The standard deviation of a distribution is 30. If each observation is increased by 5, then the new standard deviation will be

Options

A. 32
B. 28
C. 27
D. None of these

Answer

To solve the problem, we need to understand how the standard deviation behaves when we make a constant addition to each observation in a dataset.

### Step-by-Step Solution:

1. **Understand the Definition of Standard Deviation:**
The standard deviation (denoted as σ) measures the amount of variation or dispersion in a set of values. It is calculated using the formula:
\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2} \]
where \( x_i \) are the observations, \( \mu \) is the mean of the observations, and \( n \) is the number of observations.

**Hint:** Remember that standard deviation is a measure of spread, not affected by shifts in the data.

2. **Given Information:**
We know that the initial standard deviation is 30:
\[ \sigma = 30 \]

3. **Effect of Adding a Constant:**
When we increase each observation \( x_i \) by a constant value (in this case, 5), the new observations become \( x_i + 5 \). This affects the mean but does not affect the spread of the data.

4. **Calculate the New Mean:**
The new mean \( \mu' \) will be:
\[ \mu' = \mu + 5 \]
However, this does not affect the calculation of standard deviation because standard deviation is based on the deviation from the mean, which remains unchanged in terms of relative differences.

5. **New Standard Deviation Calculation:**
The standard deviation of the new set of observations \( (x_i + 5) \) is calculated as follows:
\[ \sigma' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} ((x_i + 5) - (\mu + 5))^2} \]
Simplifying this:
\[ \sigma' = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2} = \sigma \]
Thus, the new standard deviation remains:
\[ \sigma' = 30 \]

6. **Conclusion:**
Therefore, the new standard deviation after increasing each observation by 5 is still 30.

### Final Answer:
The new standard deviation will be **30**.