Using Pearson’s Product Moment Correlation for the Given Data
Introduction
Pearson’s product-moment correlation coefficient (r) is a statistical tool used to measure the strength and direction of the linear relationship between two continuous variables. In psychology and social sciences, it is often used to determine how closely two variables are related, such as test scores and performance. The coefficient ranges from -1 to +1, where +1 indicates perfect positive correlation, -1 indicates perfect negative correlation, and 0 indicates no correlation. In this task, we will calculate Pearson’s r for two datasets.
Given Data
Data 1 (X): 24, 23, 26, 25, 25, 21, 25, 26, 25, 26
Data 2 (Y): 12, 15, 22, 13, 14, 11, 16, 10, 19, 20
Step-by-Step Calculation
Pearson’s formula is:
r = Σ[(X – Mx)(Y – My)] / √[Σ(X – Mx)² * Σ(Y – My)²]
Step 1: Calculate the Means
Mean of X: (24+23+26+25+25+21+25+26+25+26) = 246 / 10 = 24.6
Mean of Y: (12+15+22+13+14+11+16+10+19+20) = 152 / 10 = 15.2
Step 2: Create a Table
| X | Y | X-Mx | Y-My | (X-Mx)(Y-My) | (X-Mx)² | (Y-My)² |
|---|---|---|---|---|---|---|
| 24 | 12 | -0.6 | -3.2 | 1.92 | 0.36 | 10.24 |
| 23 | 15 | -1.6 | -0.2 | 0.32 | 2.56 | 0.04 |
| 26 | 22 | 1.4 | 6.8 | 9.52 | 1.96 | 46.24 |
| 25 | 13 | 0.4 | -2.2 | -0.88 | 0.16 | 4.84 |
| 25 | 14 | 0.4 | -1.2 | -0.48 | 0.16 | 1.44 |
| 21 | 11 | -3.6 | -4.2 | 15.12 | 12.96 | 17.64 |
| 25 | 16 | 0.4 | 0.8 | 0.32 | 0.16 | 0.64 |
| 26 | 10 | 1.4 | -5.2 | -7.28 | 1.96 | 27.04 |
| 25 | 19 | 0.4 | 3.8 | 1.52 | 0.16 | 14.44 |
| 26 | 20 | 1.4 | 4.8 | 6.72 | 1.96 | 23.04 |
Step 3: Summation
Σ(X – Mx)(Y – My) = 26.8
Σ(X – Mx)² = 22.4
Σ(Y – My)² = 145.6
Step 4: Plug into Pearson’s Formula
r = 26.8 / √(22.4 * 145.6) = 26.8 / √3261.44 = 26.8 / 57.11 ≈ 0.47
Interpretation
The Pearson correlation coefficient is approximately 0.47, which suggests a moderate positive correlation between the two datasets. This implies that as the values in Data 1 increase, the values in Data 2 tend to increase as well, but not perfectly.
Conclusion
Pearson’s correlation is a valuable measure in psychological research, as it helps determine the strength and direction of relationships between continuous variables. In this example, the moderate positive correlation indicates a consistent but not strong relationship between the two variables. Researchers must consider the context and the significance of such correlations, especially when interpreting psychological behavior or outcomes.