grading on a curve using normal distribution suppose we have a grade list:
$
\text{grades}=\{2,3,5,7,8,10,9,9.75,8,0,11,10,10,3,5.25,13,14,20,18,9\};
$
which mean equals to 8.75 and Standard deviation  is 5.06471. we want to improve the grade average using normal distribution and increasing student grades. any idea how?
 A: Here's one possible answer:
I assume these grades are out of $20$. As one student has attained the maximum, we can't just shift all grades up by some amount, as that grade of $20$ among others might exceed the maximum. 
Note that $20$ is some number of standard deviations above the norm, $Z(20) = (20 - \mu_{old})/\sigma_{old}$.
Suppose we shift all grades up by $S$ and hence the new mean is $$\mu_{new} = \mu_{old} + S$$ Then we want $20$ to remain fixed at $20$. We do that by modifying down the standard deviation to $\sigma_{new}$ such that $Z(20)$ remains constant, i.e., choose $\sigma_{new}$ such that
$$Z(20) = \frac{20 - \mu_{new}}{\sigma_{new}} $$
That is $\displaystyle \sigma_{new} = \frac{20 - \mu_{new}}{Z(20)}$. 
Each old grade $g$ has its own $Z$ score, $Z(g) = (g - \mu_{old})/\sigma_{old}$. The rule then for rescaling a grade $g$ is
$$g \mapsto \mu_{new} + Z(g)\sigma_{new}$$
Checking, the maximum score of $20$  is fixed:
$$20 \mapsto \mu_{new} + Z(20) \cdot \frac{20 - \mu_{new}}{Z(20)} = 20$$

For example, if the mean increased by $3$, here's how applying the rule would modify the scores:

