Question

Which of the following is always true?

• A. $\sum_i (x_i - \bar{x}) = 0$
• B. $\sum_i |x_i - \bar{x}| = 0$
• C. $\sum_i (x_i - \bar{x})^2 = 0$
• D. $\sum_i f_i |x_i - \bar{x}| = 0$

Hint:

Remember this classic property: "The algebraic sum of deviations of a set of observations from their arithmetic mean is always zero."

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The arithmetic mean $\bar{x}$ acts as the center of gravity of a dataset, balancing the sum of values on both sides.
Key Formula or Approach:
Let us prove the algebraic property of the sum of deviations from the mean:
\[ \sum_i (x_i - \bar{x}) = \sum_i x_i - \sum_i \bar{x} \]

Step 2: Detailed Explanation:
Let there be $n$ observations $x_1, x_2, \dots, x_n$.
The mean is defined as:
\[ \bar{x} = \frac{\sum_i x_i}{n} \implies \sum_i x_i = n \bar{x} \]
Now evaluate the sum of the deviations:
\[ \sum_{i=1}^n (x_i - \bar{x}) = \sum_{i=1}^n x_i - \sum_{i=1}^n \bar{x} \]
Since $\bar{x}$ is a constant value across the summation:
\[ \sum_{i=1}^n \bar{x} = n \bar{x} \]
Substitute this back into the deviation sum:
\[ \sum_{i=1}^n (x_i - \bar{x}) = n \bar{x} - n \bar{x} = 0 \]
This algebraic property always holds true for any real dataset.
Options (B), (C), and (D) involve absolute values or squared terms, which can only equal zero if all individual observations are identical to the mean.

Step 3: Final Answer
The correct option is (A).