Question

The Fairfield Smith's Variance law used in uniformity trials to obtain the relationship between plot size and plot variance is of the form \((V_x\) is the variance of yield per unit area from plots of size \(x\), \(V_1\) is the variance among plots of size unity and \(b\) is the coefficient representing soil characteristic):

• A. \(V_x = x^b V_1\)
• B. \(V_x = x^b + V_1\)
• C. \(V_x = \frac{V_1}{x^b}\)
• D. \(V_x = \frac{2V_1}{x^b}\)

Hint:

Exam Tip:
Fairfield Smith's Law: \(V_x = \frac{V_1}{x^b}\).
\(b\) is the soil heterogeneity coefficient.
Larger \(b\) means faster decrease in variance with increasing plot size.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Fairfield Smith's Variance Law describes how plot variance decreases as plot size increases.

Step 2: Key Formula or Approach:
The relationship is: \[ V_x = \frac{V_1}{x^b} \] where \(V_x\) is the variance per unit area for plots of size \(x\), \(V_1\) is the variance for unit plots, and \(b\) is a soil heterogeneity index.

Step 3: Detailed Explanation:
As plot size increases, the variance per unit area decreases.
The parameter \(b\) is typically between 0 and 1.
When \(x = 1\), \(V_x = V_1\).
This law is used to determine the optimal plot size for field experiments.

Step 4: Final Answer:
Therefore, option (C) is correct.