Spearman's Rank Correlation Coefficient
- Spearman's rank correlation coefficient, denoted as $r_s$, is a non-parametric measure of rank correlation between two variables.
- It assesses how well the relationship between two variables can be described using a monotonic function, without making any assumptions about the frequency distribution of the variables.
Calculation of $r_s$
The formula for Spearman's rank correlation coefficient is:
$$ r_s = 1 - \frac{6\sum d_i^2}{n(n^2-1)} $$
Where:
- $d_i$ is the difference between the ranks of corresponding values
- $n$ is the number of pairs of values
In practice, students are expected to use technology to calculate $r_s$ rather than performing manual calculations.
Handling Tied Ranks
When two or more data points have the same value, they are assigned the average of the ranks they would have received if they had been distinct.
ExampleIf we have the data set: 7, 9, 9, 10, 10, 11 The ranks would be: 1, 2.5, 2.5, 4.5, 4.5, 6
This method ensures that the sum of the ranks remains the same as it would be for untied data.
Comparison with Pearson's Correlation Coefficient
While both Spearman's and Pearson's correlation coefficients measure the strength and direction of a relationship between two variables, they have distinct characteristics:
- Linearity: Pearson's coefficient is specifically designed to detect linear relationships, while Spearman's can identify any monotonic relationship (including non-linear).
- Data type: Pearson's works with continuous variables, while Spearman's can be used with ordinal data.
- Outlier sensitivity: Spearman's is less sensitive to outliers compared to Pearson's.
Students often confuse when to use Pearson's vs. Spearman's correlation. Remember: Use Pearson's for linear relationships between continuous variables, and Spearman's for monotonic relationships or when working with ordinal data.
What Is an Outlier?
- An outlier is a data point that is significantly different from other observations in a dataset.
- Outliers can arise due to measurement errors, natural variability, or unusual conditions.
If most students in a class score between 60 and 80 on a test, but one student scores 20, that score is an outlier.
Effect of Outliers
Outliers can significantly impact correlation coefficients:
- Pearson's correlation: Highly sensitive to outliers, as it uses the actual values of the data points.
