The top button on the Normal Tool says "Normal Tool." This allows us to find probabilities for any normal distribution. The bottom button says "Standard Normal Tool." This allows us to find probabilities for a special case of the normal distribution (sometimes known as the z-distribution or unit normal).

Instructions will be given for the general case first (Normal Tool). Then we will go on to instructions for the simpler case (Standard Normal Tool).

1. Normal Tool: Finding probabilities for any normal distribution

We will use an example to illustrate the step by step use of the Normal Tool. As a big picture overview, let's suppose that scientists doing research take some interesting phenomena in nature, reduce it to numbers by measurement operations, and then model those numbers as a random variable. A frequently used random variable is the normal probability distribution.

Height Example. In this example we will be interested in the heights of northern European males. We take such a person and reduce him to a single number via the usual operations for measuring someone's height. Then we model the height of northern European males as a normal population with mu = 150 cm and sigma = 30 cm. In other words, our model is N(150, 30).

Click on the top button--Normal Tool.

What is the Probability Between 140 and 170? We have modeled the heights of northern European males as N(150, 30). If that model is true, and if we sample one man from that population, what are the chances he has a height between 140 cm and 170 cm?

Total Area under the Normal curve. Remember that we can interpret the area below a normal curve as probability. The total area below the normal curve (from negative infinity up to positive infinity) is assumed to be 1. That is, the probability that a man's height will fall between negative and positive infinity is 1.

Area Between. Since the total area under the curve is 1, the area between 140 and 170 must be some fraction of 1. On the Normal Tool the first thing you must do is make sure that the little icon indicating "area between" is clicked (see lecture graphic). "Between" is the default setting for the Normal Tool, so when you open it up it automatically gives you the area between two values.

NOTE: Do not use the 'ENTER' key on your keyboard to enter values.
DO enter values by clicking on the Buttons next to the field where you enter the value.

Set mu. On the lecture graphic, arrows point to little boxes where you can set mu and sigma. First type in the mu which is relevant to whatever example you are working on. Then click the "ENTER MU (50 - 500)" button right next to the box where you entered the value of mu. (Note: The Normal Probability Tool only accepts values of mu between 50 and 500.) For our height example, I have entered mu = 150.

Set sigma. The lecture graphic also shows where to enter the value of sigma (toward the lower right-hand corner of the tool). For our height example, I have entered sigma = 30. You must type in the value of sigma and then press the "ENTER SIGMA" button next to it. DO NOT use the 'Enter' or 'Return' key on your keyboard to enter scores.

Set lower value. We are looking for the area (probability) between two values. The lecture graphic shows you where you can enter the lower of the two values. Once you type in the number, click on the button which says "ENTER LOWER SCORE." For the height example, the lower value is 140 cm, so on the lecture graphic I have set the lower value to 140.

Set upper value. Similarly, as you can see on the lecture graphic, there's a box where you can enter the upper score. Once you type in the upper value number, click on the button which says "ENTER UPPER SCORE." Following the height example, I have set the upper score to 170 on the lecture graphic.

Find probability. You have entered mu, sigma, upper score and lower score. Now you are ready to find the answer to the question. The lecture graphic points to a box where the probability will appear. All you have to do is read it and record it. For the height example, the probability that a northern European man's height will fall between 140 and 170 cm is .3747.

Black Area. Probability is represented by the black area under the curve. Look at the normal distribution on Normal Probability Tool. The black area between 140 and 170 represents a probability of .3747.

We have set up a correspondence between area on a picture we can see and the concept of probability. This allows us to picture probability clearly and simply.

Click and drag. Play with the Normal Tool. You'll notice that there are two blue pointers just below the normal curve. One is labeled "lower score" and the other "upper score." If you click on either of them, you can drag the black area to whatever value you want. The upper or lower score changes accordingly. The probability changes also accordingly. Try it and watch how the black area and the probability change together.

Positive and Negative Infinity. Play with the Normal Tool some more. You'll notice that to the right of the white boxes where you enter the upper and lower scores there are buttons labeled "-oo" and "+oo." This is as close as we could get to the symbols for negative infinity (-oo) and positive infinity (+oo). If you click on the minus infinity button (-oo) the lower score will become minus infinity. If you click on the plus infinity button (+oo) the upper score will become plus infinity. Try this out now. Find the probability that a height will fall between minus and plus infinity. (Answer: 1.) What is the probability that a height will fall between minus infinity and 150 cm? (Answer: .5.)

Now we will turn to a related question--what is the area probability outside of two values?

What is the Probability Outside 140 and 170? If we sample one northern European male, what's the probability that his height will fall outside of 140 and 170? In other words, what are the chances that he'll be either below 140, or he'll be above 170 in height? That's what we mean by the word "outside."

Area Outside. The first thing you have to do is click the icon for "Area Outside" on the Normal Tool. The Normal Tool will now show you the area outside 140 and 170. It will also change the probability.

And then you do exactly the same thing that you did before. For our current example, you set mu at 150, set sigma at 30, set the lower value at 140, set the upper value at 170.

Find probability. Then you simply read the probability. This time it is .6253. The probability that a height will fall outside (above or below) 140 and 170 cm is .6253.

Now we will turn to finding the area above a certain value.

What is the Probability Above 170? Perhaps a basketball coach is interested in tall men. We have modeled the heights of northern European males as N(150, 30). If that model is true, and if we sample one man from that population, what are the chances he has a height above 170 cm? This question implies that the lower score will be 170 and the upper score will be plus infinity. All scores above 170 will fall between 170 (on the low end) and plus infinity (on the high end).

Set mu: 150.

Set sigma: 30

Click Between Icon.

Set lower score: 170

Set upper score: +oo.

Read probability: .2546. There's about a 25% chance that the man would have a height above 170 cm. That's represented by the black area under the normal curve.

Now we will turn to a special case of the normal distribution: The Standard Normal.

Click on "Back to Menu" at the bottom of the tool.

2. Standard Normal Tool: Finding probabilities for N(0, 1)

N(0, 1): There is a particular form of the normal distribution which is very commonly used in statistics. It is called unit normal or the standard normal or the z distribution. The unit normal is simply a normal distribution which has a mean (mu) = 0, and a standard deviation (sigma) = 1. In more compressed symbols the unit normal is N(0, 1).

Everything works exactly the same with the unit normal as it does for any normal. So everything we've already learned applies to this topic. We will just be using a particular member of the normal family of distributions. This member of the family has mu = 0 and sigma = 1 and is sometimes called the z distribution.

z-Tables in Stat Books. The unit normal is the particular form of the normal that is found in z-tables in the back of stat books. "In the old days" before we had interactive programs like Normal Tool, we had to convert all questions to z scores and look up probabilities in z-tables.

[If you are still using the Standard Normal Tool, click on "Back to Menu" at the bottom so you see a simple page with two buttons.]

Click on the lower button--Standard Normal Tool.

Question. Suppose that we have N(0, 1) as our probability model. What is the probability of a score between -1 and +1 on (N(0, 1)?

Don't need to set mu and sigma. On the unit normal, N(0, 1), mu is always 0 and sigma is always 1. So you don't need to set them.

Click on the Area Between Icon.

Set lower and upper scores. Set the lower and upper score as we did above. In this case the lower score is -1 and the upper score is +1. When you start the Unit Normal option, it will come up with minus one and plus one as the lower and upper scores. So we don't have to do anything to solve the particular question we have asked.

Read the probability. The answer is .6827. This should be familiar to you. If it's not, it soon will be.