Which of the following is correct relations a symmetrical distribution is

Options

A. a) `A.M. - M_(o) = 3(A.M. - M_(d))`
B. b) `A.M. - M_(o) = 2(A.M. - M_(d))`
C. c) `M_(d) = 2 A.M. - 3M_(d)`
D. d) `A.M. + M_(o) = 3(A.M. - M_(d))`

Answer

To solve the question regarding the correct relation of a symmetrical distribution, we need to analyze the relationships between the mean (Am), median (Md), and mode (Mo).

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Mean (Am): The average of the data set.
- Median (Md): The middle value when the data set is ordered.
- Mode (Mo): The value that appears most frequently in the data set.

2. **Recall the Relationship in Symmetrical Distributions**:
- For a symmetrical distribution, there is a known relationship between the mean, median, and mode. The relationship is given by:
\[ \text{Mean} - \text{Median} = \frac{1}{3} (\text{Mean} - \text{Mode}) \]

3. **Rearranging the Formula**:
- We can rearrange the above relationship to express it in a different form. Multiplying both sides by 3 gives:
\[ 3(\text{Mean} - \text{Median}) = \text{Mean} - \text{Mode} \]
- This can be rewritten as:
\[ \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median}) \]

4. **Identifying the Correct Option**:
- From the rearranged formula, we can identify that:
\[ \text{Am} - \text{Mo} = 3(\text{Am} - \text{Md}) \]
- This corresponds to option A in the question.

5. **Conclusion**:
- Therefore, the correct relation for a symmetrical distribution is:
\[ \text{Am} - \text{Mo} = 3(\text{Am} - \text{Md}) \]
- Thus, the correct option is **A**.