The Normal Distribution is a family of continuous distributions that can model many histograms of real-life data which are mound-shaped (bell-shaped) and symmetric (for example, height, weight, etc.).

A normal curve has two parameters:

1. mean $\mu$ (center of the curve)
2. standard deviation $\sigma$ (spread about the center) (..and variance $\sigma^2$)

The mean can be any real number and the standard deviation is greater than zero. The normal curve ranges from negative infinity to infinity. The image below shows the effect of the mean and standard deviation on the shape of the normal curve.

A special case of the normal distribution has mean \(\mu = 0\) and a variance of \(\sigma^2 = 1\). The 'standard normal' is an important distribution.

As we mentioned previously, calculus is required to find the probabilities for a Normal random variable.  Fortunately, we have tables and software to help us.

For any normal random variable, we can transform it to a standard normal random variable by finding the Z-score. Then we can find the probabilities using the standard normal tables.

Most statistics books provide tables to display the area under a standard normal curve. Look in the appendix of your textbook for the Standard Normal Table. We include a similar table, the Standard Normal Cumulative Probability Table so that you can print and refer to it easily when working on the homework.

Most standard normal tables provide the “less than probabilities”.  For example, if \(Z\) is a standard normal random variable, the tables provide \(P(Z\le a)=P(Z<a)\), for a constant, \(a\).

Recall from Lesson 1 that the \(p(100\%)^{th}\) percentile is the value that is greater than  \(p(100\%)\) of the values in a data set. We can use the standard normal table and software to find percentiles for the standard normal distribution.

The intersection of the columns and rows in the table gives the probability. If we look for a particular probability in the table, we could then find its corresponding Z value.

Find the 10th percentile of the standard normal curve.

The standard normal is important because we can use it to find probabilities for a normal random variable with any mean and any standard deviation.

But first, we need to explain Z-scores.

We can convert any normal distribution into the standard normal distribution in order to find probability and apply the properties of the standard normal. In order to do this, we use the z-value.

Let's take a look at the idea of a z-score within context.

For a recent final exam in STAT 500, the mean was 68.55 with a standard deviation of 15.45.

• If you scored an 80%: \(Z = \dfrac{(80 - 68.55)}{15.45} = 0.74\), which means your score of 80 was 0.74 SD above the mean.
• If you scored a 60%: \(Z = \dfrac{(60 - 68.55)}{15.45} = -0.55\), which means your score of 60 was 0.55 SD below the mean.

Is it always good to have a positive Z score? It depends on the question. For exams, you would want a positive Z-score (indicates you scored higher than the mean). However, if one was analyzing days of missed work then a negative Z-score would be more appealing as it would indicate the person missed less than the mean number of days.

From Z-score to Probability

For any normal random variable, if you find the Z-score for a value (i.e standardize the value), the random variable is transformed into a standard normal and you can find probabilities using the standard normal table.

For instance, assume U.S. adult heights and weights are both normally distributed. Clearly, they would have different means and standard deviations. However, if you knew these means and standard deviations, you could find your z-score for your weight and height.

You can now use the Standard Normal Table to find the probability, say, of a randomly selected U.S. adult weighing less than you or taller than you.

The Empirical Rule is sometimes referred to as the 68-95-99.7% Rule. The rule is a statement about normal or bell-shaped distributions.

Although the normal distribution is important, there are other important distributions of continuous random variables. Some we will introduce throughout the course, but there are many others not discussed. Here are a few distributions that we will see in more detail later.

 t-distribution

The t-distribution is a bell-shaped distribution, similar to the normal distribution, but with heavier tails. It is symmetric and centered around zero. The distribution changes based on a parameter called the degrees of freedom. We will discuss degrees of freedom in more detail later.

The graph shows the t-distribution with various degrees of freedom. The standard normal distribution is also shown to give you an idea of how the t-distribution compares to the normal. As you can see, the higher the degrees of freedom, the closer the t-distribution is to the standard normal distribution.

 Chi-square distribution

The chi-square distribution is a right-skewed distribution. The distribution depends on the parameter degrees of freedom, similar to the t-distribution. Here is a plot of the Chi-square distribution for various degrees of freedom.

We will see the Chi-square later on in the semester and see how it relates to the Normal distribution.

 F-distribution

The F-distribution is a right-skewed distribution. The distribution depends on the two parameters both are referred to as degrees of freedom. The first is typically called the numerator degrees of freedom ($d_1$) and the second is typically referred to as the denominator degrees of freedom ($d_2$). Here is a plot of the F-distribution with various degrees of freedom.

The F-distribution will be discussed in more detail in a future lesson.