Correlation Analysis

What is Correlation?

Correlation measures the strength and direction of a linear relationship between two quantitative variables.

Pearson Correlation Coefficient (r)

The Pearson correlation coefficient is the most common measure of linear correlation.

Properties of r

Property	Description
Range	-1 ≤ r ≤ +1
r = +1	Perfect positive linear relationship
r = -1	Perfect negative linear relationship
r = 0	No linear relationship
Sign	Indicates direction (positive/negative)
Magnitude	Indicates strength

Interpreting Correlation Strength

Absolute Value of r	Interpretation
0.00 - 0.19	Very weak
0.20 - 0.39	Weak
0.40 - 0.59	Moderate
0.60 - 0.79	Strong
0.80 - 1.00	Very strong

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Coefficient of Determination (r²)

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Visualizing Correlation

Scatter Plots

The scatter plot is essential for understanding correlation:

Strong Positive (r ≈ 0.9)     No Correlation (r ≈ 0)
     *  *                          *    *
    *  *                        *      *
   *  *                      *    *    
  *  *                          *    *
 *  *                        *    *    

Strong Negative (r ≈ -0.9)    Nonlinear (r ≈ 0)
*  *                               * *
 *  *                           *       *
  *  *                        *           *
   *  *                        *       *
    *  *                          * *

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Hypothesis Testing for Correlation

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Spearman Rank Correlation

When data is ordinal or not normally distributed, use Spearman’s correlation.

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Correlation Pitfalls

Third Variable Problem

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Special Correlations

Point-Biserial Correlation

For one continuous and one binary variable (e.g., gender and test score).

Phi Coefficient

For two binary variables (equivalent to Pearson r calculated on 0/1 data).

Partial Correlation

Correlation between X and Y after controlling for variable Z.

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Correlation vs Regression

Correlation	Regression
Measures strength of relationship	Predicts values
Symmetric: r(X,Y) = r(Y,X)	Asymmetric: different equations for Y from X vs X from Y
No dependent/independent	Has dependent (Y) and independent (X)
Single number	Equation (slope, intercept)

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Summary

In this lesson, you learned:

• Pearson r measures linear relationship strength (-1 to +1)
• r² = proportion of variance explained
• Spearman correlation for ordinal or non-normal data
• Always visualize with scatter plots
• Correlation ≠ causation — beware third variables
• Outliers and restricted range affect correlation
• Hypothesis test: t = r√(n-2) / √(1-r²)

Practice Problems

1. For data with r = 0.6 and n = 30: a) Calculate r² b) Test if the correlation is significant at α = 0.05

2. Two variables have r = 0.9. If we remove an outlier, r drops to 0.4. What does this suggest?

3. Rank correlation data:

Item	Ranking A	Ranking B
1	1	1
2	2	3
3	3	2
4	4	4

Calculate Spearman correlation.

4. Why might two variables have a strong relationship but r ≈ 0?

Click to see answers

1. a) r² = 0.36 or 36% of variance explained b) t = 0.6√28 / √0.64 = 0.6(5.29)/0.8 = 3.97 df = 28, critical t ≈ 2.05 Since 3.97 > 2.05, significant

2. The original high correlation was driven by the outlier. Without it, the relationship is only moderate. This shows the importance of checking for outliers!

3. d values: 0, -1, 1, 0; d² values: 0, 1, 1, 0; Sum = 2

$r_s = 1 - \frac{6(2)}{4(16-1)} = 1 - \frac{12}{60} = 1 - 0.2 =$ 0.8

4. The relationship might be nonlinear (curved). Pearson r only measures linear relationships. A U-shaped or inverted-U pattern would show r ≈ 0 despite a clear pattern. Always plot your data!

Next Steps

Continue with regression analysis:

• Linear Regression - Predicting values from relationships
• Multiple Regression - Multiple predictors
• Histogram Generator - Visualize your data