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.
Spearman's Rank Correlation 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.
Spearman's Rank Correlation Coefficient Calculator
Spearman's Rank correlation coefficient is used to measure the strength of association or relationship between two variables.
| Spearman's Rank Correlation Coefficient Calculator | ||
|---|---|---|
| Data 1 : X | Data 2 : Y | |
| Enter Data (Separated by comma ,) | ||
| Results | ||
| Number of Observations (n): | ||
| Ranks for X: | ||
| Ranks for Y: | ||
| Spearman's Rank Correlation Coefficient: ($\rho_{xy}$) | ||
How to calculate Spearman's Rank Correlation Coefficient?
Step 1 - Enter the $X$ values separated by commas
Step 2 - Enter the $Y$ values separated by commas
Step 3 - Click calculate button to find spearman rank correlation coefficient
Step 4 - Gives the number of pairs of observations
Step 5 - Gives the Rank for $X$
Step 6 - Gives the Rank for $Y$
Step 7 - Gives the sample Spearman's Rank correlation coefficient.
Example 1 - Find Spearman's Rank correlation coefficient
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 spearman 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.
Conclusion
In this tutorial, you learned to find Spearman's rank correlation coefficient with steps by steps example. You also learned about how to interpret the Spearman's rank correlation coefficient.
To learn more about other correlation and regression, please refer to the following tutorials:
Let me know in the comments if you have any questions on Spearman's rank correlation coefficient calculator with examples and your thought on this article.
