Normal Distribution Calculator

Calculate z-scores or probabilities from the Normal Distribution.

Use the Normal Distribution z-score calculator below to find z-scores and/or probabilities from the Normal Distribution.

You will find a description of the Normal Distribution below the calculator.

Input

z-score Probability

Tails

Right Tail

Left Tail

Two Tails

z-score

Result

Overview of the Normal Distribution

The Normal Distribution (a.k.a. the normal curve or Gaussian Distribution) is a symmetric, unimodal, bell-shaped, continuous probability distribution. It is one of the most common probability distributions in statistics as it describes many phenomena in the natural world.

The Normal Distribution is defined by two parameters: a mean (μ) and a standard deviation (σ). If a random variable has a normal or nearly normal distribution, you can calculate the probability of observing values within a specified range using the Normal Distribution's cumulative distribution function or calculators like the one here.

Using the Normal Distribution

For a variable that follows the Normal Distribution, the normal curve describes how likely you are likely to observe a value as or more extreme than one you pick.

For example, you can use the normal curve to say how likely you are to observe a value that is two or more standard deviations from the mean.

z-scores

A z-score is a measure of how many standard deviations a value is from its mean. You can calculate the score of a value x as z = (x - μ)/σ

Generally, to analyze the likelihood of observing a value from a Normal Distribution, you would calculate its z-score and use a calculator like the one above to see how likely that z-score or a more extreme value is to occur.

Central Limit Theorem

The Normal Distribution's usefulness largely stems from the Central Limit Theorem, which says that the distribution of independently sample means is approximately Normal.

The Central Limit Theorem has many uses including hypothesis testing, such as a hypothesis test for a proportion or a hypothesis test for a mean.

68-95-99.7 Rule

The 68-95-99.7 Rule is a rule of thumb to remember how values vary under the Normal Distribution. It states that approximately...

68% of the Normal Distribution falls within one standard deviation of the mean (between z-scores -1 and 1)
95% of the Normal Distribution falls within two standard deviation of the mean (between z-scores -2 and 2)
99.7% of the Normal Distribution falls within three standard deviation of the mean (between z-scores -3 and 3)

While these values are inexact, they provide a short-cut to understanding how likely you are to observe a value under the Normal Distribution.