17.1.2 Spearman's Rank Correlation Coefficient

Spearman's Rank Correlation Coefficient:

Sometimes, instead of using raw data, it's more appropriate to analyze the ranking of the data, especially when comparing subjective judgments or when raw scores are inconsistent.

Calculate Spearman's Rank Correlation Coefficient:

Ranking the Data:

Using a consistent ranking system:

Contestant	1	2	3	4	5
Judge A (Rank)	3	4	5	1	2
Judge B (Rank)	1	2	3	5	4

Differences and Squared Differences:

Calculate the difference in ranks, $d$, and then the square of the differences, $d^2$:

Contestant	1	2	3	4	5
$d$	2	2	2	-4	-2
$d^2$	4	4	4	16	4

Total $\sum d^2 = :highlight[32]$.

Apply the Formula:

The formula for Spearman's Rank Correlation Coefficient $r_s$:

$$r_s = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}$$

Substituting the values:

$$\sum d^2 = 32 \Rightarrow r_s = 1 - \frac{6(32)}{5(5^2 - 1)} = :highlight[-0.6]$$

Conclusion:

• Conclude (using the word "agreement" rather than "correlation".
• There seems to be fairly strong disagreement between the two judges about the relative performance of the dancers.

Note that "no agreement" is different than "disagreement".

Calculating $r_s$ on a calculator:

Follow the method for '$r$' but simply input rankings.

Least Squares Regression Line for Coded Data

Coding is the transformation of data to make it easier to work with, read, and analyze.