Spearman’s Rank Correlation Coefficient is a method used to measure the strength and direction of the association between two ranked variables. It is particularly useful when data is non-parametric, qualitative, or does not meet the assumptions required by Pearson's correlation method. It ranks the data and then finds the correlation between the ranks, rather than the raw values themselves.
Why Study Spearman's Rank Correlation?
- Suitable for ordinal data and subjective measures such as preferences or satisfaction levels.
- Does not assume normal distribution of variables.
- Simple and intuitive for small datasets or datasets with ranks.
- Widely used in social sciences, psychology, education, and market research.
Rank Correlation: Handling Non-Parametric Data
In many practical scenarios, especially in behavioral sciences, variables are not measured numerically but by rank orders (e.g., 1st, 2nd, 3rd). In such cases, Spearman’s Rank Correlation is more appropriate than Pearson’s method.
Formula of Spearman's Rank Correlation
ρ (rho) = 1 - [ (6 × ΣD²) / (n × (n² - 1)) ]
Where:
- ρ = Spearman’s rank correlation coefficient
- D = Difference between the two ranks of each observation
- n = Number of observations
Step-by-Step Procedure
- Rank each variable separately.
- Calculate the difference (D) between the ranks of each observation.
- Square each difference (D²).
- Sum up all the D² values.
- Apply the formula to calculate ρ.
Treatment of Tied Ranks
When two or more values are the same (i.e., tied), they are given the average of the ranks they occupy.
Example: If two values are tied for 2nd and 3rd place, each gets a rank of (2 + 3)/2 = 2.5.
Adjustment in Formula for Ties:
The presence of tied ranks requires a correction factor. The general practice is to compute corrected ranks and proceed with the normal formula. However, if ties are excessive, Kendall’s Tau may be more appropriate.
Interpretation of Spearman's Coefficient (ρ)
- ρ = +1: Perfect positive correlation (ranks increase together).
- ρ = -1: Perfect negative correlation (as one rank increases, the other decreases).
- ρ = 0: No correlation between ranks.
- Closer the value of ρ to ±1, stronger the association.
Example:
Problem: Calculate the rank correlation between the ranks given by two judges to 6 students.
| Student | Judge A | Judge B |
|---|---|---|
| A | 1 | 2 |
| B | 2 | 1 |
| C | 3 | 4 |
| D | 4 | 3 |
| E | 5 | 6 |
| F | 6 | 5 |
Solution:
| Student | Rank A (R₁) | Rank B (R₂) | D = R₁ - R₂ | D² |
|---|---|---|---|---|
| A | 1 | 2 | -1 | 1 |
| B | 2 | 1 | 1 | 1 |
| C | 3 | 4 | -1 | 1 |
| D | 4 | 3 | 1 | 1 |
| E | 5 | 6 | -1 | 1 |
| F | 6 | 5 | 1 | 1 |
| Total ΣD² | 6 | |||
Apply the formula:
ρ = 1 - [ (6 × 6) / (6 × (36 - 1)) ] = 1 - (36 / 210) = 1 - 0.1714 = 0.8286
Interpretation: There is a strong positive correlation between the rankings of the two judges.
Application of Spearman's Rank Correlation
- Used in education to compare rankings of students given by different teachers.
- Applied in psychology to relate behavior rankings across contexts.
- Market research: Consumer product preferences ranked across demographic groups.
- Useful in any subjective assessment scenario involving ranks or orders.
Merits and Demerits of Spearman’s Rank Correlation
Merits:
- Simple to understand and calculate.
- Ideal for small samples and non-parametric data.
- Does not require interval scale or normal distribution.
- Can handle qualitative and subjective data effectively.
Demerits:
- Not suitable for large data sets with many tied ranks.
- Does not reflect linearity or strength in precise numerical terms like Pearson’s method.
- Loss of information due to conversion to ranks from actual values.
Clues for Direction and Strength
- Positive Correlation: Ranks rise together; lines or dots in scatterplot slope upward.
- Negative Correlation: One rank increases while the other decreases; downward trend in scatterplot.
- Weak Correlation: Scattered ranks with no discernible pattern.
- Perfect Correlation: Identical ranking patterns between variables.
Spearman vs. Pearson
| Basis | Spearman's Rank | Pearson's Correlation |
|---|---|---|
| Data Type | Ordinal / Ranked | Interval / Ratio |
| Distribution Assumption | Non-parametric | Assumes normal distribution |
| Sensitivity | Less affected by outliers | Highly affected by outliers |
| Interpretation | Based on ranks | Based on actual values |
Conclusion
Spearman's Rank Correlation is a powerful and accessible method for measuring the association between two ranked variables. It is particularly effective when dealing with non-parametric or ordinal data and is widely applicable in educational, psychological, and social research settings. Its simplicity, flexibility, and practical relevance make it a vital tool for students of business statistics and an essential concept for UGC NET Commerce exam preparation.