The spearman’s rank coefficient of correlation was developed by Charles Edward Spearman.

Let us suppose that a group of n individuals is arranged in order of merit or proficiency in possession of two characteristics A and B. these ranks in two characteristics will, in general, be different. For example, if we consider the relation between intelligence and beauty, it is not necessary that a beautiful individual is intelligent also. Let (xi , yi ); i=1,2,………,n be the ranks of the i^{th} individual in two characteristics A and B respectively. Pearsonian coefficient of correlation between the ranks xi’s and yi ‘s is called the Spearman rank correlation coefficient between A and B for that group of individuals.

rk = 1 – [6 ∑D^{2} / N^{3} – N]

Where D = R_{1} – R_{2}, between the paired items in the two rank series.

The value of rank correlation coefficient tells us about the degree of agreement between the 2 ranks.

Assuming that no two individuals are bracketed equal in either classification, each of the variables X and Y takes the values 1,2,………..,n

Hence

Which is the spearman's formula for the rank correlation coefficient.

-1 ≤ r_{k} ≤ +1

Rank Correlation Coefficient Example

Calculate the Rank Correlation Coefficient in each of the following cases:

X | R_{1} |
Y | R_{2} |
---|---|---|---|

10 | 1 | 5 | 1 |

20 | 2 | 6 | 2 |

30 | 3 | 7 | 3 |

To calculate the rank correlation coefficient, first we will determine the value of D = R_{1} – R_{2} in each of the entries:

X | R_{1} |
Y | R_{2} |
D | D^{2} |
---|---|---|---|---|---|

10 | 1 | 5 | 1 | 0 | 0 |

20 | 2 | 6 | 2 | 0 | 0 |

30 | 3 | 7 | 3 | 0 | 0 |

Then the Spearman’s rank correlation coefficient is calculated using the formula as:

rk = 1 – [6 ∑D^{2} / N^{3} – N]

= 1- 6(0)

= +1

Thus the value of rank correlation coefficient equal to +1 implies that there is complete agreement in the order of ranks and the ranks are in the same direction.

Let us calculate the rank correlation coefficient in another example:

X | R_{1} |
Y | R_{2} |
---|---|---|---|

10 | 1 | 7 | 3 |

20 | 2 | 6 | 2 |

30 | 3 | 5 | 1 |

To calculate the rank correlation coefficient, first we will determine the value of D = R1 – R2 in each of the entries:

X | R_{1} |
Y | R_{2} |
D | D^{2} |
---|---|---|---|---|---|

10 | 1 | 7 | 3 | -2 | 4 |

20 | 2 | 6 | 2 | 0 | 0 |

30 | 3 | 5 | 1 | +2 | 4 |

Then the Spearman’s rank correlation coefficient is calculated using the formula as:

r_{k} = 1 – [6 ∑D^{2} / N^{3} – N]

= 1 – (6*8)/9-3

= -1

Thus the value of rank correlation coefficient equal to -1 implies that there is complete agreement in the order of ranks and the ranks are in opposite direction

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- Bivariate Normal Distribution
- Multiple And Partial Correlation
- Plane Of regression
- Properties If residuals
- Coefficient Of Multiple Correlation
- Properties of Multiple Correlation coefficient
- Coefficient Of Partial Correlation
- Multiple correlation In terms Of Total And Partial Correlation
- Coefficient In Terms Of Regression
- Expression For Partial Correlation

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