Normal Distribution Calculator

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.

A random variable is a numerical outcome of a random process or experiment.

When you use the normal distribution calculator to find probability, the random variable can be expressed as a z-score from a standard normal distribution or as a raw score from a general normal distribution.

Given a normal random variable, the calculator can compute its associated cumulative probability. Or given a cumulative probability, the calculator can find its associated normal random variable.

A z-score (aka, a standard score) is a standardized value that tells you how many standard deviations a raw score is from the mean. To transform a raw score (x) into an equivalent z-score (z), use the following formula:

z = (x - μ) / σ

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

Every z-score has a mean of 0 and a standard deviation of 1. If you use a z-score with this calculator, the calculator automatically sets the mean equal to 0 and the standard deviation equal to 1.

A raw score is the original value from a distribution that has any mean (μ) and any standard deviation (σ). For example, if you're working with SAT scores, a raw score might be 1200, with a mean equal to 1050 and a standard deviation equal to 100.

When you input raw scores into a normal distribution calculator, you also need to specify the mean and standard deviation of the normal distribution from which the raw score was selected.

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 fall within a specified range.

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 within a range of values (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 mean of a normal distribution is 0 for a standard normal distribution. This is a special case of the normal distribution, which has a mean of 0 and a standard deviation of 1. The random variable for a standard normal distribution is called a z-score.

Unless it has been explicitly standardized, the mean of a general normal distribution will usually not be 0. For example, the height of adult males will follow an approximate normal distribution, but the mean height (about 70 inches) will not be 0.

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.

The standard deviation of a standard normal distribution is 1. The random variable for a standard normal distribution is called a z-score.

Unless a random variable has been explicitly standardized, its standard deviation will usually not be 1. For example, the height of adult males follows an approximate normal distribution, but the standard deviation (about 3 inches) is not 1.

The related probabilities are probabilities of events that can be derived from knowledge of other events. Here's the logic for computing related probabilities:

• The normal distribution is symmetric. Therefore, P(Z ≤ -z) = P(Z ≥ z).
• From the rule of subtraction, we know P(Z ≤ z) = 1 - P(Z ≥ z).

Given these relationships, if we know the cumulative probability for one event, we can compute cumulative probabilities for other related events. For example, suppose we know P(Z ≤ z). Then, we can compute the following:

• P(Z ≥ z) = 1 - P(Z ≤ z)
• P(Z ≤ -z) = P(Z ≥ z)
• P(-z ≤ Z ≤ z) = P(Z ≤ z) - P(Z ≤ -z)

When the normal distribution calculator computes the probability for one event, it also displays probabilities for other related events.