The normal distribution (or Gaussian distribution) is the most important probability distribution in statistics. It describes how many natural phenomena are distributed โ€” test scores, heights, measurement errors, stock returns โ€” and is the foundation of most statistical inference and hypothesis testing.

The Formula

The probability density function for a normal distribution is:

f(x) = (1 / (ฯƒโˆš(2ฯ€))) ร— e^(-(x-ฮผ)ยฒ/(2ฯƒยฒ))

Where:

  • ฮผ (mu) = mean (center of the distribution)
  • ฯƒ (sigma) = standard deviation (spread of the distribution)
  • x = the value you're evaluating
  • e โ‰ˆ 2.71828
  • ฯ€ โ‰ˆ 3.14159

The shape is bell-curved, and about 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations (the 68-95-99.7 rule).

Worked Example

A standardized test has mean 100 and standard deviation 15. What is the probability that a random score is less than 115?

First, convert to a z-score:

z = (115 - 100) / 15 = 1.0

A z-score of 1.0 means 115 is one standard deviation above the mean. Using a standard normal table or calculator, P(z โ‰ค 1.0) โ‰ˆ 0.8413 or 84.13%.

So about 84% of test-takers score below 115.

Key Properties

The normal distribution is defined entirely by its mean and standard deviation. Shifting the mean moves the curve left or right; increasing the standard deviation flattens and widens it. The total area under the curve always equals 1.

Any normal distribution can be converted to the standard normal distribution (mean 0, standard deviation 1) using the z-score formula above. This standardization allows you to use one universal normal table.

When to Use

Use the normal distribution when:

  • Data clusters around a central value
  • Values follow a bell-shaped histogram
  • The Central Limit Theorem applies (sample means from any distribution approximate normal)
  • You're doing hypothesis testing or confidence intervals

Most real-world continuous data roughly follows a normal distribution, making this the workhorse of applied statistics.

Tips

Check for normality using a histogram or Q-Q plot before assuming data is normal. If data is heavily skewed or has outliers, the normal distribution may not be appropriate. For non-normal data, use non-parametric tests or data transformation.

Use our Normal Distribution Calculator to find probabilities, percentiles, and z-scores instantly.