Monday, April 9, 2012

Pengiraan Pekali Korelasi

Pekali Korelasi digunakan semasa menganalisa hubungan diantara 2 pemboleh ubah Y dan X. Selalunya digunakan dalam analisa punca masalah dengan mengunakan gambarajah tulang ikan. Pembolehubah Y bermaksud Masalah, manakala pembolehubah X adalah salah satu punca masalah. Oleh itu, hubungan untuk menentukan samada X adalah punca sebenar atau tidak kepada masalah dapat ditentukan dengan menilai bacaan pekali korelasi tesebut.


Correlation Co-efficient Definition:
     A measure of the strength of linear association between two variables. Correlation will always between -1.0 and +1.0. If the correlation is positive, we have a positive relationship. If it is negative, the relationship is negative.

  Formula:

Correlation Co-efficient :
Correlation(r) =[ NΣXY - (ΣX)(ΣY) / Sqrt([NΣX2 - (ΣX)2][NΣY2 - (ΣY)2])] 
where 
              N = Number of values or elements 
              X = First Score
              Y = Second Score
              ΣXY = Sum of the product of first and Second Scores
              ΣX = Sum of First Scores
              ΣY = Sum of Second Scores
              ΣX2 = Sum of square First Scores
              ΣY2 = Sum of square Second Scores



Correlation Co-efficient Example
: To find the Correlation of 


X Values
Y Values
60
3.1
61
3.6
62
3.8
63
4
65
4.1


  Step 1: Count the number of values.
            N = 5

  Step 2: Find XY, X2, Y2
            See the below table


X Value
Y Value
X*Y
X*X
Y*Y
60
3.1
60 * 3.1 = 186
60 * 60 = 3600
3.1 * 3.1 = 9.61
61
3.6
61 * 3.6 = 219.6
61 * 61 = 3721
3.6 * 3.6 = 12.96
62
3.8
62 * 3.8 = 235.6
62 * 62 = 3844
3.8 * 3.8 = 14.44
63
4
63 * 4 = 252
63 * 63 = 3969
4 * 4 = 16
65
4.1
65 * 4.1 = 266.5
65 * 65 = 4225
4.1 * 4.1 = 16.81


  Step 3: Find ΣX, ΣY, ΣXY, ΣX2, ΣY2.
            ΣX = 311 
            ΣY = 18.6 
            ΣXY = 1159.7 
            ΣX2 = 19359 
            ΣY2 = 69.82 

  Step 4: Now, Substitute in the above formula given.
            Correlation(r) =[ NΣXY - (ΣX)(ΣY) / Sqrt([NΣX2 - (ΣX)2][NΣY2 - (ΣY)2])]
            = ((5)*(1159.7)-(311)*(18.6))/sqrt([(5)*(19359)-(311)2]*[(5)*(69.82)-(18.6)2])
            = (5798.5 - 5784.6)/sqrt([96795 - 96721]*[349.1 - 345.96])
            = 13.9/sqrt(74*3.14)
            = 13.9/sqrt(232.36)
            = 13.9/15.24336
            = 0.9119 

This example will guide you to find the relationship between two variables by calculating the Correlation Co-efficient from the above steps.

The correlation coefficient, denoted by r, is a measure of the strength of the straight-line or linear relationship between two variables. The correlation coefficient takes on values ranging between +1 and -1

The following points are the accepted guidelines for interpreting the correlation coefficient: 

1.     0 indicates no linear relationship.
2.     +1 indicates a perfect positive linear relationship: as one variable increases in its values, the other variable also increases in its values via an exact linear rule.
3.     -1 indicates a perfect negative linear relationship: as one variable increases in its values, the other variable decreases in its values via an exact linear rule.
4.     Values between 0 and 0.3 (0 and -0.3) indicate a weak positive (negative) linear relationship via a shaky linear rule.
5.     Values between 0.3 and 0.7 (0.3 and -0.7) indicate a moderate positive (negative) linear relationship via a fuzzy-firm linear rule.
6.     Values between 0.7 and 1.0 (-0.7 and -1.0) indicate a strong positive (negative) linear relationship via a firm linear rule.
7.     The value of r squared is typically taken as “the percent of variation in one variable explained by the other variable,” or “the percent of variation shared between the two variables.”
8.     Linearity Assumption. The correlation coefficient requires that the underlying relationship between the two variables under consideration is linear. If the relationship is known to be linear, or the observed pattern between the two variables appears to be linear, then the correlation coefficient provides a reliable measure of the strength of the linear relationship. If the relationship is known to be nonlinear, or the observed pattern appears to be nonlinear, then the correlation coefficient is not useful, or at least questionable.

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