Data Analysis for Upland Landscapes

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?

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