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 \(\frac{\lambda}{1000}\) given the mean, standard deviation, and number of observations. Let's break down the steps:

### Step 1: Understand the given data
- Number of observations (n) = 50
- Mean (\(\bar{x}\)) = 25
- Standard deviation (\(\sigma\)) = 4

### Step 2: Calculate the variance
The variance (\(\sigma^2\)) is the square of the standard deviation:
\[ \sigma^2 = 4^2 = 16 \]

### Step 3: Use the variance formula
The formula for variance is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]
Where:
- \(\sum x_i^2\) is the sum of the squares of all observations.
- \(n\) is the number of observations.
- \(\bar{x}\) is the mean of the observations.

### Step 4: Substitute the known values into the variance formula
We can rearrange the variance formula to find \(\sum x_i^2\):
\[ \sum x_i^2 = n \cdot \sigma^2 + \bar{x}^2 \]
Substituting the known values:
\[ \sum x_i^2 = 50 \cdot 16 + 25^2 \]

### Step 5: Calculate \(\sum x_i^2\)
Calculating each term:
\[ 50 \cdot 16 = 800 \]
\[ 25^2 = 625 \]
Adding these together:
\[ \sum x_i^2 = 800 + 625 = 1425 \]

### Step 6: Relate \(\sum x_i^2\) to \(\lambda\)
From the problem, we know that \(\lambda = \sum x_i^2\). Therefore:
\[ \lambda = 1425 \]

### Step 7: Calculate \(\frac{\lambda}{1000}\)
Now we can find \(\frac{\lambda}{1000}\):
\[ \frac{\lambda}{1000} = \frac{1425}{1000} = 1.425 \]

### Final Answer
\[ \frac{\lambda}{1000} = 1.425 \]