Normal Probabilities on the TI-83
Tom Linton, http://www.central.edu/homepages/lintont/

This file is a brief summary of how to utilize the TI-83 calculator to:

  1. Calculate probabilities associated with intervals of values from a normal distribution;
  2. Draw plots of the "area under a normal density function"; and
  3. Calculate inverse values (from a normal distribution) which correspond to a given percentage, such as "what value has 60% of the data below it?"
The three main commands we'll use are all located on the distribution menu (or its DRAW sub-menu). The distribution menu is the [2nd] function of the [VARS] key. It is located near the top right corner of the keypad and is labeled [DISTR].

Calculating Normal Probabilities

Two sided intervals
The normalcdf command, [2nd] [DISTR] [2], makes calculating probabilities for intervals of normal values straightforward. You must tell this command where the interval begins (i.e. give the left or lower endpoint of the interval), where the interval ends (its right or upper endpoint) and optionally both the mean and standard deviation (in that order) of the distribution. If you leave off the mean and standard deviation parameters, the TI-83 assumes the mean is zero and the standard deviation is 1 (i.e. the standard normal distribution is used). The general syntax of the normalcdf command is:
normalcdf( start, end, mean, std_dev)
The TI-83 will return the percentage (as a decimal number from 0 to 1) of data points x in a normal distribution with the given mean and standard deviation which lie between x = start and x = end. For example, if the heights of 6 year old girls are normally distributed with a mean of 46 inches and a standard deviation of 2.7 inches, then we can calculate the likelihood that a randomly chosen 6 year old girl has height between 44 and 48 inches with the command
normalcdf(44,48,46,2.7)
The TI-83 returns 0.54115, indicating that about 54% of six year old girls have heights between 44 and 48 inches. Use your calculator to verify this command by entering the centered command above, pressing [ENTER] and checking that your calculator returns the answer .54115 (roughly).
One sided intervals
Many questions dealing with normally distributed values have only one endpoint. For example, what percentage of six year girls have a height less than 44 inches? or greater than 42 inches? These questions involve plus or minus infinity as the left or right endpoint of the interval, and are called one sided intervals. If you are looking for the likelihood that a six year old girl has a height which is less than 44 inches, you are really looking for the percentage of six year old girls with a height between minus infinity inches and 44 inches. There are several ways to deal with one sided intervals like this (where either the start is minus infinity, or the end is positive infinity). One such method relies on the fact that there is virtually no probability associated to normal values which are more than 5 standard deviations below, or 5 standard deviations above the mean. Using this fact, we can freely replace minus infinity with any value which is 5 (or more) standard deviations below the mean, and instead of using plus infinity, simply use a number which is 5 (or more) standard deviations above the mean. Here is an example using the normal distribution of the heights of 6 year old girls ( i.e. normal with a mean of 46 inches and a std. dev. of 2.7 inches).

A certain water-slide park requires that all sliders must be 48 inches or taller to go down their water slides. What percentage of 6 year old girls are allowed to slide in this park? The correct answer would correspond to the area under the normal density function (the one with a mean of 46 and a standard deviation of 2.7) from x = 48 to infinity. We can essentially calculate this with the command normalcdf(48, 59.5, 46, 2.7) = 0.2294. Thus, only about 23% of six year old girls are tall enough for this water park.

NOTE: 59.5 = 46 + 5*2.7 (mean + 5 * std_dev) and you can let the calculator do this arithmetic. Rather than figuring out what 46 + 5*2.7 is, you can simply enter the "ending value" as

46 + 5*2.7
the calculator will do the arithmetic automatically. In a similar fashion, the percentage of 6 year old girls whose heights are less than 40 inches can be accessed by pressing
[2nd][DISTR] [2] [4][6][-][5][*][2][.][7][,] 40, 46, 2.7 )
and then pressing [ENTER]. The result is 0.01313..., so only about 1.3% of six year old girls are less than 40 inches tall.

Normal areas plots

It is always a good idea to visualize the areas you are calculating. The ShadeNorm command (located on the [DRAW] sub-menu of the [DISTR] menu) does just this. You pass the ShadeNorm command exactly the same values as you pass to the normalcdf command (start, end, and an optional mean and std. dev.), but you should set up your plot window before giving the ShadeNorm command. In general, set X to run from 4 std. dev.'s below the mean to 4 std. dev.'s above the mean (so Xmin = mean - 4 * std_dev, Xmax = mean + 4*std_dev). You can run the y values from a small negative number (or zero) to 0.4 / std_dev. The value 0.4 / std_dev will normally give a good view in the y direction. Because the ShadeNorm command actually displays text on your screen, I prefer setting YMin = -YMax, which usually avoids the text overlapping the picture. Here are the steps I would use to draw a picture of the percentage of six year old girls with heights from 44 to 48 inches.
  1. Turn off all STAT PLOTS and functions on the Y= screen.
  2. From the WINDOW screen, set XMin = 46 - 4*2.7, XMax = 46 + 4*2.7, YMin = -0.4 / 2.7, and YMax = 0.4 / 2.7.
  3. Press [2nd][DISTR][right arrow key] to access the DRAW submenu of the [DISTR] menu. Press [ENTER] to paste a copy of the ShadeNorm command to the homescreen. Finally, give the same arguments as you would to the normalcdf command, namely 44, 48, 46, 2.7) and press [ENTER]. You should see a plot similar to the one below.
Normally, the drawing displayd will dissappear once you alter any plotting value. You can also erase the drawing by pressing [2nd][DRAW][ENTER] which executes the ClrDraw command.

Inverse values

The inverse normal command, InvNorm, allows you to compute the value of a normal distribution which lies at a certain percentage in the distribution, for example, which x has 92% of the distribution below it? The first argument must be a decimal between 0 and 1. It represents the percentile value you are looking for (0.25 for the first quartile, .99 for the data point with only 1% of the distribution to its right). You can optionally pass the InvNorm command the mean and standard deviation of the distribution. If you leave these values off, a mean of 0 and standard deviation of 1 are used. As an example, the start of the third quartile (75% mark) for heights of six year old girls is obtained with the command
InvNorm(.75,46,2.7)
by pressing [2nd][DISTR] [3] and then entering the arguments and closing parenthesis. The result (check for yourself) is 47.82 inches, so 75% of all six year old girls are shorter than 47.82 inches, equivalently, 25% of all six year old girls are 47.82 inches or taller.