Each
![](https://latex.codecogs.com/png.latex?\bg_white z)
-score is calculated using the formula
![](https://latex.codecogs.com/png.latex?\bg_white z = \cfrac{x - \mu}{\sigma})
where
![](https://latex.codecogs.com/png.latex?\bg_white x)
is the student's score,
![](https://latex.codecogs.com/png.latex?\bg_white \mu)
is the class average, and
![](https://latex.codecogs.com/png.latex?\bg_white \sigma)
is the class standard deviation.
So, if you substitute Carol's results, you get that
![](https://latex.codecogs.com/png.latex?\bg_white 0.46 = \cfrac{32 - \mu}{\sigma})
.
Doing the same for Zachary and Matilda, you will have three equations with only two unknowns. So, you can find what value of
![](https://latex.codecogs.com/png.latex?\bg_white \mu)
and
![](https://latex.codecogs.com/png.latex?\bg_white \sigma)
work for two of them, and then figure out for the third whether the problem is an incorrect
![](https://latex.codecogs.com/png.latex?\bg_white x)
or
![](https://latex.codecogs.com/png.latex?\bg_white z)
and then find the correction that is needed.