The mean and standard deviation of 10 observations `x_(1), x_(2), x_(3)…….x_(10)` are `barx` and `sigma`respectively. Let 10 is added to `x_(1),x_(2)…..x_(9) and 90` is substracted from `x_(10)`. If still, the standard deviation is the same, then `x_(10)-barx` is equal to

Options

A. 35
B. 45
C. 55
D. 50

Answer

To solve the problem step by step, we will analyze the changes made to the observations and how they affect the mean and standard deviation.

### Step 1: Understand the initial conditions
Let the 10 observations be \( x_1, x_2, x_3, \ldots, x_{10} \). The mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_{10}}{10} \]
The standard deviation \( \sigma \) is given by:
\[ \sigma = \sqrt{\frac{\sum_{i=1}^{10} (x_i - \bar{x})^2}{10}} \]

### Step 2: Apply the changes to the observations
We add 10 to each of the first 9 observations and subtract 90 from the last observation:
- New observations: \( x_1 + 10, x_2 + 10, \ldots, x_9 + 10, x_{10} - 90 \)

### Step 3: Calculate the new mean
The new mean \( \bar{x}' \) after the changes is:
\[ \bar{x}' = \frac{(x_1 + 10) + (x_2 + 10) + \ldots + (x_9 + 10) + (x_{10} - 90)}{10} \]
\[ = \frac{(x_1 + x_2 + \ldots + x_9 + x_{10}) + 90 - 90}{10} = \bar{x} \]
Thus, the mean remains unchanged.

### Step 4: Calculate the new standard deviation
The new standard deviation \( \sigma' \) is given by:
\[ \sigma' = \sqrt{\frac{\sum_{i=1}^{9} ((x_i + 10) - \bar{x})^2 + ((x_{10} - 90) - \bar{x})^2}{10}} \]
This can be simplified as:
\[ = \sqrt{\frac{\sum_{i=1}^{9} (x_i - \bar{x} + 10)^2 + (x_{10} - 90 - \bar{x})^2}{10}} \]
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ = \sqrt{\frac{\sum_{i=1}^{9} (x_i - \bar{x})^2 + 100 + 20 \sum_{i=1}^{9} (x_i - \bar{x}) + (x_{10} - 90 - \bar{x})^2}{10}} \]

### Step 5: Set the new standard deviation equal to the old standard deviation
Since the standard deviation remains the same, we have:
\[ \sum_{i=1}^{10} (x_i - \bar{x})^2 = \sum_{i=1}^{9} (x_i - \bar{x} + 10)^2 + (x_{10} - 90 - \bar{x})^2 \]
Expanding the right side gives:
\[ \sum_{i=1}^{9} (x_i - \bar{x})^2 + 900 + 20 \sum_{i=1}^{9} (x_i - \bar{x}) + (x_{10} - 90 - \bar{x})^2 \]

### Step 6: Simplify the equation
Let \( S = \sum_{i=1}^{9} (x_i - \bar{x})^2 \):
\[ S = S + 900 + 20 \sum_{i=1}^{9} (x_i - \bar{x}) + (x_{10} - 90 - \bar{x})^2 \]
This simplifies to:
\[ 0 = 900 + 20 \sum_{i=1}^{9} (x_i - \bar{x}) + (x_{10} - 90 - \bar{x})^2 \]

### Step 7: Solve for \( x_{10} - \bar{x} \)
Rearranging gives:
\[ (x_{10} - 90 - \bar{x})^2 = -900 - 20 \sum_{i=1}^{9} (x_i - \bar{x}) \]
Since the left side is non-negative, the right side must also equal zero:
\[ x_{10} - 90 - \bar{x} = 0 \implies x_{10} - \bar{x} = 90 \]

### Final Result
Thus, we find:
\[ x_{10} - \bar{x} = 90 \]