Question Type 4: Calculating correlation coefficients with and without outliers Exercises

Calculate the Spearman rank correlation coefficient $r_s$ for the dataset including the outlier:

$(1,3), (3,5), (4,7), (6,10), (7,12), (9,200).$

Calculate the Spearman rank correlation coefficient $r_s$ for the dataset after removing the outlier:

$(1,3), (3,5), (4,7), (6,10), (7,12).$

Calculate the Pearson correlation coefficient $r$ for the dataset after removing the outlier, i.e.

$(1,3), (3,5), (4,7), (6,10), (7,12).$

For the dataset without the outlier $(1,3), (3,5), (4,7), (6,10), (7,12)$ calculate both the Pearson correlation $r$ and the Spearman correlation $r_s$. Compare and comment on the two values.

Calculate the Pearson correlation coefficient $r$ for the dataset including the extreme outlier:

$(1,3), (3,5), (4,7), (6,10), (7,12), (9,200).$

Calculate the percentage change in the Pearson correlation coefficient when the outlier $(9,200)$ is removed, using $r_{\text{with}}\approx0.706$ and $r_{\text{without}}\approx0.994$.

Calculate and compare the Spearman correlation coefficient before and after removing the outlier. Quantify the change and discuss the impact of the outlier on rank-based association.

Calculate and compare the Pearson correlation coefficient before and after removing the outlier. Quantify the change and discuss the impact of the outlier on linear association.

For the dataset including the outlier $(1,3), (3,5), (4,7), (6,10), (7,12), (9,200)$ calculate both the Pearson correlation $r$ and the Spearman correlation $r_s$. Compare and comment on the two values.

Explain why the Spearman rank correlation coefficient is less sensitive to the extreme outlier than the Pearson correlation coefficient, using mathematical reasoning based on ranks vs. aw values.

Compare the Pearson correlation coefficients for the three datasets:

(a) with outlier $(9,200)$,
(b) with reduced outlier $(9,20)$,
(c) without any outlier.

Comment on how each change in the outlier affects $r$.

For the modified dataset where the outlier’s $y$-value is reduced to 20, i.e.

$(1,3), (3,5), (4,7), (6,10), (7,12), (9,20),$

calculate the Pearson correlation coefficient $r$.