The document discusses correlation and correlation coefficients. It defines correlation as the numerical measure of the linear relationship between two variables. A positive correlation means high values in one variable are associated with high values in the other, while a negative correlation means high values in one variable are associated with low values in the other. Correlation coefficients range from -1 to 1, where 0 is no correlation, 1 is total positive correlation, and -1 is total negative correlation. The document contrasts parametric and non-parametric correlation tests and discusses the Pearson product-moment correlation and Spearman's rank correlation tests.

2. Correlation
By Rama Krishna Kompella

3. Correlation
• The coefficient of correlation is a numerical
measure of the strength of the linear
relationship between 2 variables.
• X, Y must be measurable quantitatively.

4. Correlation
• Two variables X, Y have a positive correlation if large
values of X tend to be associated with large values of
Y; similarly, for small values.
• Two variables X, Y have a negative correlation if
large values of X tend to be associated with small
values of Y, and vice-versa.

5. What Are Correlation Tests?
• Provides a quantitative perspective on the strength of
the relationship if one is found, denoted by an “r” value.
• Values of r are always between -1 & 1; i.e., between 0
and 1 in absolute value.
• r = 0 means no correlation; r = +-1 means perfect
correlation; both rare.
• Tests can be run for both parametric and non-parametric
data sets.

6. Correlation Relationships
Negative Relationship No Relationship Positive Relationship

7. Parametric vs. Non-Parametric
• Parametric data:
– Assumes normal distribution, homogenous
variance, and data sets are typically ratio or
interval.
– Can draw more conclusions.
• Pearson Correlation
• Non-Parametric data:
– No assumption on distribution or variance
relationship, and data sets are typically ordinal or
nominal.
– More simple and less affected by outliers.
• Spearman Correlation

8. Pearson Product Moment Correlation
• More simply known as Pearson Correlation,
designated by the Greek letter rho (ρ) but
reported as a “r” value.
• Depicts linear relationships.
• Data sets must meet assumptions of
parametric data.
• Like Spearman’s, Hypothesis testing states;
– Ho : ρ = 0
– Ha: ρ ≠ 0

9. Correlation Coefficient
Pearson Product-moment Correlation Coefficient
1 X i −X (Yi − )
Y
rxy = *∑ *
(n − )
1 Sx Sy
Cov xy
rxy =
Sx * S y
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10. Determining Sample Correlation
Coefficient
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11. Testing the Significance of the
Correlation Coefficient
• Null hypothesis: Ho : p = 0
• Alternative hypothesis: Ha : p ≠ 0
• Test statistic
6 −2
t = .70 2
= 1.96
1 − 0.70
Example: n = 6 and r = .70
At α = .05 , n-2 = 4 degrees of freedom,
Critical value of t = 2.78
Since 1.96<2.78, we fail to reject the null hypothesis.
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12. Questions?