How Do You Calculate Normal and Inverse Normal Probabilities Using Technology?

Use normal CDF when you know the boundary value and need a probability. Use inverse normal when you know a cumulative probability and need the corresponding value of the variable.

Both routines use the area under the normal curve. Normal CDF accumulates area between chosen bounds, whereas inverse normal reverses this process: it starts with a left-tail area and returns the boundary. First identify whether the unknown is a probability or a variable value.

Using Normal CDF

For $X\sim N(\mu,\sigma^2)$, enter the lower bound, upper bound, mean $\mu$, and standard deviation $\sigma$ into your GDC’s normal cumulative distribution function.

For example, suppose $X\sim N(60,8^2)$. To calculate $P(X<70)$, use normal CDF with lower bound $-\infty$, upper bound , mean , and standard deviation :

$$P(X<70)=0.894350\ldots\approx 0.894.$$

On calculators without an infinity symbol, use the device’s recommended very large positive or negative bound.

Using Inverse Normal

The inverse normal function returns the value $x$ for a given cumulative area to the left:

$$P(X<x)=p.$$

For the same distribution, the value below which $90\%$ of observations lie is found by entering left-tail area $0.90$, mean $60$, and standard deviation $8$:

$$x=70.252\ldots\approx 70.3.$$

For a right-tail probability, convert it first. If $P(X>x)=0.10$, enter the left-tail area $1-0.10=0.90$.

For a central $80\%$ interval, split the remaining $20\%$ equally between the tails. The lower endpoint uses left-tail area $0.10$ and the upper endpoint uses $0.90$; apply inverse normal twice.

A common misconception is entering the variance $\sigma^2$ into the GDC. Although the distribution is written $N(\mu,\sigma^2)$, the calculator normally requires the standard deviation $\sigma$.

Exam Technique

This is SL 4.9 Normal distribution. Show the distribution and GDC setup before the answer, retain full precision, and round only the final value, normally to three significant figures. At SL, do not standardize using $z$; use technology directly.