The normal distribution is important because it describes the statistical behavior of many real-world events. The shape of the normal distribution is completely described by the mean and the standard deviation.

Thus, given the mean and standard deviation, you can use the properties of the normal distribution to quickly compute the cumulative probability for any value. This process is illustrated in the Sample Problems below.

There are an infinite number of normal distributions. Although every normal distribution has a bell-shaped curve, some normal distributions have a curve that is tall and narrow; while others have a curve that is short and wide.

The exact shape of a normal distribution is determined by its mean and its standard deviation. The standard normal distribution is the normal distribution that has a mean of zero and a standard deviation of one.

The normal random variable of a standard normal distribution is called a standard score or a z-score. The normal random variable X from any normal distribution can be transformed into a z score from a standard normal distribution via the following equation:

z = (X - μ) / σ

where X is a normal random variable, μ is the mean, and σ is the standard deviation.

Because any normal random variable can be "transformed" into a z score, the standard normal distribution provides a useful frame of reference. In fact, it is the normal distribution that generally appears in the appendix of statistics textbooks.

The normal distribution is defined by the following equation:

In this equation, the random variable X is called a normal random variable. A unique cumulative probability can be associated with every normal random variable. Given the normal random variable, the standard deviation of the normal distribution, and the mean of the normal distribution, we can compute the cumulative probability (i.e., the probability that a random selection from the normal distribution will be less than or equal to the normal random variable.)

A standard score (aka, a z-score) is the normal random variable of a standard normal distribution.

To transform a normal random variable (x) into an equivalent standard score (z), use the following formula:

z = (x - μ) / σ

where μ is the mean, and σ is the standard deviation.

A probability is a number expressing the chances that a specific event will occur. This number can take on any value from 0 to 1. A probability of 0 means that there is zero chance that the event will occur; a probability of 1 means that the event is certain to occur.

Numbers between 0 and 1 quantify the uncertainty associated with the event. For example, the probability of a coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent of the time, the coin flip would result in Heads; and fifty percent of the time, it would result in Tails.

A cumulative probability is a sum of probabilities. In connection with the normal distribution, a cumulative probability refers to the probability that a randomly selected score will be less than or equal to a specified value, referred to as the normal random variable.

Suppose, for example, that we have a school with 100 first-graders. If we ask about the probability that a randomly selected first grader weighs exactly 70 pounds, we are asking about a simple probability - not a cumulative probability.

But if we ask about the probability that a randomly selected first grader is less than or equal to 70 pounds, we are really asking about a sum of probabilities (i.e., the probability that the student is exactly 70 pounds plus the probability that he/she is 69 pounds plus the probability that he/she is 68 pounds, etc.). Thus, we are asking about a cumulative probability.

A mean score is an average score. It is the sum of individual scores divided by the number of individuals.

The standard deviation is a numerical value used to indicate how widely scores in a set of data vary. It is a measure of the average distance of individual observations from the group mean.