Question

The sum of deviation of values $x_i$ from their mean is:

• A. Zero
• B. Less than zero
• C. More than zero
• D. Infinity

Hint:

Remember the two key properties of the arithmetic mean:
1. The sum of deviations from the mean is always zero: \( \sum(x_i - \bar{x}) = 0 \).
2. The sum of squared deviations from the mean is a minimum: \( \sum(x_i - a)^2 \) is minimized when \( a = \bar{x} \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An algebraic property of the arithmetic mean is that it acts as the center of gravity of a dataset.
The sum of the positive deviations of values above the mean is exactly balanced by the sum of the negative deviations of values below the mean.
Key Formula or Approach:
The sum of deviations of $n$ observations \( x_1, x_2, \dots, x_n \) from their sample mean \( \bar{x} \) is mathematically written as:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) \]

Step 2: Detailed Explanation:
Let us expand the summation expression algebraically:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x} \]
Since \( \bar{x} \) is a constant value for a given sample, summing it $n$ times gives \( n\bar{x} \):
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - n\bar{x} \]
By definition, the arithmetic mean is \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \), which means \( \sum_{i=1}^{n} x_i = n\bar{x} \).
Substituting this back into our equation:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = n\bar{x} - n\bar{x} = 0 \]
This proof demonstrates that the sum of the deviations of a set of values from their arithmetic mean is always exactly zero.

Step 3: Final Answer:
The sum of deviation of values \( x_i \) from their mean is Zero.