The mean of 200 items is 48 and their standard deviation is 3. Find the sum of squares of all items.

Answer

To find the sum of squares of all items given the mean and standard deviation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Number of items (n) = 200
- Mean (x̄) = 48
- Standard Deviation (σ) = 3

2. **Understand the Relationship Between Variance, Mean, and Sum of Squares:**
- The formula for variance (σ²) is given by:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \left(\bar{x}\right)^2 \]
- Rearranging this formula to find the sum of squares (Σx_i²):
\[ \sum_{i=1}^{n} x_i^2 = n \cdot \sigma^2 + n \cdot \left(\bar{x}\right)^2 \]

3. **Calculate Variance:**
- Since standard deviation (σ) is given as 3, we calculate variance (σ²):
\[ \sigma^2 = 3^2 = 9 \]

4. **Calculate Mean Squared:**
- Calculate the square of the mean:
\[ \left(\bar{x}\right)^2 = 48^2 = 2304 \]

5. **Substitute Values into the Sum of Squares Formula:**
- Now substitute n, σ², and (x̄)² into the rearranged formula:
\[ \sum_{i=1}^{200} x_i^2 = 200 \cdot 9 + 200 \cdot 2304 \]

6. **Perform the Calculations:**
- Calculate each term:
\[ 200 \cdot 9 = 1800 \]
\[ 200 \cdot 2304 = 460800 \]
- Now add these results together:
\[ \sum_{i=1}^{200} x_i^2 = 1800 + 460800 = 462600 \]

### Final Answer:
The sum of squares of all items is **462600**.

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