If M is the mean of n observations `x_(1)-k, x_(2)-k, x_(3)-k,…., x_(n)-k`, where k is any real number, then what is the mean of `x_(1), x_(2), x_(3) …., x_(n)`?

Options

A. M
B. `M+k`
C. `M-k`
D. kM

Answer

To find the mean of the observations \( x_1, x_2, x_3, \ldots, x_n \) given that \( M \) is the mean of the observations \( x_1 - k, x_2 - k, x_3 - k, \ldots, x_n - k \), we can follow these steps:

### Step 1: Define the mean of the modified observations
The mean \( M \) of the observations \( x_1 - k, x_2 - k, \ldots, x_n - k \) is given by the formula:
\[ M = \frac{(x_1 - k) + (x_2 - k) + (x_3 - k) + \ldots + (x_n - k)}{n} \]

### Step 2: Simplify the expression for \( M \)
We can rewrite the numerator:
\[ M = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) - nk}{n} \]
where \( nk \) is the sum of \( k \) subtracted \( n \) times.

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate the sum of the original observations:
\[ M = \frac{S - nk}{n} \]
where \( S = x_1 + x_2 + x_3 + \ldots + x_n \).

Multiplying both sides by \( n \) gives:
\[ Mn = S - nk \]

### Step 4: Solve for \( S \)
Now, we can solve for \( S \):
\[ S = Mn + nk \]

### Step 5: Find the mean of the original observations
The mean of the original observations \( x_1, x_2, \ldots, x_n \) is given by:
\[ \text{Mean} = \frac{S}{n} = \frac{Mn + nk}{n} \]
This simplifies to:
\[ \text{Mean} = M + k \]

### Final Answer
Thus, the mean of the observations \( x_1, x_2, x_3, \ldots, x_n \) is:
\[ \text{Mean} = M + k \]