Correlation

When data consist of pairs (X, Y) of related measurements it is often useful to study whether there is at least an approximate linear relationship between X and Y. The strength of such a relationship is measured by the linear correlation coefficient ρ which always lies between −1 and + 1. ρ = 0 indicates no linear relationship, and ρ = +1 and ρ = −1 respectively indicate exact linear relationships with +ve and −ve slopes. More generally, values of ρ close to 0 indicate little linear relationship, and values near +1 or −1 indicate strong linear relationships. Tests etc concerning ρ are formulated using the sample linear correlation coefficient https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203848944/755cc168-5d48-428f-a06b-9e70e9ae2f1e/content/eqnu14_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> r = ∑ ( X − X ¯ ) ( Y − Y ¯ ) ∑ ( X − X ¯ ) 2 ( Y − Y ¯ ) 2 where https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203848944/755cc168-5d48-428f-a06b-9e70e9ae2f1e/content/inline219_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> X ¯ and https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203848944/755cc168-5d48-428f-a06b-9e70e9ae2f1e/content/inline113_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> Y ¯ are the sample mean values of X and Y. The first table on page 36 is for testing H₀: ρ = 0. Critical regions are | r | ≥ tabulated value if H₁ is the two-sided alternative hypothesis ρ ≠ 0 (using α₂ significance levels); r ≥ tabulated value if H₁ is ρ > 0; or r ≤ −tabulated value if H₁ is ρ < 0 (using α₁ levels in these one-sided cases).