Wiki Page, Math 13 - Perez Sandbox : Module 12 - Introduction to Normal Probability Distributions (1 of 5)

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2026-01-17T21:15:13-08:00

In the unit on Summarizing Data Graphically and Numerically, we examined data sets, such as height and weight, with distributions that bell-shaped.

Many variables that measure human physical characteristics are bell-shaped. We model these distributions with a mathematical equation that has a bell-shaped curve. These models are called normal curves. They are called “normal” because the bell-shaped pattern occurs so often.

The normal curve looks like this:

Observations About Normal Distributions

There are many normal curves. Even though all normal curves have the same bell shape, they vary in their center and spread.

Because normal curves are mathematical models, we use Greek letters to represent the mean and standard deviation of a normal curve.

Normal distributions are symmetric. In symmetric distributions the mean and the median are equal. Therefore, the mean of a normal distribution divides the distribution in half. We use the Greek letter (pronounced “mu,” rhymes with "mew") to represent the mean.

We use the Greek letter (pronounced “sigma”) to represent the standard deviation of a normal distribution. The standard deviation determines the spread of the distribution about the mean. In fact, the shape of a normal curve is completely determined by specifying its standard deviation. As we will see, if two normal distributions have the same standard deviation, then the shapes of their normal curves will be identical.

Following are some observations we can make as we look at the figure above:

The black and the red normal curves have means at . However, the red curve is more spread out and thus has a larger standard deviation. Notice that the red normal curve is also shorter. This makes sense because these curves are probability density curves, so the area under each curve has to be 1.

The black and the green normal curves have the same standard deviation but different means.

Comment About Notation

We use to represent the mean of data in a sample. We use to represent the mean of a probability curve defined by a mathematical equation.

We use SD or to represent the standard deviation of data in sample. We use to represent the standard deviation of a probability curve defined by a mathematical equation.

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