A set of n values `x_(1),x_(2),…….,x_(n)` has standard deviation `sigma`. The standard deviation of n values `x_(1)-k,x_(2)-k,…….,x_(n)-k` is

Options

A. `sigma-k`
B. `sigma+k`
C. `sigma`
D. `ksigma`

Answer

To solve the problem, we need to understand the concept of standard deviation and how it is affected by transformations of the data set.

### Step-by-Step Solution:

1. **Understanding Standard Deviation**:
The standard deviation (σ) of a set of values measures the amount of variation or dispersion in the values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation means that the values are spread out over a wider range.

2. **Given Values**:
We have a set of n values: \( x_1, x_2, \ldots, x_n \) with a standard deviation of \( \sigma \).

3. **Transformation of Values**:
We are transforming the original values by subtracting a constant \( k \) from each value. The new set of values will be: \( x_1 - k, x_2 - k, \ldots, x_n - k \).

4. **Effect of Subtracting a Constant**:
When we subtract a constant from each value in a data set, the mean of the data set changes, but the standard deviation remains unchanged. This is because standard deviation is a measure of spread relative to the mean, and subtracting a constant shifts all values equally without affecting their relative distances from each other.

5. **Conclusion**:
Therefore, the standard deviation of the new set of values \( x_1 - k, x_2 - k, \ldots, x_n - k \) is still \( \sigma \).

### Final Answer:
The standard deviation of the values \( x_1 - k, x_2 - k, \ldots, x_n - k \) is \( \sigma \).

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