The average of n numbers a$_1$, a$_2$,..., a$_n$ is $\overline{a}$. The value of $\sum_{i=1}^{n} (a_i - \bar{a})$is :

Question

Accepted Answer

To determine the value of $\sum_{i=1}^{n} (a_i - \overline{a})$, we need to understand the concept of the average and how it relates to the sum of deviations from the average.

1. **Explanation for Option 1 (Correct Answer):**

The average $\overline{a}$ of the numbers $a_1, a_2, \ldots, a_n$ is defined as:
   $$
   \overline{a} = \frac{1}{n} \sum_{i=1}^{n} a_i
   $$
   This implies that:
   $$
   n \cdot \overline{a} = \sum_{i=1}^{n} a_i
   $$
   Now, consider the expression $\sum_{i=1}^{n} (a_i - \overline{a})$:
   $$
   \sum_{i=1}^{n} (a_i - \overline{a}) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} \overline{a}
   $$
   Since $\overline{a}$ is a constant, $\sum_{i=1}^{n} \overline{a} = n \cdot \overline{a}$. Therefore:
   $$
   \sum_{i=1}^{n} (a_i - \overline{a}) = \sum_{i=1}^{n} a_i - n \cdot \overline{a} = n \cdot \overline{a} - n \cdot \overline{a} = 0
   $$
   Hence, the value of $\sum_{i=1}^{n} (a_i - \overline{a})$ is 0, which confirms that Option 1 is correct.

2. **Explanation for why the other options are incorrect:**

- **Option 2 (1):** This option suggests that the sum of deviations from the average is 1. However, as shown above, the sum of deviations from the average is always 0 because the positive and negative deviations cancel each other out.

- **Option 3 (n):** This option implies that the sum of deviations equals the number of terms, $n$. This is incorrect because the sum of deviations from the average is not dependent on the number of terms but rather on how the terms are distributed around the average. The sum of deviations is always 0, regardless of $n$.

- **Option 4 ($\overline{a}$):** This option suggests that the sum of deviations equals the average itself. This is incorrect because the average is a measure of central tendency, not a sum of deviations. The sum of deviations from the average is always 0, as the deviations balance out.

In conclusion, the correct answer is Option 1, as the sum of deviations from the average is always 0, which aligns with the definition and properties of an average.