Normal distribution: normalcdf, invNorm and normalpdf in Scientific calculator plus 991 for Android

The normal distribution has three functions on the calculator, one for each of three questions about the bell curve: how likely is a value between two bounds (normalcdf), which value has a given share of the curve to its left (invNorm), and how high is the curve at a point (normalpdf). They need no data entered in STAT mode: you give them the mean and the standard deviation directly.

The distribution menu

Press SHIFT 3 DISTR. The menu holds sixteen functions covering the normal, Student-t, chi-square, F, binomial, Poisson and geometric distributions. Type part of a name in the search box at the top of the menu to narrow the list, then choose one. Its whole name goes into the calculation; you type its arguments, and = gives the answer like any other calculation.

The names can also be typed letter by letter on the calculation screen, without opening the menu. The menu is quicker, since it inserts the whole name in one tap.

Every entry shows the arguments it takes, in order. Arguments inside [ ] are optional: leave them out and a standard value is used. For the normal functions, mean and sigma are the mean and the standard deviation of the distribution; left out, they are 0 and 1, which is the standard normal.

normalcdf: the probability between two bounds

normalcdf(lowerbound, upperbound[, mean, sigma]) gives the probability that a normally distributed value falls between the two bounds.

The probability that a standard normal value falls between 2 and 3:

The same interval on a distribution with mean 0.2 and standard deviation 0.6:

Below a value: the one-sided trick

A probability below a value needs a lower bound all the same. Put a number far below the mean where −∞ would go, such as −10⁹⁹, as the manual does. The probability that a value from a distribution with mean 35 and standard deviation 2 falls below 36:

invNorm: the value behind an area

invNorm(area[, mean, sigma]) runs normalcdf backwards: given an area between 0 and 1, it returns the value of x with that area to its left. Feeding the answer above back in, shortened, returns the 36 it came from:

The standard normal value with 97.5% of the area to its left leaves 2.5% in each tail, so it is the number behind a 95% confidence interval:

normalpdf: the height of the curve

normalpdf(x[, mean, sigma]) gives the probability density at x: the height of the bell curve at that point, not a probability. For a probability over an interval, use normalcdf.

With the mean and sigma left out, normalpdf(0) is the peak of the standard normal curve:

normalcdf is the area under this curve between two bounds, the kind of area a definite integral finds under a curve you write yourself.

normalcdf or P, Q and R

STAT mode has its own normal areas: P(t), Q(t) and R(t) on the Distribution tab of the STAT menu, SHIFT 1 STAT. They work on the standard normal only, and are the natural next step after →t turns a value from your data into a standard score; Z-scores, P, Q and R covers them. normalcdf needs no data, and takes any mean and standard deviation.

At a glance