Finding Probabilities in a Normal Distribution

Lesson Plan - Get It!

Audio:

The shaded region shown above is a difficult one to find the area of without the right tools. In this lesson, we will learn what those tools are and why it matters that we find that area.

The image above is of a normal distribution.

A normal distribution, often referred to as a bell curve, is a symmetric distribution that comes up often in statistics. Many data sets are distributed in normal distribution.

When we find probabilities in a normal distribution, we are always going to look at ranges. The area under the curve in a particular range is the probability that a randomly selected value from the distribution falls in that range. Therefore, the shaded region is representing a probability.

Let's look more closely at these ideas through an example problem:

Summer monsoons bring about 80% of the yearly rainfall in India. The exact amount of rain varies from year to year and follows a normal distribution with a mean of 862 milliliters and a standard deviation of 82 milliliters.

• What is the probability that the rainfall amount during monsoon season for a random year is between 750 and 800 milliliters?

Step 1:

We will draw a normal curve, indicating the mean in the center.

Use the Normal Curves Template, found under the Downloadable Resources in the right-hand sidebar, to draw your own.

The normal curve will look like this:

Step 2:

Place the upper and lower bounds of the range for which we want the probability on the curve.

It will look like this:

Step 3:

Shade that region. This is the region that you want to know the area of, since the area under the curve is the probability.

Your shaded region will look like this:

Step 4:

Now that you have a visual representation, let's calculate the area using a TI-84 calculator.

You can find Normal Probability on a TI-84 Calculator Instructions under the Downloadable Resources in the right-hand sidebar, if you need help.

You will enter the lower bound, upper bound, mean, and standard deviation into the normalcdf function on your TI-84 (found in the distr menu).

normalcdf(750,800,852,82) = 0.1562

This tells us that the probability that the rainfall amount during monsoon season for a randomly selected year in India was between 750 ml and 800 ml of water is about .1562 or 15.62%.

You try a couple to check your understanding. Select the correct probability for each problem below.

Sometimes, we want to know the probability that the data point fall less than a certain number or more than a certain number instead of in a range.

To do that, you need to know one more thing about using your calculator. This is also included in the Normal Probability on a TI-84 Calculator Instructions (Downloadable Resources).

Let's look at an example using the monsoons in India data again:

Summer monsoons bring about 80% of the yearly rainfall in India. The exact amount of rain varies from year to year and follows a normal distribution with a mean of 862 milliliters and a standard deviation of 82 milliliters.

• What is the probability that the rainfall amount during monsoon season for a random year is over 800 milliliters?

The shaded region represented by this problem looks like this:

In your calculator, input normalcdf (800,EE99,852,82). The answer you get is 0.7370.

This means that the probability that a randomly selected year in India saw monsoon rains totaling over 800 milliliters is 0.7370 or 73.7%.

Sometimes, we want to look at the other end of a normal distribution to find a probability. Consider this question:

Summer monsoons bring about 80% of the yearly rainfall in India. The exact amount of rain varies from year to year and follows a normal distribution with a mean of 852 milliliters and a standard deviation of 82 milliliters.

• What is the probability that the rainfall amount during monsoon season for a random year is under 750 milliliters?

The shaded region represented by this problem looks like this:

In your calculator, input normalcdf (-EE99,750,852,82). The answer you get is 0.1068.

This means that the probability that a randomly selected year in India saw monsoon rains totaling under 750 milliliters is 0.1068 or 10.68%.

Try a couple of problems like this in order to check your understanding. Select the correct probability for each problem below.

If you feel like you understand this concept, move on to the Got It? section to quiz yourself on your understanding.