Spearman’s Rank Correlation: Distinct & Tied Ranks
1. What Is Spearman’s Rank Correlation?
Formulated in 1904 by British psychologist and statistician Charles Edward Spearman, the Rank Correlation Coefficient (denoted by R or the Greek letter ρ, read as “Rho”) measures the strength and direction of association between two ranked variables.
When Is Spearman’s Method Preferred Over Pearson’s?
- Qualitative Attributes: When characteristics cannot be measured with numerical meters or scales — such as beauty, honesty, employee morale, leadership ability, singing talent, or consumer taste preferences.
- Ranked Data: When respondents simply assign rank orders (1st, 2nd, 3rd…).
- Extreme Outliers: When data contains severe outliers that would distort Karl Pearson’s product moment calculations.
2. Case 1: When Ranks Are Distinct (No Repeated Ranks)
When every item has a unique rank without any ties, Spearman’s famous formula applies:
Step-by-Step Procedure:
- Identify the rank given to variable
X(R1) and variableY(R2). If raw values are given, assign Rank 1 to the highest value, Rank 2 to the second highest, and so on. - Calculate the rank difference for each pair:
d = R1 - R2. - Square each difference:
d². - Calculate the sum:
Σd². (Note: As a verification check,Σdmust ALWAYS equal zero!) - Substitute into the master formula.
Solved Numerical Example (Activity 1: Height vs. Shoe Size)
Problem: In a BBA classroom activity, 5 students measured their Height (X in cm) and Shoe Size (Y in Indian sizes):
| Student | Height (X) | Shoe Size (Y) |
|---|---|---|
| A | 170 | 8 |
| B | 165 | 7 |
| C | 180 | 10 |
| D | 160 | 6 |
| E | 175 | 9 |
Solution Table:
Assigning ranks (Rank 1 to highest value):
- Heights: 180 (Rank 1), 175 (Rank 2), 170 (Rank 3), 165 (Rank 4), 160 (Rank 5).
- Shoe Sizes: 10 (Rank 1), 9 (Rank 2), 8 (Rank 3), 7 (Rank 4), 6 (Rank 5).
| Student | Height Rank (R1) | Shoe Rank (R2) | d = R1 - R2 | d² |
|---|---|---|---|---|
| A | 3 | 3 | 0 | 0 |
| B | 4 | 4 | 0 | 0 |
| C | 1 | 1 | 0 | 0 |
| D | 5 | 5 | 0 | 0 |
| E | 2 | 2 | 0 | 0 |
| Total | — | — | Σd = 0 | Σd² = 0 |
Calculation:
N = 5,Σd² = 0R = 1 - [ (6 × 0) ÷ (5 × (25 - 1)) ]
R = 1 - 0 = +1.0- Interpretation: There is a Perfect Positive Rank Correlation (R = +1.0) between height and shoe size in this group!
3. Case 2: When Ranks Are Repeated (Tied Ranks)
When two or more observations have the exact same numerical value, they cannot be given different ranks.
The Average Rank Rule:
If two students tie for the 3rd and 4th position, each is awarded the average rank:
Average Rank = (3 + 4) ÷ 2 = 3.5!
The Correction Factor:
For every tie, a correction factor of [ m × (m² - 1) ] ÷ 12 is added to Σd² (where m is the number of times that specific value repeats).
Solved Example with Tied Ranks:
Suppose N = 5, Σd² = 8, and the score 40 appeared twice (m = 2):
- Compute Correction Factor:
CF = [ 2 × (2² - 1) ] ÷ 12 = [ 2 × 3 ] ÷ 12 = 6 ÷ 12 = 0.5 - Adjusted Sum:
Adjusted Σd² = 8 + 0.5 = 8.5 - Apply Formula:
R = 1 - [ 6 × 8.5 ] ÷ [ 5 × (25 - 1) ]
R = 1 - [ 51 ÷ 120 ] = 1 - 0.425 = +0.575
- Conclusion: A moderate positive rank correlation of +0.575.
4. Merits & Demerits of Spearman’s Rank Correlation
Merits:
- The only available method when data is qualitative (judges’ scores, brand loyalty rankings, interviews).
- Much simpler and quicker to compute than Karl Pearson’s covariance formula.
- Not distorted by extreme wild numbers because values are converted to uniform ordinal ranks (1, 2, 3…).
Demerits:
- Cannot be applied to grouped continuous frequency distributions.
- Slightly less mathematically efficient than Pearson’s
rwhen dealing with precise, normal quantitative measurements.