Introduction to the Normal Distribution

This photo shows many different pairs of shoes in various colors. The shoes appear to be hanging from a wall by cords.
If you ask enough people about their shoe size, you will find that your graphed data (histogram) is shaped like a bell curve and can be described as normally distributed. (credit: Ömer Ünlϋ)

The normal distribution 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 chapter, 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

This is a frequency curve for a normal distribution. It shows a single peak in the center with the curve tapering down to the horizontal axis on each side. The distribution is symmetrical; it represents the random variable X having a normal distribution with a mean, m, and standard deviation, s.

The curve is symmetrical about a vertical line drawn through the mean, μ. As the notation indicates, the normal distribution depends only on the mean and the standard deviation. 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 are an infinite number of normal probability distributions. One of special interest is called the standard normal distribution.