Normal Distribution Calculator

Calculate probabilities, visualize the bell curve, and find critical values for any normal distribution. Free online tool with the 68-95-99.7 rule.

What is a Normal Distribution?

The normal distribution (bell curve) is a symmetric, unimodal probability distribution described by its mean (μ) and standard deviation (σ). About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

Many natural phenomena — heights, test scores, measurement errors — approximate a normal distribution, which is why it is central to statistics and hypothesis testing.

Any normal distribution can be converted to the standard normal distribution (mean 0, standard deviation 1) using the Z transformation z = (x − μ) / σ, allowing probabilities to be looked up in a Z-table.

Probability Density Function

The height of the curve at any value x is given by:

f(x) = (1/(σ√(2π)))·e^(-(x−μ)²/(2σ²))

How to Use This Calculator

Enter the mean and standard deviation of your distribution, then:

Type a value x to find P(X ≤ x) and P(X ≥ x).
Type a lower bound a and upper bound b to find P(a ≤ X ≤ b).
Use the visualization tab to see the shaded area, and the inverse tab to find x from a probability.

Worked Example

IQ scores are normally distributed with μ = 100 and σ = 15. What percentage of people score below 115?

z = (115 − 100) / 15 = 1.00 P(X < 115) = P(Z < 1.00) = 0.8413 About 84.1% of people score below 115.

Frequently Asked Questions

Have another question? Let us know.

What does a normal distribution look like?

It is symmetric and bell-shaped, with the mean, median and mode all equal and located at the center. The curve is defined by μ (center) and σ (spread).

How do I calculate the probability of a value in a normal distribution?

Convert x to a Z score using z = (x − μ)/σ, then look up the cumulative probability in a Z-table or use the standard normal CDF. This tool does it automatically.

What is the 68-95-99.7 rule?

Also called the empirical rule: about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

When can I use the normal distribution?

When data is approximately symmetric and bell-shaped, or when a sample mean is large enough (central limit theorem, typically n ≥ 30).