Intro to Business Statistics: Correlation and Regression Spearman Rank Correlation rho

What This Is

The Spearman Rank Correlation (rho) is a non-parametric test used to measure the strength and direction of the relationship between two variables. It's particularly useful when the data doesn't meet the assumptions of a parametric test, such as normality or equal variances. A retail chain wants to know if the average daily sales of its online store are related to the average daily website traffic. They collect data on sales and website traffic for 10 consecutive days and want to determine if there's a significant correlation between the two variables.

Key Formulas & Symbols

• ρ = 1 - (6 * Σd²) / (n² - 1) where ρ = Spearman Rank Correlation Coefficient, d = difference between ranks, n = sample size.
• ρ measures the strength and direction of the relationship between two variables.
• d is the difference between the ranks of the two variables.
• Σd² is the sum of the squared differences between the ranks.
• n is the sample size.
• ρ ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.
• ρ is a non-parametric test, meaning it doesn't assume a normal distribution or equal variances.
• ρ is sensitive to outliers and non-linear relationships.
• ρ is often used in business to measure the relationship between variables such as sales and advertising, or customer satisfaction and loyalty.

Step-by-Step Procedure

1. State hypotheses: H₀: ρ = 0 (no correlation), H₁: ρ ≠ 0 (correlation exists).
2. Choose test: Spearman Rank Correlation (rho) test.
3. Compute test statistic: Calculate ρ using the formula ρ = 1 - (6 * Σd²) / (n² - 1).
4. Find p-value or critical value: Use a Spearman Rank Correlation table or calculator to find the p-value or critical value.
5. Compare to α: Compare the p-value to α (default = 0.05) or compare the test statistic to the critical value.
6. Conclude: If p-value < α or test statistic > critical value, reject H₀ and conclude that a significant correlation exists.

Common Mistakes

• Mistake: Misinterpreting ρ as a probability.
• Correction: ρ is a correlation coefficient, not a probability. It measures the strength and direction of the relationship between two variables.
• Mistake: Failing to check for non-linear relationships.
• Correction: ρ is sensitive to non-linear relationships, so it's essential to check for non-linear relationships before interpreting the results.
• Mistake: Ignoring outliers.
• Correction: ρ is sensitive to outliers, so it's essential to check for outliers before interpreting the results.

Quick Practice Problems

1. A marketing firm wants to know if the average daily website traffic is related to the average daily sales of its online store. They collect data on website traffic and sales for 15 consecutive days. What is the Spearman Rank Correlation Coefficient (ρ)?

ρ = 0.75

Explanation: The marketing firm collected data on website traffic and sales for 15 consecutive days and calculated the Spearman Rank Correlation Coefficient (ρ) using the formula ρ = 1 - (6 * Σd²) / (n² - 1).

1. A retail chain wants to know if the average daily sales of its online store are related to the average daily website traffic. They collect data on sales and website traffic for 10 consecutive days. What is the p-value of the Spearman Rank Correlation test?

p-value = 0.01

Explanation: The retail chain collected data on sales and website traffic for 10 consecutive days and calculated the p-value of the Spearman Rank Correlation test using a Spearman Rank Correlation table or calculator.

1. A company wants to know if the average daily customer satisfaction is related to the average daily sales. They collect data on customer satisfaction and sales for 20 consecutive days. What is the critical value of the Spearman Rank Correlation test?

Critical value = 1.725

Explanation: The company collected data on customer satisfaction and sales for 20 consecutive days and calculated the critical value of the Spearman Rank Correlation test using a Spearman Rank Correlation table or calculator.

Last-Minute Cram Sheet

1. ρ measures the strength and direction of the relationship between two variables.
2. ρ ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.
3. ρ is a non-parametric test, meaning it doesn't assume a normal distribution or equal variances.
4. ρ is sensitive to outliers and non-linear relationships.
5. ρ is often used in business to measure the relationship between variables such as sales and advertising, or customer satisfaction and loyalty.
6. ρ is calculated using the formula ρ = 1 - (6 * Σd²) / (n² - 1).
7. ρ is used to test the null hypothesis H₀: ρ = 0 (no correlation) against the alternative hypothesis H₁: ρ ≠ 0 (correlation exists).
8. ρ is compared to the critical value or p-value to determine if a significant correlation exists.
9. ρ is sensitive to sample size, so larger samples are preferred.
10. ρ is often used in conjunction with other statistical tests, such as regression analysis.