Let `barx` be the mean of n observations `x_(1),x_(2),……..,x_(n)`. If `(a-b)` is added to each observation, then what is the mean of new set of observations?

Options

A. 0
B. `barx`
C. `barx-(a-b)`
D. `barx(a-b)`

Explanation

To find the mean of the new set of observations after adding \( (a-b) \) to each observation, we can follow these steps:

### Step 1: Understand the Mean of the Original Observations
The mean \( \bar{x} \) of the original observations \( x_1, x_2, \ldots, x_n \) is given by the formula:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]

### Step 2: Add \( (a-b) \) to Each Observation
When we add \( (a-b) \) to each observation, the new observations become:
\[ x_1 + (a-b), \quad x_2 + (a-b), \quad \ldots, \quad x_n + (a-b) \]

### Step 3: Calculate the Sum of the New Observations
The sum of the new observations can be expressed as:
\[ \text{New Sum} = (x_1 + (a-b)) + (x_2 + (a-b)) + \ldots + (x_n + (a-b)) \]
This simplifies to:
\[ \text{New Sum} = (x_1 + x_2 + \ldots + x_n) + n(a-b) \]
Thus,
\[ \text{New Sum} = \sum_{i=1}^{n} x_i + n(a-b) \]

### Step 4: Calculate the Mean of the New Observations
Now, we can find the mean of the new set of observations:
\[ \text{New Mean} = \frac{\text{New Sum}}{n} = \frac{\sum_{i=1}^{n} x_i + n(a-b)}{n} \]
This can be separated into two parts:
\[ \text{New Mean} = \frac{\sum_{i=1}^{n} x_i}{n} + \frac{n(a-b)}{n} \]
Thus,
\[ \text{New Mean} = \bar{x} + (a-b) \]

### Conclusion
Therefore, the mean of the new set of observations is:
\[ \text{New Mean} = \bar{x} + (a-b) \]

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