Pearson Product Moment Correlation Coefficient PDF

THE PEARSON PRODUCT MOMENT COEFFICIENT OF CORRELATION, R The Pearson Product Moment Coefficient of Correlation r is an index of relationship between two variables. The independent variable can be represented by x while the dependent variable can be represented by y. The value of r...

THE PEARSON PRODUCT MOMENT COEFFICIENT OF CORRELATION, R The Pearson Product Moment Coefficient of Correlation r is an index of relationship between two variables. The independent variable can be represented by x while the dependent variable can be represented by y. The value of r is +1,zero to -1. If the value of r is +1 or -1, there is a perfect correlation between x and y.  y High r=+ x Low High If the line graph is going upward, the value of r is positive. As the value of x increases the value of y increases. Likewise, if the value of x decreases, the value of y also decreases.  y High r=− x Low High If the line graph is going downward, the value of r is negative. As the value of x increases the corresponding value of y decreases.  y High r=0 x Low High If the trend line graph cannot be established either upward or downward, then r = 0, indicating that there is no correlation between the x and y variables.  𝑛 σ 𝑥𝑦−σ 𝑥 σ 𝑦 r= 2 𝑛 σ 𝑥2− σ 𝑥 2 𝑛 σ 𝑦 − σ 𝑦 2 Where: r = the Pearson Product Moment Coefficient of Correlation n = sample size σ 𝑥𝑦 = the sum of the product of x and y σ 𝑥 σ 𝑦 = the product of the sum of σ 𝑥 and the sum σ 𝑦 σ 𝑥 2 = sum of square of x. σ 𝑦 2 = sum of square of y. EXAMPLE: Below are the midterm (x) and final (y) grades. x 75 70 65 90 85 85 80 70 65 90 y 80 75 65 95 90 85 90 75 70 90 PROBLEM: Is there a significant relationship between the midterm and final examinations of 10 students in mathematics? HYPOTHESIS: 𝑯𝟎 : There is no significant relationship between the midterm grades and the final examination/grades of 10 students in mathematics. 𝑯𝟏 : There is a significant relationship between the midterm grades and the final examination/grades of 10 students in mathematics. STATISTICS: Pearson Product Moment Coefficient of Correlation. x y 𝑿𝟐 𝒚𝟐 xy 75 80 5625 6400 6000 70 75 4900 5625 5250 65 65 4225 4225 4225 90 95 8100 9025 8550 85 90 7225 8100 7650 85 85 7225 7225 7225 80 90 6400 8100 7200 70 75 4900 5625 5250 65 70 4225 4900 4550 90 90 8100 8100 8100 σ 𝑥 = 775 σ 𝑦 = 815 σ 𝑥 2 = 60925 σ 𝑦 2 = 67325 σ 𝑥𝑦 = 64000 x = 77.5 y = 81.5  𝑛 σ 𝑥𝑦−σ 𝑥 σ 𝑦 r= 2 𝑛 σ 𝑥2− σ 𝑥 2 𝑛 σ 𝑦 − σ 𝑦 2 10 64000 −(775)(815) r= 10 60925 −(775)2 10(67325)− 815 2 8,375 r= 609,250−600,625 673,250−664,225 8,375 r= 8625 9025 8,375 r= 77840625 r =.949 VHR COEFFICIENT OF DETERMINATION: The coefficient of determination is 𝑟 2 times 100%. This explains the extent to which the independent variable x influences y or the extent to which y depends on x. 𝐶𝐷 = 𝑟 2 𝑥 100% = (.949)2 𝑥100% =.9006 x 100% = 90.06% This 90.06% indicates that the final examination grade depends on the midterm grades. Thus, the final grade influenced by the midterm grade.  𝑡𝑐𝑣 = 8.514 > 𝑡𝑇𝑉 = 2.306 ; 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0.05−two tailed ; df = 10−2  There is a significant relationship between the X & Y. Interpretation: ±.00 - ±.20 No Relationship ±.21 - ±.40 Low Relationship ±.41 - ±.60 Moderate Relationship ±.61 - ±.89 High Relationship ±.90 - ±.99 Very High Relationship ± 1.00 Perfect Relationship Validation Test for Correlation (r) using t- test Formula: 𝑛−2 𝑡=𝑟 1 − 𝑟2