Consider the following statements:
1.The algebraic sum of deviations of a set of values from their arithmetic mean is always zero.
2 Arithmetic mean `gt` median `gt` mode for a symmetric distribution
Which of the above statements are corrects?

Options

A. A) 1only
B. B) 2only
C. C) Both 1 and 2
D. D) Neither I nor 2

Answer

To determine which of the given statements is correct, we will analyze each statement step by step.

### Statement 1: The algebraic sum of deviations of a set of values from their arithmetic mean is always zero.

1. **Definition of Arithmetic Mean**:
The arithmetic mean (denoted as \( \bar{x} \)) of a set of values \( x_1, x_2, \ldots, x_n \) is calculated as:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]

2. **Sum of Deviations**:
We need to find the sum of deviations from the mean:
\[ A = \sum_{i=1}^{n} (x_i - \bar{x}) \]

3. **Expanding the Sum**:
We can expand the sum:
\[ A = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x} \]
Since \( \bar{x} \) is constant for all \( n \) observations, we can rewrite the second term:
\[ A = \sum_{i=1}^{n} x_i - n \cdot \bar{x} \]

4. **Substituting the Mean**:
From the definition of the mean, we know that:
\[ n \cdot \bar{x} = \sum_{i=1}^{n} x_i \]
Therefore, substituting this into our equation for \( A \):
\[ A = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} x_i = 0 \]

5. **Conclusion for Statement 1**:
The algebraic sum of deviations from the arithmetic mean is indeed always zero. Thus, Statement 1 is **correct**.

### Statement 2: Arithmetic mean > Median > Mode for a symmetric distribution.

1. **Understanding Symmetric Distribution**:
In a symmetric distribution, the mean, median, and mode are all equal. This means:
\[ \text{Mean} = \text{Median} = \text{Mode} \]

2. **Analyzing the Inequality**:
The statement suggests that the arithmetic mean is greater than the median, and the median is greater than the mode. However, since all three measures are equal in a symmetric distribution, this inequality does not hold.

3. **Conclusion for Statement 2**:
Therefore, Statement 2 is **incorrect**.

### Final Conclusion:
- **Statement 1** is correct.
- **Statement 2** is incorrect.

Thus, the answer is that only the first statement is correct.