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Question 1

Skill question

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.

Question 2

Skill question

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.

Question 3

For the dataset $\{(1,3), (3,5), (4,7), (6,10), (7,12), (9,200)\}$ , which includes an outlier:

(a) calculate the Pearson product-moment correlation coefficient, $r$ ;

(b) calculate the Spearman's rank correlation coefficient, $r_s$ ;

(c) compare the two values and comment on the effect of the outlier $(9, 200)$ .

[6]

Question 4

Calculate the Pearson correlation coefficient $r$ for the following dataset, which includes an extreme outlier:

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

[5]

Question 5

Calculate the percentage change in the Pearson correlation coefficient when the outlier $(9, 200)$ is removed, using $r_{\text{with}} \approx 0.706$ and $r_{\text{without}} \approx 0.994$ .

[3]

Question 6

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 $r$ .

[4]

Question 7

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

[4]

Question 8

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

[4]

Question 9

Calculate the Pearson correlation coefficient $r$ for the dataset after removing the outlier: $(1, 3), (3, 5), (4, 7), (6, 10), (7, 12)$ Give your answer to three significant figures.

[4]

Question 10

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

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

[4]

Question 11

A researcher calculates the Pearson correlation coefficient, $r$ , for three datasets based on the same nine base observations, but with different treatments of a tenth point $(x, y)$ .

The calculated values of $r$ for each scenario are: (a) including the point $(9, 200)$ , $r \approx 0.706$ (b) including the point $(9, 20)$ , $r \approx 0.961$ (c) excluding the tenth point entirely, $r \approx 0.994$

Compare the Pearson correlation coefficients for the three datasets and comment on how each treatment of the outlier affects the value of $r$ .

[4]

Question 12

This question assesses the calculation of Spearman's rank correlation coefficient and the understanding of its robustness against outliers compared to the Pearson product-moment correlation coefficient.

1.

A student collects bivariate data to investigate the relationship between two variables, $X$ and $Y$ . The following dataset is obtained:

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

Calculate the Spearman's rank correlation coefficient, $r_s$ , for this dataset.

[4]

2.

Comment on the effect of the outlier $(9, 200)$ on the value of $r_s$ and explain why $r_s$ might be a more appropriate measure of correlation for this dataset than the Pearson product-moment correlation coefficient, $r$ .

[2]

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