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Unit A: Biostatistics

3. Correlation
Understanding how variables interact and co-vary in biological systems.
In zoology, we rarely study an organism's traits in isolation. We want to know how traits co-vary—for example, how does the metabolic rate of a mammal scale with its body mass?
A. Concept of Correlation

Correlation measures the strength and direction of a linear relationship between two quantitative variables. It does not imply causation; it simply tells us if two variables move together.

B. Types of Correlation

Positive Correlation: As X increases, Y increases (e.g., ambient temperature and the chirping rate of crickets).

Negative Correlation: As X increases, Y decreases (e.g., water pollution levels and dissolved oxygen concentration).

Linear vs. Non-linear: Linear correlation forms a straight line on a scatter plot. Non-linear (curvilinear) relationships follow a curve (e.g., enzyme activity across a wide temperature gradient—increasing to an optimum, then crashing due to denaturation).

C. Pearson’s Correlation Coefficient (r)

Parametric method used for continuous, normally distributed data.

r = Σ(X - X̄)(Y - Ȳ) / √[Σ(X - X̄)² Σ(Y - Ȳ)²]

• Dimensionless, bounded between -1.0 and +1.0.

r = +1: Perfect positive linear correlation.

r = -1: Perfect negative linear correlation.

r = 0: No linear correlation (Note: a strong non-linear relationship can still yield r = 0).

D. Spearman’s Rank Correlation (ρ)

Non-parametric alternative when data violates normality, contains outliers, or is ranked/ordinal (e.g., animal dominance ranks).

ρ = 1 - [6Σd² / n(n² - 1)]

d = difference between ranks; n = number of pairs.

E. Coefficient of Determination (r²)

CSIR/NET High-Yield Concept

Measures the proportion of variance in the dependent variable that is predictable from the independent variable.

Example: If r = 0.8 (Bird tail length vs mating success), then r² = 0.64. This means 64% of mating success variation is explained by tail length; the remaining 36% is due to other factors.