Module 6: Probability and Probability Distributions
Module 6: Probability and Probability Distributions
Normal Random Variables (1 of 6)
Normal Random Variables (1 of 6)
Learning OUTCOMES
- Use a normal probability distribution to estimate probabilities and identify unusual events.
In Summarizing Data Graphically and Numerically, we encountered data sets, such as height and weight, with distributions that are fairly symmetric with a central peak. We call these bell-shaped.
Many variables, such as weight, shoe sizes, foot lengths, and other human physical characteristics, exhibit these properties. The symmetry indicates that the variable is just as likely to take a value a certain distance below its mean as it is to take a value that same distance above its mean. The bell shape indicates that values closer to the mean are more likely, and it becomes increasingly unlikely to take values far from the mean in either direction.
We use a mathematical model with a smooth bell-shaped curve to describe these bell-shaped data distributions. These models are called normal curves or normal distributions. They were first called “normal” because the pattern occurred in many different types of common measurements.
The general shape of the mathematical model used to generate a normal curve looks like this:
Observations of 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. The mean of a normal distribution locates its center. We use the Greek letter μ (pronounced “mu” ) to represent the mean. We use the Greek letter σ (pronounced “sigma”) to represent the standard deviation of a normal distribution. The standard deviation determines the spread of the distribution. 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 or centers at μ = 10. 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 or spread.
Comment
- We use [latex]\bar{x}[/latex] to represent the mean of data in a sample. We use μ to represent the mean of a density curve defined by a mathematical model.
- We use SD or [latex]s_x[/latex] to represent the standard deviation of data in sample. We use σ to represent the standard deviation of a density curve defined by a mathematical model.
The normal curve has a central role in statistical inference, as we’ll see in Linking