if the variance of a dataset is 50 and all data points are incresases by 100% then what will be the variance

To solve this problem, let's first understand what happens to the variance when all data points in a dataset are increased by a certain percentage.

### Understanding the Impact on Variance

1. **Variance Definition**: The variance of a dataset is a measure of the spread of the data points around the mean. It is defined as the average of the squared differences from the mean.

2. **Scaling Data Points**: If each data point in a dataset is scaled by a factor \( k \), the new variance \( \text{Var}(kX) \) of the dataset will be \( k^2 \times \text{Var}(X) \). This is because variance is affected by the square of the scaling factor.

3. **Applying the Scaling to the Problem**:
   - In this case, each data point is increased by 100%. 
   - Increasing a data point by 100% means multiplying it by 2 (since \( 1 + 100\% = 1 + 1 = 2 \)).

4. **Calculating the New Variance**:
   - Original variance, \( \text{Var}(X) = 50 \).
   - Scaling factor, \( k = 2 \).
   - The new variance will be \( \text{Var}(2X) = 2^2 \times \text{Var}(X) \).

\[
\text{New Variance} = 2^2 \times 50 = 4 \times 50 = 200
\]

### Conclusion

If all data points in a dataset are increased by 100%, the variance of the dataset will increase by a factor of 4. Therefore, the new variance will be **200**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What happens to the standard deviation when all data points are increased by 100%?
2. How does adding a constant to all data points affect the variance?
3. What is the difference between variance and standard deviation?
4. If the original dataset had a mean of 10, what would the new mean be after increasing each data point by 100%?
5. How is variance affected if all data points are multiplied by a negative number?

**Tip:** Remember that variance is sensitive to changes in data scale but not to shifts (adding or subtracting a constant).

Understand how increasing all data points in a dataset by 100% affects variance. Explore the variance formula, calculation process, and the impact of scaling data. Suitable for advanced high school and college students.

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