If each observation of a data is incre ased by 7, then their mean

Options

A. remains the same
B. becomes 7 times the original mean
C. is decreased by 7
D. is increased by 7

Answer

To solve the problem, we need to understand how the mean of a dataset is affected when each observation in that dataset is increased by a constant value. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Mean**:
The mean (average) of a dataset is calculated by summing all the observations and dividing by the number of observations. If we have a dataset with observations \( x_1, x_2, x_3, \ldots, x_n \), the mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

2. **Increasing Each Observation by 7**:
If we increase each observation by 7, the new observations will be \( (x_1 + 7), (x_2 + 7), (x_3 + 7), \ldots, (x_n + 7) \).

3. **Calculating the New Mean**:
The new mean \( \bar{x}' \) can be calculated as follows:
\[ \bar{x}' = \frac{(x_1 + 7) + (x_2 + 7) + (x_3 + 7) + \ldots + (x_n + 7)}{n} \]
This simplifies to:
\[ \bar{x}' = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) + 7n}{n} \]
\[ \bar{x}' = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} + \frac{7n}{n} \]
\[ \bar{x}' = \bar{x} + 7 \]

4. **Conclusion**:
Therefore, when each observation in the dataset is increased by 7, the mean of the dataset also increases by 7.

### Final Answer:
The new mean will be the original mean plus 7.