1. TOPIC 17 Spearman’s Rank Correlation Co-efficient

2. In some situations it is simply the order or rank that is important, for example, judges at a dance competition might rank the dances in order rather than awarding marks for each dance. The formula for Spearman’s Rank Correlation Co-efficient is: r = 1 - 6 ∑d 2 n(n 2 – 1) Where:d = difference between the ranking of each item n = the number of paired observations

3. Spearman’s Rank Correlation Co-efficient EXAMPLE 1 Eight dances are performed at a local dance competition. After seeing all the dances, two judges rank the dances in order so the dance each judge likes best has rank 1. Calculate Spearman’s rank correlation co-efficient for this data. DanceJudge 1, xJudge 2, yd=x–yd2d2 A14-39 B73416 C26-416 D37-416 E514 F48-416 G6511 H82636 ∑d 2 =126

4. Spearman’s Rank Correlation Co-efficient ANSWER 1 r = 1 - 6 ∑d 2 n(n 2 – 1) = 1 – 6 x 126 8(8 2 – 1) = 1 – 756 8(64 – 1) = 1 – 756 8 x 63 = 1 – 756 504 = 1 – 1.5 = -0.5

5. Spearman’s Rank Correlation Co-efficient Spearman’s Rank Correlation Co-efficient can take any value between +1 and -1. +1 = Perfect positive correlation 0 = No correlation -1 = Perfect negative correlation So for Example 1, there is some negative correlation between the judges’ scores. Dealing With Tied Ranks Sometimes more than one item is given the same rank. In this case new ranks are allocated to the items that have tied ranks. When asked to calculate Spearman’s Rank Correlation Co-efficient of data given as numerical values, it is important to rank the data first.

6. Example 2 The marks of 12 pupils in geography and history essays as follows: Geography, xHistory, yRank, xRank, yd = x - yd2d2 15109.590.50.25 16127.543.512.25 191214-39 17135.51.5416 17115.57-1.52.25 1599.510-0.50.25 181137-416 137.51.5636 181137-416 1812341 14811 00 10712 00 ∑d 2 =109

7. Calculate Spearman’s Rank Correlation Co-efficient ANSWER 2 GeographyHistory 19 = 113 = ½(1 + 2) = 1.5 18 = ⅓(2 + 3 + 4) = 312 = ⅓(3 + 4 + 5) = 4 17 = ½(5 + 6) = 5.511 = ⅓(6 + 7 + 8) = 7 16 = ½(7 + 8) = 7.510 = 9 15 = ½(9 + 10) = 9.59 = 10 14 = 118 = 11 10 = 127 = 12

8. Spearman’s Rank Correlation Co-efficient r = 1 - 6 ∑d 2 n(n 2 – 1) = 1 – 6 x 109 12(12 2 – 1) = 1 – 654 12(144 – 1) = 1 – 654 12 x 143 = 1 – 654 1716 = 1 – 0.381 = 0.619 Some positive correlation between the geography and history results.