Contents

## 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.

## 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 Corr. Coeff.: ($\rho_{xy}$) | ||

## 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 calculate 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.

## Spearman's Rank correlation coefficient 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.

## Conclusion

In this tutorial, you learned about the step by step procedure for calculating Spearman's rank correlation coefficient. 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.