Introduction to Statistics
V - Significance of Correlation Coefficient
Real Life Example
Here we compare the frequency of storms off the east coast of the United States with ocean temperatures off the coast of Peru.
| Year | Anomaly Niño 3 | Frequency anomaly of all storms |
|---|---|---|
| 1) 79-80 | +0.44 | +1.2 |
| 2) 80-81 | -.006 | +3.2 |
| 3) 81-82 | +0.35 | -1.8 |
| 4) 83-84 | +3.04 | +5.2 |
| 5) 84-85 | -0.19 | -2.8 |
| 6) 85-86 | -0.98 | -3.8 |
| 7) 86-87 | -0.45 | -1.8 |
| 8) 87-88 | +1.21 | +4.2 |
| 9) 88-89 | +0.68 | +3.2 |
| 10) 89-90 | -1.11 | -6.8 |
Since the points are pretty close to a straight line, and the line goes up to the right, it's a good guess that "r" is positive and close to 1.
Let's calculate "r":
Niño 3 Anomaly
S = Square root of (1/(N-1)) [sum of (each data point - mean)2]
S = Square root of 1/(10-1) * [(0.44 - 0.29)2 + (-0.006 - 0.29)2 + (0.35 - 0.29)2 + (3.04 - 0.29)2 + (-0.19 - 0.29)2 + (-0.98 -0.29)2 + (-0.45 -0.29)2 + (1.21 - 0.29)2 + (0.68 - 0.29)2 + (-1.11 - 0.29)2]
S = Square root of 13.06/9
S = 1.2046
All Storms Anomaly
S = Square root of ((1/9) * (141.6))
S = 3.9665
r = [ (0.44-0.29) (1.2-0) (-.006-0.29) (3.2-0) +...] / [ (10-1) (1.2046) (3.9665) ]
r = [ +0.18-1.12-0.11+1.34+4.38+1.33+3.86+1.25+9.52 ] / [ (9) (1.20) (3.97) ]
r = 0.825
This is a very high r! But with only 10 data points, is it statistically significant? Use the Fisher test to find out:
1/2 ln ( (1 + .83)/(1 - .83)) ± 1.96 * Square root of (1/7) = 1/2 ln ((1 + ρ)/(1 - ρ))
1.19 ± 0.74 = 1/2 ln ((1 + ρ)/(1 - ρ))
1.93 = 1/2 ln ((1 + ρ)/(1 - ρ))
47.47 = (1 + ρ)/(1 - ρ)
ρ high = 0.96
.45 = 1/2 ln ((1 + ρ)/(1 - ρ))
2.46 = (1 + ρ)/(1 - ρ)
ρ low = 0.42
Since 0 is not even close to the interval for ρ, then we can conclude that there is a statistically significant relationship between El Niño and storm frequency off the East coast of the United States. Since r is positive, the relationship is: Strong El Niño means more storms. Our figure shows this relationship, but the statistical test confirms that it is a significant relationship. That's why statistics is important and useful!
