If each observation of the data is increased by 5, then their mean

Options

A. remains the same
B. becomes 5 times the original mean
C. is decreased by 5
D. is increased by 5

Answer

To solve the problem step by step, we will analyze how the mean changes when each observation in a dataset is increased by a constant value.

### Step-by-Step Solution:

1. **Understanding the Mean**:
- The mean (average) of a set of observations is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
- Let's denote the observations as \(x_1, x_2, x_3, x_4, x_5\). The mean of these observations can be represented as:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} \]

2. **Increasing Each Observation**:
- According to the problem, each observation is increased by 5. Therefore, the new observations will be:
\[ x_1 + 5, \quad x_2 + 5, \quad x_3 + 5, \quad x_4 + 5, \quad x_5 + 5 \]

3. **Calculating the New Mean**:
- The new mean (let's call it \(\bar{x}_{new}\)) can be calculated as follows:
\[ \bar{x}_{new} = \frac{(x_1 + 5) + (x_2 + 5) + (x_3 + 5) + (x_4 + 5) + (x_5 + 5)}{5} \]
- This simplifies to:
\[ \bar{x}_{new} = \frac{x_1 + x_2 + x_3 + x_4 + x_5 + 25}{5} \]

4. **Separating the Terms**:
- We can separate the sum of the original observations and the constant:
\[ \bar{x}_{new} = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} + \frac{25}{5} \]
- This can be simplified to:
\[ \bar{x}_{new} = \bar{x} + 5 \]

5. **Conclusion**:
- From the calculation, we see that the new mean is equal to the original mean plus 5:
\[ \bar{x}_{new} = \bar{x} + 5 \]
- Therefore, the mean has increased by 5.

### Final Answer:
The mean is increased by 5.