# The Difference between Spearmans rho and Pearsons r

**The Difference between Spear mans rho and Pearson**

**r**

Spearmans rho and Pearsons r differ in various ways as explained below. The Spear mans rho ranks the values for each variable then uses the ranks in computing the correlation while Pearsons use the values of the variables to calculate the correlation. Spear mans rho is affected by ties due to the need for ranking the values hence this is the major limitation for this method.

This limitation factor makes spearmans rho the appropriate method when the values of the variables have a wide range while for values with a small range Pearsons r is the appropriate method. When the relationship of the two variables is curvilinear Spear mans rho is more useful than Pearson r. This is because the correlation of the variables is underestimated by Pearson r thus weakening the relationship

Example: The following table shows the scores of five students in two exams.

Student | Exam 1 | Rank 1(r1) | Exam 2 | Rank 2 (r2) | di=r1-r2 | (di)2 |

1 | 56 | 1 | 86 | 4 | -3 | 9 |

2 | 78 | 3 | 58 | 2 | 1 | 1 |

3 | 95 | 5 | 85 | 3 | 2 | 4 |

4 | 65 | 2 | 95 | 5 | -3 | 9 |

5 | 86 | 4 | 56 | 1 | 3 | 9 |

Total | 32 |

Rs=âˆš6*32 / (125-5) =âˆš1.6 = +1.26

The positive sign shows that there is a positive relation between the first exam and the second exam. It means that if a student passed in the first exam it is most likely that the student will also pass exam two. The answer is bound to be a positive answer since all differences (di) in ranks are squired to eliminate the negative sign thus giving a positive answer.

**Reference**

- Kendall, (1973)
*Advanced Theory of Statistics, Volume 2: Inference and Relationship*, Griffin, ISBN.

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