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Spearman Rank Correlation is a statistical measure used to determine the strength and direction of the relationship between two variables based on their ranks rather than their actual values.
It is particularly useful when the relationship between variables is not necessarily linear or when the data is better represented by rankings.
This article explains what Spearman's correlation is, how it differs from the standard (Pearson) correlation, its formula, and how to calculate it.
Key Takeaways
- Spearman Rank Correlation measures the strength and direction of a monotonic relationship between two variables, not just a linear one.
- It works on the ranks of data points rather than their actual values, making it less sensitive to outliers.
- The coefficient ranges from −1 to +1, where values near +1 or −1 indicate a strong relationship and values near 0 indicate little to no consistent relationship.
- It is commonly used in investing to compare fund manager rankings, factor performance, or the consistency of an indicator’s predictive power over time.
- Unlike Pearson correlation, it doesn’t assume normally distributed data, making it more forgiving of real-world financial datasets that are often skewed.
What is Spearman Rank Correlation?
Spearman Rank Correlation, denoted as ρ (rho) or r_s, is a statistical measure that assesses how well the relationship between two variables can be described using a monotonic function.
Spearman’s method first converts raw data into ranks, then measures the correlation between those ranks. This ranking step is what makes it resistant to outliers and non-linear patterns that would distort a standard correlation calculation.
What Happens When Two Values Are Identical
When two or more data points have identical values (for example, two mutual funds returning exactly 8.5%), they cannot be given distinct consecutive ranks. Instead, they receive a fractional rank, the average of the ranks they would have occupied if they differed slightly.
Example: If two funds tie for the 3rd and 4th ranks, both are assigned an average rank of 3.5 (calculated as (3 + 4) / 2), and the next unique value receives rank 5. When ties are present, a modified formula that includes a correction factor for ties is technically required to keep the coefficient accurate.
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Spearman vs Pearson Correlation
| Feature | Spearman Rank Correlation | Pearson Correlation |
| Measures | Monotonic relationship | Linear relationship |
| Uses | Ranks of data | Raw data values |
| Sensitivity to outliers | Low | High |
| Assumes normal distribution | No | Yes (for hypothesis testing) |
| Best suited for | Skewed or non-linear data, ordinal rankings | Data with a roughly linear, normally distributed relationship |
What is the Formula for Spearman rank correlation?
ρ = 1 − (6 × Σd² / n(n² − 1))
| Symbol | Meaning |
| ρ | Spearman rank correlation coefficient |
| d | Difference between the ranks of corresponding values in the two datasets |
| Σd² | Sum of the squared rank differences |
| n | Number of data pairs (observations) |
The result, ρ, always falls between −1 and +1:
|
Value of ρ |
Interpretation |
|
+1 |
Perfect positive monotonic relationship |
|
0 |
No consistent monotonic relationship |
|
−1 |
Perfect negative monotonic relationship |
|
0.7 to 0.99 (or −0.7 to −0.99) |
Strong relationship |
|
0.3 to 0.69 (or −0.3 to −0.69) |
Moderate relationship |
|
Below 0.3 (or above −0.3) |
Weak or negligible relationship |
How to Calculate Spearman Rank Correlation?
| Month | Fund Return Rank | Benchmark Return Rank | d (difference) | d² |
| 1 | 2 | 1 | 1 | 1 |
| 2 | 1 | 2 | −1 | 1 |
| 3 | 4 | 3 | 1 | 1 |
| 4 | 3 | 4 | −1 | 1 |
| 5 | 6 | 5 | 1 | 1 |
| 6 | 5 | 6 | −1 | 1 |
Σd² = 1+1+1+1+1+1 = 6
n = 6
ρ = 1 − (6 × 6) / (6 × (36 − 1))
ρ = 1 − 36 / 210
ρ = 1 − 0.171
ρ ≈ 0.83
A value of 0.83 indicates a strong positive relationship. Investors should make a note of the fact that the fund’s return rankings closely track the benchmark’s rankings month to month, even if the exact return values differ.
Statistical Significance
When working with small datasets, such as a limited number of monthly observations or a short track record, a high or low Spearman coefficient may occur by chance due to random market noise. Analysts can use a p-value or t-statistic to assess whether the observed relationship is statistically significant.
t = ρ × √[(n − 2) / (1 − ρ²)]
Here, ρ (rho) represents the Spearman correlation coefficient, while n represents the number of observations.
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Where Spearman Correlation is Used in Investing?
- Comparing fund manager rankings: Checking whether a fund consistently ranks well against peers across different time periods.
- Factor investing: Testing whether a factor (like momentum or value) consistently ranks stocks in a way that correlates with future returns.
- Backtesting indicators: Measuring whether an indicator’s signal-strength ranking consistently aligns with actual price outcomes, even if the relationship isn’t perfectly linear.
- Portfolio diversification checks: Assessing whether two assets tend to move in the same relative direction without assuming a strict linear link.
| Advantages | Disadvantages |
| Captures monotonic relationships, not just straight-line ones, making it more flexible than Pearson correlation | Only measures monotonic relationships. Misses complex patterns where variables rise then reverse direction |
| Less sensitive to outliers, since it works on ranks rather than raw values | Loses some information by converting raw values into ranks, which can obscure the actual magnitude of differences |
| Doesn't require normally distributed data, making it suitable for skewed financial datasets | Can become unstable with small sample sizes, where a single rank change may swing the coefficient significantly |
| Works well with ordinal data (rankings, ratings) as well as continuous data | Requires an adjusted formula when there are tied ranks, adding a layer of complexity |
| Simple to calculate and interpret, with a clear −1 to +1 scale | Does not indicate causation. A strong coefficient shows consistent ranking, not that one variable drives the other |
| Useful for comparing consistency across fund rankings, factor models, or indicator signals | Less powerful than Pearson correlation when the underlying relationship genuinely is linear and data is well-behaved |
Conclusion
Spearman Rank Correlation gives investors a way to measure how consistently two variables move together in rank order, even when their relationship isn’t a straight line. Its resistance to outliers and lack of a normal-distribution assumption make it a practical tool for real-world financial data, which is often messier than textbook examples suggest.
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