The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused.

Its graph is bell-shaped. You see the bell curve in almost all disciplines.

Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics.

Some of your instructors may use the normal distribution to help determine your grade. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution.

The normal distribution is extremely important, but it cannot be applied to everything in the real world.

In this article, you will study the normal distribution, the standard normal distribution, and applications associated with them.

The normal distribution has two parameters (two numerical descriptive measures), the mean (μ), and the standard deviation (σ).

If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ), we designate this by writing.

The cumulative distribution function is P(X < x). It is calculated either by a calculator or a computer, or it is looked up in a table.

Technology has made the tables virtually obsolete. For that reason, as well as the fact that there are various table formats, we are not including table instructions.

The curve is symmetrical about a vertical line drawn through the mean, μ.

In theory, the mean is the same as the median, because the graph is symmetric about μ.

As the notation indicates, the normal distribution depends only on the mean and the standard deviation.

Since the area under the curve must equal one, a change in the standard deviation, σ, causes a change in the shape of the curve; the curve becomes fatter or skinnier depending on σ.

A change in μ causes the graph to shift to the left or right. This means there is an infinite number of normal probability distributions.

One of special interest is called the standard normal distribution.

### The standard Normal Distribution

The standard normal distribution is a normal distribution of standardized values called z-scores.

A z-score is measured in units of the standard deviation. For example, if the mean of a normal distribution is five and the standard deviation is two, the value 11 is three standard deviations above (or to the right of) the mean. The calculation is as follows:

x = μ + (z)(σ) = 5 + (3)(2) = 11

The z-score is three.

The mean for the standard normal distribution is zero, and the standard deviation is one.

The transformation z = x − μ/σ produces the distribution Z ~ N(0, 1).

The value x comes from a normal distribution with mean μ and standard deviation σ.