91Ó°ÊÓ

Before transformations:

Mean, $\mathrm{μ}=12$

Standard deviations, $\mathrm{σ}=3$

After transformations: ${\mathrm{μ}}_N=75$

New Standard deviations,${\mathrm{σ}}_N=12$

The standard deviation was used to account for the distance between the numbers in the first stage.

When each score is multiplied by four, the distance between the numbers increases by four times.

When the distance between two numbers is multiplied by four, the rate of deviations is multiplied by four.

As a result, the standard deviation is converted to $12$ instead of $3$

In the next step,

According to the first phase, the mean has been changed to $48$ instead of $12$

However, we must reach a $75th$ percentile average.

To do this, we must add $27$ to each score, resulting in a $75-$point average.

Whereas,

The standard deviation and distance between numbers will remain unchanged.

Therefore,

To carry out the modifications,

Ms. Martin must multiply each score by four and then add $27$to each result.