How Do You Calculate Normal Probabilities Using Technology?

Use your GDC’s normal cumulative distribution function (normal CDF), entering the lower bound, upper bound, mean, and standard deviation. This is IB Mathematics: Analysis and Approaches SL, sub-topic SL 4.9: Normal distribution and calculations, and is normally tested on Paper 2.

If $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, write

$$X \sim N(\mu,\sigma^2).$$

Although the distribution notation contains the variance $\sigma^2$, most GDC normal CDF commands require the standard deviation $\sigma$, not the variance.

The CDF returns the accumulated area under the normal curve, so the bounds must match the region described by the inequality. Sketching a quick bell curve can help you identify whether the required region is a left tail, right tail, or central interval.

Because a normal distribution is continuous, $P(X<a)=P(X\leq a)$; including or excluding an endpoint does not change the probability.

For one-sided probabilities, many GDCs accept $-\infty$ or $\infty$; otherwise use values such as $-10^{99}$ and $10^{99}$.

For example, suppose

$$X \sim N(70,8^2).$$

To calculate $P(62<X<78)$, enter:

• lower bound: $62$
• upper bound: $78$
• mean: $70$
• standard deviation: $8$

The GDC gives

$$P(62<X<78)=0.682689\ldots \approx 0.683.$$

The interval is one standard deviation either side of the mean, so the result also agrees with the approximate $68\%$ normal-distribution rule.

For $P(X>82)$, use $82$ as the lower bound and a very large positive upper bound. This gives

$$P(X>82)\approx 0.0668.$$

At SL 4.9, direct normal-distribution technology is expected; standardizing with a $z$-value is unnecessary.

Exam technique: Write the distribution and probability statement before using the GDC, then give the answer to three significant figures unless instructed otherwise. A common misconception is entering $\sigma^2$ into the GDC: enter $\sigma$ instead.