Understanding Pearson Correlation Coefficient

Oct 13, 2025 by sorusgentrification.com 42 views

**Which of the following Pearson correlations shows the greatest strength or consistency of relationship?**

Understanding Pearson Correlation Coefficient

The Pearson correlation coefficient is a statistical measure that calculates the strength and direction of the linear relationship between two continuous variables. It is denoted by the symbol 'r' and ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, while a value of -1 indicates a perfect negative linear relationship. A value of 0 indicates no linear relationship between the variables.

Calculating the Strength of the Relationship

The strength of the relationship between two variables can be calculated using the absolute value of the Pearson correlation coefficient. The closer the absolute value of 'r' is to 1, the stronger the relationship between the variables. Conversely, the closer the absolute value of 'r' is to 0, the weaker the relationship between the variables.

Comparing the Options

Let's compare the options given to determine which one shows the greatest strength or consistency of relationship.

A. $r=-0.90$

This option shows a strong negative linear relationship between the variables. The absolute value of 'r' is 0.90, which is close to 1. This indicates a strong relationship between the variables.

B. $r=-0.33$

This option shows a weak negative linear relationship between the variables. The absolute value of 'r' is 0.33, which is far from 1. This indicates a weak relationship between the variables.

C. $r=+0.85$

This option shows a strong positive linear relationship between the variables. The absolute value of 'r' is 0.85, which is close to 1. This indicates a strong relationship between the variables.

D. $r=+0.74$

This option shows a moderate positive linear relationship between the variables. The absolute value of 'r' is 0.74, which is not as close to 1 as option C. This indicates a moderate relationship between the variables.

Conclusion

Based on the calculations, option C ( $r=+0.85$ ) shows the greatest strength or consistency of relationship between the variables.

Frequently Asked Questions

Q: What is the Pearson correlation coefficient?

A: The Pearson correlation coefficient is a statistical measure that calculates the strength and direction of the linear relationship between two continuous variables.

Q: What does the value of 'r' represent?

A: The value of 'r' represents the strength and direction of the linear relationship between two variables. A value of 1 indicates a perfect positive linear relationship, while a value of -1 indicates a perfect negative linear relationship. A value of 0 indicates no linear relationship between the variables.

Q: How is the strength of the relationship calculated?

A: The strength of the relationship between two variables can be calculated using the absolute value of the Pearson correlation coefficient. The closer the absolute value of 'r' is to 1, the stronger the relationship between the variables.

Q: What is the difference between a strong and weak relationship?

A: A strong relationship is indicated by an absolute value of 'r' close to 1, while a weak relationship is indicated by an absolute value of 'r' close to 0.

Q: Can the Pearson correlation coefficient be used to determine causality?

A: No, the Pearson correlation coefficient cannot be used to determine causality. It only measures the strength and direction of the linear relationship between two variables.

Q: What are the limitations of the Pearson correlation coefficient?

A: The Pearson correlation coefficient assumes a linear relationship between the variables and is sensitive to outliers. It also does not account for non-linear relationships or non-normal distributions.

Q: Can the Pearson correlation coefficient be used with categorical variables?

A: No, the Pearson correlation coefficient is used with continuous variables. For categorical variables, other measures such as the chi-squared test or the phi coefficient can be used.

Q: How is the Pearson correlation coefficient calculated?

A: The Pearson correlation coefficient is calculated using the following formula:

r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])

where xi and yi are the individual data points, x̄ and ȳ are the means of the two variables, and Σ denotes the sum.