Question

The correlation coefficient between packing density and porosity of a set of yarn is

• A. 1
• B. 0
• C. $-1$
• D. Can’t be determined

Hint:

Perfect inverse relationships between two physical parameters always have a correlation coefficient of $-1$.

Answer 1

The Correct Option is C

Step 1: Understanding packing density and porosity.
Packing density represents the fraction of space occupied by fibres in a yarn, whereas porosity represents the fraction of void space present in the yarn structure. Both are complementary structural parameters.
Step 2: Establishing the relationship.
As packing density increases, the void space within the yarn decreases proportionally. Hence, porosity decreases whenever packing density increases, and vice versa.
Step 3: Correlation interpretation.
Since one parameter increases exactly when the other decreases in a linear manner, the relationship is perfectly inverse.
Step 4: Conclusion.
Therefore, the correlation coefficient between packing density and porosity is $-1$.

Answer 2

Packing density and porosity describe the same yarn cross-section from opposite sides, so the question really asks whether these two quantities are tied together in a fixed, predictable way. Checking each candidate correlation value against this relationship settles it.

1. 1: A correlation of \( +1 \) would mean the two quantities always rise and fall together. That cannot be true here, because packing more fibre into a given yarn cross-section leaves less unoccupied space, so porosity must fall as packing density rises, not rise with it.
2. 0: A correlation of \( 0 \) would mean the two quantities are unrelated, varying independently of each other. But porosity is simply the fraction of the yarn cross-section not occupied by fibre, and packing density is the fraction that is occupied, so one is fixed the moment the other is known; they cannot be independent.
3. \(-1\): A correlation of \(-1\) means the quantities move in exactly opposite directions in a fixed linear proportion. Since porosity equals \(1\) minus packing density (as a fraction of yarn volume), every rise in packing density produces an exactly equal fall in porosity. This is precisely a perfect negative linear relationship.
4. Can't be determined: This would apply only if the relationship between the two varied unpredictably from sample to sample. Because porosity is mathematically fixed once packing density is known (they are complements of the same total volume), the relationship is always determinable, and always perfectly negative.

Only the perfectly negative, fixed relationship matches how packing density and porosity are defined, so the correlation coefficient is \(-1\).

Therefore, the correct answer is \(-1\).