Standard Deviation Variance Rule Explained

The problem asks for the variance of a new set of values after each original value is increased by a constant. We are given the standard deviation of the original 12 values.

Key Statistical Property

A fundamental property in statistics is that adding a constant value to every data point in a dataset does not change the dataset's spread or variability. This means the standard deviation and variance remain unchanged.

• Let the original standard deviation be $\sigma_{old}$.
• Let the new standard deviation be $\sigma_{new}$.
• If each value is increased by a constant $k$, then $\sigma_{new} = \sigma_{old}$.

Calculating the New Variance

We are given:

• Original standard deviation, $\sigma_{old} = 3$.
• Each value is increased by $k = 4$.

According to the property mentioned above, the new standard deviation is the same as the old standard deviation:

$ \sigma_{new} = \sigma_{old} = 3 $

Variance is the square of the standard deviation ($\sigma^2$).

Therefore, the variance of the new set of values is:

$ \text{Variance}_{new} = (\sigma_{new})^2 $

$ \text{Variance}_{new} = (3)^2 $

$ \text{Variance}_{new} = 9 $

Thus, the variance of the new set of values is 9.