Normal Curve Calculations

Using the TI-83/TI-84 Graphing Calculators

Dr. John C. Nardo

Division of Mathematics & Computer Science

Oglethorpe University

Atlanta, GA 30319
(404) 364-8327
jnardo@oglethorpe.edu

www.oglethorpe.edu/faculty/~j_nardo/

### Introduction

In order to work with a normal curve, you must first identify its two parameters:  its mean and its standard deviation .  These two pieces of information uniquely identify the particular normal distribution, denoted , with which you will work.  In traditional, pencil-and-paper statistics, each situation/problem is transformed via z-scores to a situation/problem about the standard normal distribution ( and ).

Such work is unnecessary when using technology for normal curve calculations.

## Main Example

The fall deer population in Mesa Verde National Park is approximately normal with mean 4400 deer and standard deviation 620 deer.

You may sketch a graph of this normal distribution via technology; however, in this case, it is easier and quicker to sketch by hand. WARNING:           In order to turn this screen dump into an appropriate sketch of this normal distribution, we would need to:  (1) mark the mean, (2) note the standard deviation to the side, (3) put arrows on the axis and the normal curve, (4) write the variable underneath the axis, and (5) put several numbers/ values on the axis to orient the reader.

The “normalcdf” Command

The normalcdf command calculates the area under the normal curve between two given points.  It requires four inputs/numbers for it to function – in this precise order – minimum number/starting point, maximum number/ending point, mean, and standard deviation.  The normalcdf command is found in the distribution menu.  Press [2nd] [VARS] to access this menu.  You should see the following screen: .

The first command, normalPdf, sketches Pictures of normal curves (after much tedious window searching).  As mentioned above, sketch the curves by hand!  The second command, normalCdf, performs Calculations.  Press the down arrow once to highlight/select the normalcdf command.  Then press the [ENTER] key to paste it back to the home screen. The calculator is now ready for you to input the four parameters/inputs – separated by commas.

Fall Seasons in which the Deer Population will be between 4000 and 4500

The four parameters/inputs for the normalcdf command in this case are:  minimum=4000, maximum=4500. mean=4400, and standard deviation=620.  Type these four numbers in order, separated by commas.  After pressing the [ENTER] key to execute the command, your screen should look like this: .

This number represents the area under this normal curve between 4000 and 4500.  More importantly for this situation, it represents the proportion of Fall Seasons during which the deer population will be between 4000 and 4500.  We may conclude that 30.47% of Fall Seasons will have a deer population between 4000 and 4500!

Fall Seasons in which the Deer Population will be above 5000

If we wish to know the percentage of Fall Seasons for which the deer population is above 5000, we must be careful.  We still must give four inputs to the calculator!  We envision starting at 5000 and ending at , positive infinity.  We will represent positive infinity with the largest positive number that the calculator can process:  +1 followed by 99 zeros!  We write this in scientific notation by +1E99.  The “E” from scientific notation is accessed by shifting the comma key:  pressing [2nd] [,] .

Thus, the command for this normal curve calculation is: .

The proportion of Fall Seasons with a deer population above 5000 is 0.1665866273; the percentage is 16.66%.

Fall Seasons in which the Deer Population will be below 5000

Clearly, if 16.66% of the time the population is above 5000, then we would deduce that 100% – 16.66% = 83.34% of the time, the population would be below 5000.  Let’s check this deduction using similar reasoning as above.

In order to calculate the percentage of Fall Seasons with deer populations below 5000, we envision starting at negative infinity and stopping at 5000.  We represent negative infinity by – 1 followed by 99 zeros, i.e. – 1 E 99. Indeed, the direct command gives the same result as our deduction!

WARNINGS:        (1)  Be sure to use the negative button beneath the number 3 key and not the subtraction button!

(2)     Be sure that the negative sign goes in front of the one and not the 99!

## Direct versus Indirect Problems

So far, we have been solving direct problems.  Given one or two numbers/values of the variable, calculate the percentage of data associated with the number(s).

We could turn this problem around to solve an indirect problem.  Given a percentage, find the number/value of the variable associated with that percentage!

## The “InvNorm” Command

The command for solving inverse problems, handily named “InvNorm,” is found in the same menu as our previous command, the distributions menu.  Press [2nd] [VARS] to access this menu.  The third command from the top is the inverse command.  Press down arrow twice to highlight/select this command. Then press [ENTER] to paste it to the home screen: .

It requires three parameters/inputs to make it work:  percentage of area to the LEFT (written as a decimal), mean, and standard deviation.

### “Lean” Fall Seasons

A “lean” Fall Season is one in which the deer population is unnaturally low.  We must define what we mean mathematically.  If the population is at its lowest 5%, then we will define a Fall Season to be “lean.”  Find the number associated with being a “lean” Fall Season.

If we are in the lowest 5%, then we would be on the left side of the normal distribution; indeed, 5% of the curve will be to our LEFT.  The InvNorm command will need these parameters:  0.05, 4400, 620. Thus, if we count 3380 or fewer deer in a particular Fall Season, we will be in a “lean” year!

NOTE:                    You could check that this works out!  If you calculate the area under the curve from negative infinity to 3380, you should get five percent.  You can always check inverse problems via a direct calculation. ###### “Abundant” Fall Seasons

An “abundant” Fall Season is one in which the deer population is unnaturally high.  We must define what we mean mathematically.  If the population is at its highest 5%, then we will define a Fall Season to be “abundant.”  Find the number associated with being an “abundant” Fall Season.

If we are in the highest 5%, then we would be on the right side of the normal distribution.  Since 5% of the curve will be to our RIGHT, we know that 95% will be on our LEFT.  The InvNorm command will need these parameters:  0.95, 4400, 620. Thus, if we count 5420 deer or more in a particular Fall Season, we may classify it as an “abundant” year!

As before we can check this indirect calculation via a direct one.  The percentage of Fall Seasons from 5420 to positive infinity should be 5%.  Indeed it is! 