normal distribution probability calculator

What this normal distribution probability calculator does

This tool helps you find probabilities for values that follow a normal distribution (also called a Gaussian distribution). It answers questions like:

• What is the probability that a value is below a threshold?
• What is the probability that a value is above a threshold?
• What is the probability that a value falls between two limits?

These calculations are common in statistics, quality control, finance, psychology, exam scoring, and scientific research.

How to use the calculator

Step-by-step

• Enter the mean (μ) of your distribution.
• Enter the standard deviation (σ). It must be greater than zero.
• Select the probability type: P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b).
• Enter the needed value(s): one cutoff for left/right-tail, two bounds for interval probability.
• Click Calculate Probability to get the result instantly.

Behind the scenes: the math

If a random variable X is normally distributed as N(μ, σ²), the standardized z-score is:

z = (x - μ) / σ

Then probabilities are computed using the standard normal cumulative distribution function Φ(z):

• P(X ≤ x) = Φ(z)
• P(X ≥ x) = 1 - Φ(z)
• P(a ≤ X ≤ b) = Φ(z_b) - Φ(z_a)

This page evaluates those formulas numerically in JavaScript with a stable approximation to the error function.

Practical examples

Exam scores

Suppose exam scores are normally distributed with mean 70 and standard deviation 10. You can estimate the probability a student scores at least 85 by choosing P(X ≥ x) and setting x = 85.

Manufacturing tolerances

If part diameter is normal with mean 20.00 mm and standard deviation 0.05 mm, use the interval option to find the probability a part lies inside specification limits.

Service performance

If response times are approximately normal, this calculator can estimate the percentage of requests completed under a target time.

Common mistakes to avoid

• Using a non-positive standard deviation. Always use σ > 0.
• Mixing units (for example, entering mean in minutes and bounds in seconds).
• Confusing left-tail and right-tail probabilities.
• Assuming every dataset is normal without checking a histogram or QQ plot.

Quick interpretation guide

• A probability near 0 means the event is very unlikely.
• A probability near 1 means the event is very likely.
• The percentage form is simply probability × 100.

Final note

The normal model is powerful, but it is still a model. For best decisions, confirm that your data is reasonably bell-shaped and free of severe outliers. If the normal assumption is poor, consider alternative distributions or nonparametric methods.