Charles Spearman 1863 1945 British Psychologist and pioneer

Charles Spearman (1863 -1945) British Psychologist and pioneer of IQ theory

Spearman’s Rank Correlation Coefficient What can this test tell you? Ø Whethere is a statistically significant correlation between two measurements from the sample when you have 5 -30 pairs of data. ØIf the correlation is negative or positive

What is the correlation coefficient r ?

Does being good at maths make you better at biology? Student Maths exam score Biology exam score Anand 57 83 Bernard 45 37 Charlotte 72 41 Demi 78 85 Eustace 53 56 Ferdinand 63 85 Gemma 86 77 Hector 98 87 Ivor 59 70 Jasmine 71 59 Is there a statistically significant correlation between these two sets of results?

Spearman’s Rank Correlation Coefficient: rs Where: rs = 1 – 6 x ∑D 2 [N 3 ] – N) N = the number of individuals in the sample D = difference in the rank of the two measurements made on an individual rs will be a number between – 1 and +1 This number can be compared with those in a table of critical values

H 0 = there is no statistically significant correlation between Maths scores and Biology scores HA = there is a statistically significant correlation between Maths scores and Biology scores If the value for rs exceeds the critical value, then you can reject the null hypothesis A negative value for rs implies a negative correlation A positive value for rs implies a positive correlation

Step 1: Rank each set of data (lowest to highest) Student Maths exam score Maths rank Biology exam score Biology rank Alex 57 3 83 7 Bernard 45 1 37 1 Charlotte 72 7 41 2 Demi 78 8 85 8. 5 8 Eustace 53 2 56 3 Ferdinand 63 85 Gemma 86 5 9 77 8. 5 9 6 Hector 98 10 87 10 Ivor 59 4 70 5 Jasmine 71 6 59 4 Where two or more scores are tied each is assigned an average rank

Step 2: Work out the differences in ranks (maths – biology) Student Maths exam score Maths rank Biology exam score Biology rank D D 2 Alex 57 3 83 7 -4 16 Bernard 45 1 37 1 0 0 Charlotte 72 7 41 2 5 25 Demi 78 8 85 8. 5 Eustace 53 2 56 3 - 0. 5 -1 0. 25 1 Ferdinand 63 5 85 8. 5 - 3. 5 12. 25 Gemma 86 9 77 6 3 9 Hector 98 10 87 10 0 0 Ivor 59 4 70 5 -1 1 Jasmine 71 6 59 4 2 4 ∑D 2 = 68. 5 Step 3: Work out the square of the differences Step 4: Work out the sum of the square of the differences

Step 5: Work out the correlation coefficient, rs rs = 1 – 6 x ∑D 2 [N 3 ] – N) 6 x 68. 5) rs = 1 103 – 10) 411 =1990 Where: N = 10 ∑D 2 = 68. 5 6 x 68. 5) =11000 - 10 = 1 – 0. 415 = 0. 585

Step 6: Compare your calculated value of rs with the relevant critical value in your stats table Our value of rs is 0. 585 For N = 10 and P = 0. 05, the critical value of rs is 0. 65 Because this is below the critical value, we must accept H 0 There is no statistically significant correlation between Maths scores and Biology scores rs critical values (P=0. 05) No. of pairs Critical value 8 0. 74 9 0. 68 10 0. 65 12 0. 59 14 0. 54

Mad Geoff’s Chaotic Firework Factory Do peoples stress levels increase the closer they live to Mad Geoff’s Chaotic Firework Factory? Acacia Ave. Cortisol is a stress hormone The more stressed an individual is, the higher their blood cortisol levels will be

Resident Address Blood cortisol level (μg/ml) Karl (Caretaker) 2 (Factory) 13. 4 Lillie 8 22. 6 Melanie 10 23. 4 Nigel 12 18. 6 Olga 12 17. 4 Peter 14 16. 8 Quentin 16 15. 2 Rajesh 16 10. 2 Susan 18 9. 8 Toni 18 12. 6 Uri 18 8. 8 Vanessa 20 7. 5 H 0 = there is no significantly significant correlation between proximity to the fireworks factory and blood cortisol levels

Resident Address rank Blood cortisol level (μg/ml) Cortisol rank D D 2 Karl 2 1 13. 4 6 -5 25 Lillie 8 2 22. 6 11 -9 81 Melanie 10 3 23. 4 Nigel 12 4. 5 18. 6 12 10 -9 -5. 5 81 30. 25 Olga 12 4. 5 17. 4 9 -4. 5 20. 25 Peter 14 16. 8 Quentin 16 6 7. 5 15. 2 8 7 -2 0. 5 4 0. 25 Rajesh 16 7. 5 10. 2 4 3. 5 12. 25 Susan 18 10 9. 8 3 7 49 Toni 18 10 12. 6 5 5 25 Uri 18 10 8. 8 2 8 64 Vanessa 20 12 7. 5 1 11 121 ∑D 2 513

Where: rs = 1 - 6∑D 2 N 3 – N) 6(513) rs = 1 123 - 12 3078 =11716 N = 12 ∑D 2 = 513 6(513) =11728 - 12 = 1 – 1. 794 = - 0. 794

When comparing rs to the critical value ignore the sign on rs For N = 12 and P = 0. 05, the critical value of rs is 0. 587 Our value of rs is 0. 794 Because this is above the critical value, we can reject H 0 There is a statistically significant correlation between proximity to the fireworks factory and blood cortisol levels rs is negative so there is a negative correlation. . . Therefore: the further one lives from the firework factory, the lower one’s blood cortisol levels.