Question

If \(\bar{x}\) represent the mean of n observations \(x_1, x_2, x_3, \dots, x_n\), then the value of \(\sum_{i=1}^{n} (x_i - \bar{x})\) is :

• A. \(-1\)
• B. 0
• C. \(1\)
• D. \(n-1\)

Hint:

A fundamental property of the arithmetic mean (\(\bar{x}\)) is that the sum of the differences of each data point from the mean is always zero. \[ \sum (x_i - \bar{x}) = 0 \] This is because the mean is the "balancing point" of the data. The positive deviations (values above the mean) exactly cancel out the negative deviations (values below the mean).

Answer 1

The Correct Option is B

Concept: This question relates to a fundamental property of the arithmetic mean (average) of a set of observations. The sum of the deviations of each observation from the mean is always zero. Step 1: Definition of the mean (\(\bar{x}\)) The mean of \(n\) observations \(x_1, x_2, x_3, \dots, x_n\) is given by: \[ \bar{x} = \frac{x_1 + x_2 + x_3 + \dots + x_n}{n} \] This can also be written using summation notation: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] From this definition, we can write: \[ \sum_{i=1}^{n} x_i = n \bar{x} \] Step 2: Evaluate the given summation \(\sum_{i=1}^{n} (x_i - \bar{x})\) We need to find the value of \(\sum_{i=1}^{n} (x_i - \bar{x})\). Using the properties of summation, we can split the sum: \[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x} \] Step 3: Simplify the terms
The first term, \(\sum_{i=1}^{n} x_i\), is the sum of all observations. From Step 1, we know \(\sum_{i=1}^{n} x_i = n \bar{x}\).
The second term, \(\sum_{i=1}^{n} \bar{x}\), means we are summing the constant value \(\bar{x}\) for \(n\) times. So, \(\sum_{i=1}^{n} \bar{x} = \bar{x} + \bar{x} + \dots + \bar{x}\) (\(n\) times) \( = n \bar{x}\). Step 4: Substitute back and find the final value Substitute these simplified terms back into the expression from Step 2: \[ \sum_{i=1}^{n} (x_i - \bar{x}) = (n \bar{x}) - (n \bar{x}) \] \[ \sum_{i=1}^{n} (x_i - \bar{x}) = 0 \] Thus, the sum of the deviations of each observation from the mean is 0. This matches option (2).