Mean of a set of numbers is `bar(x)`. If each number is increased by `lambda`, then the mean of new set is

Options

A. `bar(x)`
B. `bar(x) - lambda`
C. `lambda bar(x)`
D. `lambda`

Answer

To solve the problem step by step, we will follow a structured approach:

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We are given that the mean of a set of numbers is denoted as \( \bar{x} \).
- We need to find the new mean when each number in the set is increased by a constant \( \lambda \).

2. **Defining the Observations**:
- Let the original set of observations be \( x_1, x_2, x_3, \ldots, x_n \).
- The mean of these observations can be expressed mathematically as:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
- This can be rearranged to give:
\[ x_1 + x_2 + x_3 + \ldots + x_n = n \bar{x} \tag{1} \]

3. **Creating the New Set of Observations**:
- If each observation is increased by \( \lambda \), the new observations will be:
\[ x_1 + \lambda, x_2 + \lambda, x_3 + \lambda, \ldots, x_n + \lambda \]

4. **Calculating the New Mean**:
- The mean of the new set of observations can be calculated as:
\[ \text{New Mean} = \frac{(x_1 + \lambda) + (x_2 + \lambda) + (x_3 + \lambda) + \ldots + (x_n + \lambda)}{n} \]
- This can be simplified to:
\[ = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) + n\lambda}{n} \]
- Substituting the sum from equation (1):
\[ = \frac{n \bar{x} + n \lambda}{n} \]

5. **Final Simplification**:
- Simplifying the expression gives:
\[ = \bar{x} + \lambda \]

6. **Conclusion**:
- Therefore, the mean of the new set of numbers after increasing each by \( \lambda \) is:
\[ \bar{x} + \lambda \]

### Final Answer:
The mean of the new set is \( \bar{x} + \lambda \).