Sophisticated data analysis will help you spot patterns, trends and relationships in your results. Data analysis can be qualitative and/or quantitative, and may include statistical tests. An example of a statistical test is outlined below.

## Spearman’s Rank Correlation Test

Spearman’s Rank Correlation is a statistical test to test whether there is a significant relationship between two sets of data.

The Spearman’s Rank Correlation test can only be used if there are at least 5 (ideally at least 8-15) pairs of data.

There are 3 steps to take when using the Spearman’s Rank Correlation Test

### Step 1. State the null hypothesis

There is no significant relationship between _______ and _______

### Step 2. Calculate the Spearman’s Rank Correlation Coefficient

\(r_s = 1-\frac{(6∑D^2)}{n(n^2-1)}\)- \(r_s\) = Spearman’s Rank correlation coefficient
- \(D\) = differences between ranks
- \(n\) = number of pairs of measurements

Step 3. Test the significance of the result

Compare the value of \(r_s\) that you have calculated *against* the critical value for \(r_s\) at a confidence level of 95% / significance value of p = 0.05.

If \(r_s\) is equal to or above the critical value (p=0.05) the REJECT the null hypothesis. There is a SIGNIFICANT relationship between the 2 variables.

A positive sign for \(r_s\) indicates a significant positive relationship and a negative sign indicates a significant negative relationship.

If \(r_s\) (ignoring any sign) is less than the critical value, ACCEPT the null hypothesis. There is NO SIGNIFICANT relationship between the 2 variables.

#### Worked example

A geographer is interested in whether there is a relationship between the area of a corrie and the height of the corrie’s backwall.

She combines fieldwork measurements with map evidence for 17 corries in an upland area in north Wales. Here are the results.

Corrie backwall height (m) | Corrie length (m) |
---|---|

1025 | 230 |

1850 | 180 |

1780 | 80 |

1320 | 255 |

440 | 200 |

700 | 170 |

1600 | 400 |

895 | 180 |

640 | 110 |

625 | 173 |

890 | 160 |

810 | 130 |

1020 | 255 |

580 | 60 |

885 | 175 |

800 | 200 |

2110 | 450 |

### Step 1. State the null hypothesis

There is no significant relationship between the length and the backwall height of a corrie.

### Step 2. Calculate the Spearman’s Rank Correlation Coefficient

(a) Rank the measurements

Corrie backwall height (m) | Corrie length (m) | ||
---|---|---|---|

Data | Rank | Data | Rank |

1850 | 2 | 180 | 8.5 |

1780 | 3 | 80 | 16 |

1320 | 5 | 255 | 3.5 |

440 | 17 | 200 | 6.5 |

700 | 13 | 170 | 12 |

1600 | 4 | 400 | 2 |

895 | 8 | 180 | 8.5 |

640 | 14 | 110 | 15 |

625 | 15 | 173 | 11 |

890 | 9 | 160 | 13 |

810 | 11 | 130 | 14 |

1020 | 7 | 255 | 3.5 |

580 | 16 | 60 | 17 |

885 | 10 | 175 | 10 |

800 | 12 | 200 | 6.5 |

2110 | 1 | 450 | 1 |

(b) Calculate \(D\) and \(D^2\)

Corrie backwall height (m) | Corrie length (m) | \(D\) | \(D^2\) | ||
---|---|---|---|---|---|

Data | Rank | Data | Rank | ||

1025 | 6 | 230 | 5 | -1 | 1 |

1850 | 2 | 180 | 8.5 | 6.5 | 42.25 |

1780 | 3 | 80 | 16 | 13 | 169 |

1320 | 5 | 255 | 3.5 | -1.5 | 2.25 |

440 | 17 | 200 | 6.5 | -10.5 | 110.25 |

700 | 13 | 170 | 12 | -1 | 1 |

1600 | 4 | 400 | 2 | -2 | 4 |

895 | 8 | 180 | 8.5 | 0.5 | 0.25 |

640 | 14 | 110 | 15 | 1 | 1 |

625 | 15 | 173 | 11 | -4 | 16 |

890 | 9 | 160 | 13 | 4 | 16 |

810 | 11 | 130 | 14 | 3 | 9 |

1020 | 7 | 255 | 3.5 | -3.5 | 12.25 |

580 | 16 | 60 | 17 | 1 | 1 |

885 | 10 | 175 | 10 | 0 | 0 |

800 | 12 | 200 | 6.5 | -5.5 | 30.25 |

2110 | 1 | 450 | 1 | 0 | 0 |

SUM | 415.5 |

(c) Calculate \(∑D^2\) i.e. the sum of the \(D^2\)column = 945.5

(d) Calculate \(r_s\)

\(r_s = 1-\frac{(6∑D^2)}{n(n^2-1)}\) \(r_s = 1-\frac{(6\;\times\;415.5)}{17\;\times\;(17^2-1)}\) \(r_s = 0.491\)### Step 3. Test the significance of the result

The critical value at \(p=0.05\) significance level for \(17\) pairs of measurements is \(0.488\)

Since our calculated value of \(0.491> 0.488\) (ignore the minus sign), the null hypothesis can be rejected.

In conclusion, there is a significant positive relationship between the length and backwall height of corries. Although statistically significant, it is not a very strong relationship. Why might this be?