The mean and variance of 50 observations are 16 and 4, find the sum of overvatives and the sum of squares of observations.

Given the mean (\(\mu\)) and variance (\(\sigma^2\)) of 50 observations, we can calculate the sum of observations (overvatives) and the sum of squares of observations.

### Step 1: Calculate the Sum of Observations
The mean is given by:
\[
\mu = \frac{\text{Sum of observations}}{n}
\]
Where:
- \( \mu = 16 \) (mean)
- \( n = 50 \) (number of observations)

So, the sum of observations is:
\[
\text{Sum of observations} = \mu \times n = 16 \times 50 = 800
\]

### Step 2: Calculate the Sum of Squares of Observations
The variance is given by:
\[
\sigma^2 = \frac{\text{Sum of squares of observations} - \frac{(\text{Sum of observations})^2}{n}}{n}
\]
Where:
- \( \sigma^2 = 4 \) (variance)
- \( \text{Sum of observations} = 800 \)
- \( n = 50 \)

Rearranging the formula to find the sum of squares of observations:
\[
\text{Sum of squares of observations} = n \times \sigma^2 + \frac{(\text{Sum of observations})^2}{n}
\]

Substituting the known values:
\[
\text{Sum of squares of observations} = 50 \times 4 + \frac{800^2}{50}
\]

Calculating it step by step:
\[
\text{Sum of squares of observations} = 200 + \frac{640000}{50} = 200 + 12800 = 13000
\]

### Final Answer:
- **Sum of observations**: 800
- **Sum of squares of observations**: 13,000

Would you like more details or have any questions? Here are some related topics you might find interesting:

1. How to derive the formula for variance.
2. Examples of variance calculations with different sample sizes.
3. Understanding the relationship between mean and sum of observations.
4. How to interpret variance in statistical data.
5. Applications of mean and variance in real-world scenarios.

**Tip:** Always remember to square the sum of observations and divide by the number of observations when calculating variance.

Discover how to calculate the sum of observations and the sum of squares of observations from given mean and variance values. This problem involves understanding statistical concepts like mean, variance, and their formulas. Suitable for Grades 10-12.

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