Statistics How To

Normal Distribution Probability in Statisitcs

Statistics Definitions > What is normal distribution probability?

Normal Distribution Probability

A normal distribution is a symmetric distribution shaped like a bell:

normal distribution probability

A normal distribution probability curve.

Bell curves can represent lots of natural phenomena. For example, grades in class will often be bell-shaped: most students will get C in class (the average), a smaller number of students will get a B or a D, and even fewer students will get an A or an F.

class grades.

Class grades usually follow a bell curve. Most students will get a C on any test or exam.

IQ scores naturally fall into this shape; the following graph shows that the majority of people have IQ scores between 85 and 115.

IQ scores fit a bell curve shape.

IQ scores.

While the bell curve is useful, it would be tricky to try to figure out how many people have IQ scores between 110 and 130. Or perhaps you would want to know what percentage of students got grades slightly below passing (67-69). The math would get a little tricky, but not if you superimpose your information onto a normal distribution curve.

normal distribution probability 2

The area under the curve represents 100% probability. In statistics and probability, 100% is written as a decimal (100% = 1), so you will often see it mentioned that the total area under the curve is 1. The mean (the average) in a standard normal distribution is always zero. Why? Because it makes it easier to look up normal distribution probability (a z-score) using a z-table.

Normal distribution probability (and the associated z-scores) allow you to look up percentage probabilities for any set of data that is shaped like a bell.

standard normal distribution

The above normal probability distribution shows what percentage of scores fall within a number of standard deviations from the mean. Standard deviations are the same as z-scores; a standard deviation of 1 corresponds to a z-score of 1 and a standard deviation of 2 corresponds to a z-score of 2. 68% of scores (IQ scores, grades, heights, weights and a host of other phenomena) will fall between standard deviations of -1 and +1 (z-scores of -1 and 1).

How to Calculate Normal Distribution Probability

Normal distribution probability can be calculated with a little basic arithmetic and use of a z-table. You can find several examples of solving these types of problems here: Area under a normal distribution curve index.


If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Normal Distribution Probability in Statisitcs was last modified: October 12th, 2017 by Stephanie

2 thoughts on “Normal Distribution Probability in Statisitcs

  1. Amalbeta

    thanks a lot!!!!!!!!!! This helps me with my addmath project work. May God bless all of you!!!!!!!!!!!