If the first five elements of the set xi+5,i=1,2,3...5 and the next five elements are replaced by xj −5,j=6...10 then the mean will change by

Answer

To solve the problem step by step, we need to analyze the two sets of elements and how they contribute to the mean.

### Step 1: Define the first set of elements
The first five elements of the set are given as \( x_i + 5 \) for \( i = 1, 2, 3, 4, 5 \).

The elements are:
- \( x_1 + 5 \)
- \( x_2 + 5 \)
- \( x_3 + 5 \)
- \( x_4 + 5 \)
- \( x_5 + 5 \)

### Step 2: Calculate the mean of the first set
To find the mean of the first set, we sum these elements and divide by the number of elements (which is 5):

\[ \text{Mean}_1 = \frac{(x_1 + 5) + (x_2 + 5) + (x_3 + 5) + (x_4 + 5) + (x_5 + 5)}{5} \]

This simplifies to:

\[ \text{Mean}_1 = \frac{x_1 + x_2 + x_3 + x_4 + x_5 + 25}{5} \]

### Step 3: Define the second set of elements
The next five elements are replaced by \( x_j - 5 \) for \( j = 6, 7, 8, 9, 10 \).

The elements are:
- \( x_6 - 5 \)
- \( x_7 - 5 \)
- \( x_8 - 5 \)
- \( x_9 - 5 \)
- \( x_{10} - 5 \)

### Step 4: Calculate the mean of the second set
Now, we calculate the mean of the second set, which includes the first five elements and the new five elements:

\[ \text{Mean}_2 = \frac{(x_1 + 5) + (x_2 + 5) + (x_3 + 5) + (x_4 + 5) + (x_5 + 5) + (x_6 - 5) + (x_7 - 5) + (x_8 - 5) + (x_9 - 5) + (x_{10} - 5)}{10} \]

This simplifies to:

\[ \text{Mean}_2 = \frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}}{10} \]

### Step 5: Compare the means
Notice that the mean of the first set and the mean of the second set are both centered around the same average \( \bar{x} \). Thus, the difference between the two means is:

\[ \text{Difference} = \text{Mean}_2 - \text{Mean}_1 = 0 \]

### Conclusion
The mean will not change after replacing the elements. Therefore, the answer is:

\[ \text{Mean change} = 0 \]

---