1. Primary Succession Tropical Rainforest GPP ATP / NADPH Forest fire
Photosynthesis
Secondary Succession
Light dependent
NPP
Climax Community

2. Pearsons Correlation Coefficient
C/W
14/06/16
Pearsons Correlation Coefficient
To recap primary and secondary succession
To calculate Pearsons Correlation coefficient using an example set of data and our own data.

3. KARL PEARSON ( )
British mathematician, ‘father’ of statistics and a pioneer of eugenics! 3

4. Pearson’s Correlation Coefficient
What can this test tell you?
If there is a statistically significant correlation between two measured variables, X and Y, and….
If that correlation is negative or positive
Note: The data must show normal distribution

5. What is the correlation coefficient r ?

6. Mass (arbitrary units) Length (arbitrary units)
Is there a significant correlation between an animal’s nose-to-tail length and its body mass?
Animal
Mass (arbitrary units)
Length (arbitrary units) 1 2 4 5 3 8 12 14 6 9 19 7 22
If yes, then is the correlation positive (does a long tail mean a larger mass)?

7. Pearson’s Correlation Coefficient
Where: n = the number of values of X and Y
∑XY – [(∑X)(∑Y)]/n
 {∑X2-[(∑X)2/n]} {∑Y2-[(∑Y)2/n]}
r =
r will always be a number between –1 and +1
This number can be compared with those in a table of critical values using: n – 2 degrees of freedom.

8. A negative value for r implies a negative correlation
What is the null hypothesis (H0)?
H0 = there is no statistically significant correlation between length and body mass
HA = there is a statistically significant correlation between length and body mass
If the value for r exceeds the critical value, then you can reject the null hypothesis.
A negative value for r implies a negative correlation
A positive value for r implies a positive correlation

9. Construct the following results table
Animal (n = 7)
Mass (X)
Length (Y) X2 Y2 XY 1 2 4 5 3 8 12 14 6 9 19 7 22
Total
Mean 9

10. Construct the following results table
Animal (n = 7)
Mass (X)
Length (Y) X2 Y2 XY 1 2 4 5 3 8 12 14 6 9 19 7 22
Total
Mean 10

11. Construct the following table:
Animal
Mass (X)
Length (Y) X2 Y2 XY 1 2 4 5 16 3 8 9 12 14 64 6 19 81 7 22
Total
X = 37
Mean
5.29

12. Calculate values for X:
Animal
Mass (X)
Length (Y) X2 Y2 XY 1 2 4 5 16 3 8 9 12 14 64 6 19 81 7 22
Total
X = 37
X2 = 251
Mean
5.29 12

13. Calculate values for Y:
Animal
Mass (X)
Length (Y) X2 Y2 XY 1 2 4 5 16 25 3 8 9 64 12
244 14
196 6 19 81
361 7 22
484
Total
X = 37
Y = 82
X2 = 251
Y2 = 1278
Mean
5.29
11.71 13

14. Calculate values for XY:
Animal
Mass (X)
Length (Y) X2 Y2 XY 1 2 4 5 16 25 20 3 8 9 64 24 12
244 48 14
196
112 6 19 81
361
152 7 22
484
176
Total
X = 37
Y = 82
X2 = 251
Y2 = 1278
XY = 553
Mean
5.29
11.71 14

15. Use values obtained to populate the equation: X = 37 Y = 82
∑XY – [(∑X)(∑Y)]/n
 {∑X2-[(∑X)2/n]} {∑Y2-[(∑Y)2/n]}
r =
553 – (37 x 82)/7
 {251-[372/7]} {1278-[822/7]}
r =
119.57 =
132.60
r = 15

16. Compare your calculated value of r with the relevant critical value in your stats table
The + sign shows that any correlation is positive.
Our value of r =
Degrees of freedom = n = 5
D.F.
Critical Value (P = 0.05) 3
0.88 4
0.81 5
0.75 6
0.71 7
0.67
Our value for r exceeds the critical value, so we can reject the null hypothesis.
We can conclude that there is a significant positive correlation between the length of an animal and its body mass
i.e. a long tail is associated with a large body mass!

17. Construct the following results table
Person (n = 8)
Units of alcohol (X)
Reaction time (s) (Y) X2 Y2 XY 1 2 3 4 5
4.5
Total
Mean 17

18. Construct the following results table
Person (n = 8)
Units of alcohol (X)
Reaction time (s) (Y) X2 Y2 XY 1 2 3 4 5
4.5
Total
Mean 18

19. Step 1: Rank each set of data (lowest to highest)
Student
Maths exam score
Maths rank
Biology exam score
Biology rank
Alex 57 83
Bernard 45 37
Charlotte 72 41
Demi 78 85
Eustace 53 56
Ferdinand 63
Gemma 86 77
Hector 98 87
Ivor 59 70
Jasmine 71 3 7 1 1 7 2 8
8.5 8 2 3 5
8.5 9 9 6 10 10 4 5 6 4
Where two or more scores are tied...
... each is assigned an average rank