How Do You Do an Inverse Normal Calculation?

Use an inverse normal calculation when a probability or area is given and you need to find the corresponding value of the normally distributed variable. Convert the given area into a left-tail cumulative probability, then enter that probability, the mean and the standard deviation into your GDC.

This is IB Mathematics AA SL 4.9: Normal distribution and calculations and is normally tested on Paper 2 because technology is required.

If $X\sim N(\mu,\sigma^2)$, the inverse normal function finds $x$ from

$$P(X\leq x)=p.$$

The normal cumulative distribution function gives the area to the left of a chosen value. Inverse normal reverses this process: area goes in and the boundary value comes out. It does not calculate a probability from a known score; that is a normal CDF calculation. Also enter $\sigma$, not the variance $\sigma^2$.

On most GDCs, select an inverse normal command such as $\operatorname{invNorm}$, then enter the left-tail area $p$, mean $\mu$, and standard deviation $\sigma$. Calculator menus vary, but the mathematical inputs are the same.

If the calculator asks for bounds rather than a named inverse command, choose the inverse-normal distribution menu and specify the relevant tail or cumulative area.

For example, suppose examination scores satisfy $X\sim N(70,8^2)$. Find the score exceeded by the highest $10\%$ of students.

The required score $x$ satisfies

$$P(X>x)=0.10.$$

Convert this to a left-tail probability:

$$P(X\leq x)=1-0.10=0.90.$$

Using the GDC,

$$x=\operatorname{invNorm}(0.90,70,8)=80.3$$

to three significant figures. Therefore, students in the highest $10\%$ score above approximately $80.3$.

A common misconception is entering $0.10$ directly for an upper-tail question. Most inverse normal commands require the area to the left, so the correct input here is $0.90$.

In an IB response, write the probability statement and tail conversion before giving the GDC result. Use exact values or three significant figures unless the question specifies different accuracy.