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).