discussion response (respond to the post)

DISCUSSION POST 6

Nicholas Townsend – L31362956

Graduate School of Business, Liberty University

BUSI820: Quantitative Research Methods

Dr. Mark Ellis

06/28/2026

We use scatterplots and regression lines to see the relationship between two variables before relying solely on statistics. A scatterplot gives a visual picture of whether the relationship appears positive, negative, weak, strong, linear, or possibly curved. It can also show outliers or unusual cases that may affect the correlation or regression results. This matters because a correlation coefficient can summarize the relationship, but it does not show the full pattern of the data on its own.

A regression line adds another layer by showing the general direction of the relationship and how well the data points follow a linear trend. If the points fall closely around the line, the relationship is stronger. If the points are spread far from the line, the relationship is weaker. Barrett et al. (2025) explain that scatterplots and regression lines are useful because they help researchers visually inspect the data before interpreting correlation and regression results. Erdely and Rubio-Sánchez (2025) also emphasized that visual tools are important when examining relationships between variables because they can reveal patterns that may be missed by only looking at numerical summaries.

Part A: What do the correlation coefficients tell us? The correlation coefficients tell us the direction and strength of the relationship between mother’s education and math achievement. In Output 8.2, the Pearson correlation is r = .338, and the Spearman correlation is rs = .315. Both are positive, meaning that as mother’s education increases, math achievement also tends to increase. The relationship is not extremely strong, but it is meaningful enough to pay attention to. In practical terms, students whose mothers had higher levels of education tended to have somewhat higher math achievement scores.

Part B: What is r² for the Pearson correlation? What does it mean? The Pearson correlation is .338, so r² is calculated by squaring the correlation: .338 × .338 = .114, or about 11.4%. This means that approximately 11.4% of the variance in math achievement can be explained by mother’s education. That also means most of the variance is still explained by other factors. So, while mother’s education is related to math achievement, it is not the only factor influencing student performance.

Part C: Compare the Pearson and Spearman correlations on both correlation size and significance level. The Pearson and Spearman correlations are similar in size. Pearson’s r is .338, while Spearman’s rho is .315. Both are positive, and both suggest a modest relationship between mother’s education and math achievement. The significance levels are also close. Pearson is statistically significant at p = .003, and Spearman is statistically significant at p = .006. Since both p-values are below .01, both results are statistically significant. This means the relationship is unlikely to be due to random chance alone.

Part D: When should you use which type in this case? Pearson correlation is best when both variables are scale variables and the relationship is linear and reasonably normal. Spearman correlation is better when the data are ordinal, skewed, or when the assumptions for Pearson are not fully met. In this case, mother’s education is an ordinal variable and is also skewed, so Spearman is the better choice. Pearson is still useful to compare, but Spearman fits the measurement level and data pattern better here. This is one of those places where the researcher has to slow down and choose the statistic that matches the data instead of just using the default SPSS output.

The standardized regression weights, or beta coefficients, tell us the relative strength of each predictor in predicting the dependent variable. In this case, the dependent variable is math achievement. The benefit of standardized coefficients is that they put the predictors on the same scale, which makes them easier to compare. A larger absolute beta value means that predictor contributes more strongly to predicting math achievement, while a smaller beta value means the predictor contributes less.

In Output 8.5, the academic track appears to be the strongest predictor of math achievement because it has the largest standardized coefficient. This suggests that whether a student is in the fast track or regular track is a meaningful predictor of math achievement. The other predictors, such as grades in high school, father’s education, and mother’s education, may still contribute, but they do not appear to predict math achievement as strongly as academic track. Roustaei (2024) explains that standardized regression coefficients are helpful because they show which independent variables have the strongest linear relationship with the dependent variable. However, the unstandardized coefficients are still better when the goal is to write the actual prediction equation.

Incorporating a biblical perspective reminds me that research should be conducted with wisdom and integrity. Proverbs 2:6 says, “For the Lord gives wisdom; from his mouth come knowledge and understanding” (New International Version, 2011). This aligns well with correlation and regression, as numbers alone do not create understanding. A researcher has to know what the numbers mean, when to use Pearson or Spearman, and how to explain regression coefficients honestly. Proverbs 9:10 also states, “The fear of the Lord is the beginning of wisdom” (New International Version, 2011). To me, this means that research should not be used carelessly or to overstate a result. If a correlation is modest, I should describe it as modest. If a predictor is stronger than another, I should explain that clearly without stretching the conclusion.

Respectfully,

Nick Townsend

Barrett, K. C., Leech, N. L., Gloeckner, G. W., & Morgan, G. A. (2025). IBM SPSS for introductory statistics: Use and interpretation (7th ed.). Routledge.

Erdely, A., & Rubio-Sánchez, M. (2025). D-plots: Visualizations for analysis of bivariate dependence between continuous random variables. Stats, 8(2), Article 43. https://doi.org/10.3390/stats8020043

Roustaei, N. (2024). Application and interpretation of linear regression analysis. Medical Hypothesis, Discovery & Innovation in Ophthalmology, 13(3), 151–159. https://doi.org/10.51329/mehdiophthal1506

The Holy Bible: New International Version. (2011). Zondervan. (Original work published 1978).