Spearman’s Rank correlation Coefficient
Let $(x_1, y_1), (x_2, y_2), \cdots , (x_n, y_n)$ be the ranks of $n$ individuals in two characteristics $A$ and $B$ respectively.
Formula
Then the Spearman’s rank correlation coefficient is denoted by $\varrho$ and is given by
$\varrho = 1- \dfrac{6 \sum_{i=1}^{n}d_i^2}{n(n^2-1)}$
where,
$d_i = x_i - y_i$is the difference between the pairs of ranks of the $i^{th}$ individual in the two characteristics and- $n$ is the number of pairs.
Rank correlation coefficient lies between -1 and +1. i.e. $-1 \leq \varrho \leq +1$.
- If $\varrho =0$, then there is no correlation between the ranks.
- If $\varrho >0$, then there is a positive correlation between the ranks.
- If $\varrho = 1$, then there is a perfect positive correlation between the ranks.
- If $0 <\varrho < 1$, then there is a partially positive correlation between the ranks.
- If $\varrho <0$, then there is a negative correlation between the ranks.
- If $\varrho = -1$, then there is a perfect negative correlation between the ranks.
- If $-1 <\varrho < 0$, then there is a partially negative correlation between the ranks.
Example 1
The scores given by two judges to 10 participants in a competition are as follows:
| Judge A | 30 | 29 | 30 | 47 | 45 | 36 | 47 | 37 | 25 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|
| Judge B | 31 | 32 | 29 | 46 | 43 | 32 | 46 | 34 | 26 | 45 |
Determine the rank correlation coefficient.
Solution
Let $x$ denote the scores by Judge A and $y$ denote the scores by Judge B.
Let $R_x$ denote the rank of $x$ and $R_y$ denote the rank of $y$.
| $x$ | $y$ | Rank of $x (R_x)$ | Rank of $y (R_y)$ | $d=R_x-R_y$ | d^2 | |
|---|---|---|---|---|---|---|
| 1 | 30 | 31 | 7.5 | 8 | -0.5 | 0.25 |
| 2 | 29 | 32 | 9 | 6.5 | 2.5 | 6.25 |
| 3 | 30 | 29 | 7.5 | 9 | -1.5 | 2.25 |
| 4 | 47 | 46 | 2 | 1.5 | 0.5 | 0.25 |
| 5 | 45 | 43 | 4 | 4 | 0 | 0 |
| 6 | 36 | 32 | 6 | 6.5 | -0.5 | 0.25 |
| 7 | 47 | 46 | 2 | 1.5 | 0.5 | 0.25 |
| 8 | 37 | 34 | 5 | 5 | 0 | 0 |
| 9 | 25 | 26 | 10 | 10 | 0 | 0 |
| 10 | 47 | 45 | 2 | 3 | -1 | 1 |
| Total | 10.5 |
The Spearman’s Rank correlation coefficient between the ranks of $x$ and $y$ is
$$ \begin{aligned} \varrho &= 1- \frac{6 \sum_{i=1}^{n}d_i^2}{n(n^2-1)}\\ &= 1-\frac{6 \times 10.5}{10(10^2-1)}\\ &= 1-\frac{63}{990}\\ &= 1- 0.0636364\\ &= 0.9894 \end{aligned} $$
The correlation coefficient between scores by Judge A and scores by Judge B is $0.9894$. Since the value of correlation coefficient is positive, there is a strong positive relationship between scores by Judge A and scores by Judge B.
