# Shifting Marks with the Bell Curve

Don’t worry; we don’t do this sort of thing. We use criterion-referenced assessment, so your peers’ performance doesn’t affect your evaluation.

We looked at a question in class (page 431 #11) in which a professor was “belling” the marks. Madison asked if there were marks that would be shifted down as a result of the belling.

There are, sort of. Here’s the math.

Let’s call the original mark $x_1$ and the resultant mark $x_2$. We want to know for which values of $x_1$

$x_2 \textless x_1$

In the belling process the $z$-score is maintained for each grade; so

$\frac{x_1-55}{13}=\frac{x_2-70}{10}$

Solving for $x_2$ yields

$x_2 = 10\left( \frac{x_1-55}{13} \right) + 70$

Substituting into the inequality

$x_2 \textless x_1$

yields

$x_1 \textgreater 120$

which is, you know, impossible anyway. :)