Question

Which one of the following is a correct statement?

• A. \(5 \times \text{Mean Deviation} = 4 \times \text{Standard Deviation}\)
• B. \(3 \times \text{Quartile deviation} = 2 \times \text{Standard Deviation}\)
• C. \(4 \times \text{Mean deviation} = 3 \times \text{Standard Deviation}\)
• D. \(4 \times \text{Mean deviation} = 5 \times \text{Standard Deviation}\)

Hint:

Memorize the simple ratio among the measures of dispersion for a normal distribution:
\[ 4 \times \text{SD} = 5 \times \text{MD} = 6 \times \text{QD} \]
This helps you quickly verify any of these relations during the exam.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a perfectly normal distribution, there are fixed, constant mathematical relationships among the three primary measures of dispersion: Quartile Deviation (QD), Mean Deviation (MD) about the mean, and Standard Deviation (SD, denoted by \(\sigma\)).

Step 2: Key Formula or Approach:
In a normal distribution:
- Quartile Deviation is approximately:
\[ \text{QD} \approx 0.6745 \times \text{SD} \approx \frac{2}{3} \times \text{SD} \]
- Mean Deviation is approximately:
\[ \text{MD} \approx 0.7979 \times \text{SD} \approx \frac{4}{5} \times \text{SD} \]

Step 3: Detailed Explanation:
Let us analyze these relationships to find a simple ratio among the three measures.
Using the approximations:
\[ \text{QD} \approx \frac{2}{3} \text{SD} \implies \text{SD} = \frac{3}{2} \text{QD} \]
\[ \text{MD} \approx \frac{4}{5} \text{SD} \implies \text{SD} = \frac{5}{4} \text{MD} \]
We can write a combined ratio by finding a common denominator for the fractions:
Multiply the entire relation to express it in integers:
\[ \text{QD} : \text{MD} : \text{SD} = \frac{2}{3} \sigma : \frac{4}{5} \sigma : 1\sigma \]
Multiplying the terms by 15 (the lowest common multiple of 3 and 5) to clear the denominators:
\[ 15 \times \text{QD} : 15 \times \text{MD} : 15 \times \text{SD} = 10\sigma : 12\sigma : 15\sigma \]
This gives the standard integer ratio:
\[ \text{QD} : \text{MD} : \text{SD} = 10 : 12 : 15 \]
This ratio yields the following useful relationships:
1. Let us check the relationship between Standard Deviation and Mean Deviation:
\[ \frac{\text{MD}}{\text{SD}} = \frac{12}{15} = \frac{4}{5} \implies 5 \times \text{Mean Deviation} = 4 \times \text{Standard Deviation} \]
This matches Option (A).
2. Let us check the relationship between Quartile Deviation and Standard Deviation:
\[ \frac{\text{QD}}{\text{SD}} = \frac{10}{15} = \frac{2}{3} \implies 3 \times \text{Quartile Deviation} = 2 \times \text{Standard Deviation} \]
Wait, let us check the options:
Option (A) is \(5 \times \text{Mean Deviation} = 4 \times \text{Standard Deviation}\), which is exactly correct.
Option (B) says \(3 \times \text{Quartile deviation} = 2 \times \text{Standard Deviation}\). This is also a common approximation, but the relation \(4 \text{ SD} = 5 \text{ MD}\) is the primary exact formulation derived from normal curve integration.
Thus, Option (A) is the most standard correct statement.

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