Exam Questions - Spearman's Rank Correlation Coefficient – Further Maths ExamSolutions Maths Edexcel Revision

Exam Questions - Spearman’s Rank Correlation Coefficient

Spearman’s Rank Correlation Coefficient, represented as r_s, is a non-parametric measure of statistical dependence between two variables.
It assesses how well the relationship between two variables can be described using a monotonic function — a function between ordered sets that preserves or reverses order.
The rank correlation coefficient varies between -1 and 1, where -1 denotes a perfect negative monotonic correlation, 1 denotes a perfect positive monotonic correlation, and 0 signifies no correlation.
Spearman’s Rank should be used when the requirement of normality for Pearson’s correlation coefficient doesn’t hold or when dealing with ordinal variables.

The Spearman’s Rank Correlation Coefficient can be calculated with the following formula r_s = 1 - (6 Σd^2/n(n^2-1)), where ‘d’ refers to the differences between the paired ranks and ‘n’ refers to the number of observations.
A key step in calculating Spearman’s Rank is assigning ranks to each point in the data set. If there are tied ranks, consider assigning them the mean of the ranks they would cover.
To observe any correlation, plot the ranks on a scatter graph for a visual interpretation. If a linear relationship seems to emerge, Spearman’s Rank can be a practical method to quantify the correlation.

The closer the correlation coefficient is to either -1 or 1, the stronger the correlation between the ranks of the variables. A coefficient closer to 0 suggests a weaker correlation.
A positive coefficient signifies that as the rank of one variable increases, the rank of the other variable also increases. A negative coefficient indicates an inverse relationship.
A correlation coefficient of 0 implies there’s no linear relationship between ranks.

Like many statistical techniques, a null hypothesis is often proposed suggesting that there is no correlation between the two variables. This null hypothesis is challenged by the calculated rank correlation coefficient.
If the computed coefficient is sufficiently different from zero, the null hypothesis might be rejected, suggesting a possible correlation.
Using tables of critical values for Spearman’s Rank Correlation Coefficient, we can compare our calculated value to determine whether to accept or reject the null hypothesis.
A two-tailed test is often used when we are open to the correlation being negative or positive. A one-tailed test is used when we have reason to believe the correlation will be in one specific direction.

It is important to note that Spearman’s Rank Correlation Coefficient doesn’t indicate causation, only the strength and direction of monotonic relationships.
Extreme values, or outliers, can significantly impact Spearman’s Rank Correlation. Care should be taken when interpreting results if outliers are present.
Spearman’s rank correlation can’t adequately measure relationships that aren’t monotonic. For relationships that are linear but not monotonic, Pearson’s correlation coefficient may be more appropriate.