Question

The relationship between root mean square deviation (r.m.s.d.) and standard deviation (s.d.) is
1. s.d. = highest value of r.m.s.d.
2. s.d. = least value of r.m.s.d.
3. s.d. = r.m.s.d.
4. s.d. = (r.m.s.d)/2

• A. s.d. = highest value of r.m.s.d.
• B. s.d. = least value of r.m.s.d.
• C. s.d. = r.m.s.d.
• D. s.d. = (r.m.s.d)/2

Hint:

A key property of the mean is:
"The sum of squared deviations is minimum when taken about the arithmetic mean."
Consequently, standard deviation is always the minimum possible root mean square deviation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests a fundamental mathematical property of the arithmetic mean: the sum of squared deviations is minimized when the deviations are taken about the mean.

Step 2: Detailed Explanation:
Let us mathematically define both terms:
- Root Mean Square Deviation (r.m.s.d.): The square root of the arithmetic mean of the squared deviations taken about any arbitrary origin \( A \):
\[ \text{r.m.s.d.} = s' = \sqrt{\frac{1}{n} \sum_{i=1}^n (X_i - A)^2} \] - Standard Deviation (s.d.): The square root of the arithmetic mean of the squared deviations taken specifically about the arithmetic mean \( \bar{X} \):
\[ \text{s.d.} = s = \sqrt{\frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^2} \] Let us examine the algebraic relationship between the two measures.
Define the sum of squared deviations about an arbitrary origin \( A \):
\[ \sum (X_i - A)^2 = \sum (X_i - \bar{X} + \bar{X} - A)^2 \] Let \( d = \bar{X} - A \). Expanding the term:
\[ \sum (X_i - A)^2 = \sum \left[ (X_i - \bar{X}) + d \right]^2 \] \[ \sum (X_i - A)^2 = \sum (X_i - \bar{X})^2 + 2d \sum (X_i - \bar{X}) + n d^2 \] Since the sum of deviations about the mean is always zero (\( \sum (X_i - \bar{X}) = 0 \)), the middle term vanishes:
\[ \sum (X_i - A)^2 = \sum (X_i - \bar{X})^2 + n d^2 \] Divide by \( n \) and take the square root of both sides:
\[ \text{r.m.s.d.} = \sqrt{s^2 + d^2} \] Since \( d^2 \ge 0 \), the minimum value of this expression occurs when \( d = 0 \) (which means \( A = \bar{X} \)).
Under this condition, the root mean square deviation is exactly equal to the standard deviation.
Therefore, the standard deviation is the least (minimum) possible value of the root mean square deviation.

Step 3: Final Answer:
The standard deviation is the least value of the root mean square deviation.
Therefore, the correct choice is Option (B).