8 + 5*14 + 7*10 + 9*7&nbsp; =&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;44
5 + 24 + 70 + 70 + 63&nbsp; =&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;44&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp; 232&nbsp;&nbsp;&nbsp; =&nbsp; 5,3&nbsp; &nbsp; &nbsp; &nbsp;44