Normal Distribution Calculator

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

This Normal Distribution Calculator lets you work with any normal distribution by specifying the mean and standard deviation. You can compute probabilities from values or find the value corresponding to a given probability. Choose the direction of your calculation below.

Understanding the Normal Distribution

The normal distribution, also called the Gaussian distribution, is the most widely used probability distribution in statistics. It is defined by two parameters: the mean (μ), which determines the center of the distribution, and the standard deviation (σ), which controls the spread. Approximately 68% of observations fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This pattern is known as the empirical rule or the 68-95-99.7 rule.

Many natural phenomena follow a normal distribution, including measurement errors, heights and weights of populations, test scores on standardized exams, and manufacturing tolerances. When working with data that is approximately normal, this calculator lets you determine how likely specific outcomes are or find the threshold values that correspond to given probabilities.

Formulas Used

Value to Probability (CDF): To find the probability that X is less than or equal to a given value x:

P(X ≤ x) = Φ((x − μ) / σ)

Probability to Value (Inverse CDF): To find the value x such that the cumulative probability equals p:

x = μ + σ · Φ⁻¹(p)

Here, Φ is the standard normal CDF and Φ⁻¹ is its inverse (the quantile function). These functions do not have simple closed-form expressions, so numerical approximation algorithms are used internally.

Worked Example

The weights of apples from a certain orchard follow a normal distribution with mean μ = 150 grams and standard deviation σ = 20 grams.

Question 1: What is the probability an apple weighs less than 170 grams?

1. Standardize: z = (170 − 150) / 20 = 1.0
2. Find CDF: P(X ≤ 170) = Φ(1.0) ≈ 0.8413
3. About 84.13% of apples weigh less than 170 grams.

Question 2: What weight marks the 90th percentile?

1. Find the Z score: z = Φ⁻¹(0.90) ≈ 1.28155
2. Convert to X: x = 150 + 20 × 1.28155 ≈ 175.63
3. The 90th percentile weight is approximately 175.63 grams.

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