Probability Distributions Explained: Normal, Binomial, and More

Probability distributions describe how the outcomes of a random event are spread across possible values. They are the mathematical backbone of statistics, data science, quality control, and risk analysis. Choosing the right distribution is key to making accurate predictions and decisions.

This guide introduces the three most commonly used distributions: the normal distribution, the binomial distribution, and z-scores. We cover when to use each one and work through practical examples.

The Normal Distribution

The normal (Gaussian) distribution is the familiar bell curve. It is defined by two parameters: the mean (the centre) and the standard deviation (the spread).

Key properties of the normal distribution:

• It is symmetric about the mean.
• Approximately 68% of data falls within of the mean.
• Approximately 95% falls within .
• Approximately 99.7% falls within .

Example 1: Normal Distribution

Test scores in a class are normally distributed with and . What percentage of students scored between 64 and 80?

The range 64 to 80 is , which is from the mean. By the 68-95-99.7 rule, approximately 68% of students scored in this range.

Z-Scores

A z-score tells you how many standard deviations a data point is from the mean. It standardises values so you can compare across different distributions:

Example 2: Computing a Z-Score

Using the test scores above (, ), find the z-score for a student who scored 88.

A z-score of 2.0 means the student scored 2 standard deviations above the mean. Using a z-table, this corresponds to the 97.7th percentile, meaning the student scored higher than about 97.7% of the class.

The Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It is defined by:

• = number of trials
• = probability of success on each trial

The probability of exactly successes is:

where is the binomial coefficient.

Example 3: Binomial Probability

A coin is flipped 10 times. What is the probability of getting exactly 7 heads?

Here , , and .

There is roughly an 11.7% chance of getting exactly 7 heads.

Example 4: Binomial in Quality Control

A factory produces widgets with a 3% defect rate. In a batch of 20, what is the probability that exactly 0 are defective?

Here , , and .

There is roughly a 54.4% chance that the entire batch is defect-free.

When to Use Each Distribution

• Normal: Continuous data that clusters around a mean (heights, test scores, measurement errors).
• Binomial: Counting successes in a fixed number of yes/no trials (coin flips, pass/fail tests, defective items).
• Z-scores: Standardising values from any normal distribution to compare across datasets or look up probabilities in standard tables.

Key Formulas at a Glance

• Normal mean and SD: ,
• Binomial mean:
• Binomial SD:

Try It Yourself

Calculate probabilities for normal and binomial distributions with our free Probability Calculator. It computes exact probabilities, z-scores, and cumulative values with step-by-step working.