If the mean of 50 observation is 25 and their standard deviation is 4 and the sum of the squares of all the observations is `lambda`, then `(lambda)/(1000)` is

Answer

To solve the problem, we need to find the value of \(\frac{\lambda}{1000}\), where \(\lambda\) is the sum of the squares of all observations. We are given the mean, standard deviation, and the number of observations.

### Step-by-Step Solution:

1. **Identify the given values:**
- Mean (\( \bar{x} \)) = 25
- Standard Deviation (\( \sigma \)) = 4
- Number of observations (\( n \)) = 50

2. **Recall the formula for variance:**
The variance (\( \sigma^2 \)) can be expressed as:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]
where \( \sum x_i^2 \) is the sum of the squares of the observations.

3. **Substitute the known values into the variance formula:**
- We know \( \sigma = 4 \), so \( \sigma^2 = 4^2 = 16 \).
- Substitute \( n = 50 \) and \( \bar{x} = 25 \):
\[ 16 = \frac{\lambda}{50} - 25^2 \]

4. **Calculate \( 25^2 \):**
\[ 25^2 = 625 \]
Thus, the equation becomes:
\[ 16 = \frac{\lambda}{50} - 625 \]

5. **Rearrange the equation to solve for \( \lambda \):**
\[ \frac{\lambda}{50} = 16 + 625 \]
\[ \frac{\lambda}{50} = 641 \]

6. **Multiply both sides by 50 to find \( \lambda \):**
\[ \lambda = 641 \times 50 \]
\[ \lambda = 32050 \]

7. **Calculate \( \frac{\lambda}{1000} \):**
\[ \frac{\lambda}{1000} = \frac{32050}{1000} = 32.05 \]

### Final Answer:
\[ \frac{\lambda}{1000} = 32.05 \]