Normal Distribution Calculator
Find probabilities and percentiles for the normal distribution.
This tool is for informational and educational purposes only. It is not a substitute for professional financial, medical, legal, or engineering advice. See Terms of Service.
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Enter the mean and standard deviation of your normal distribution, then the x value. The calculator computes the cumulative probability P(X < x), the upper tail P(X > x), and the z-score. Results update instantly.
About This Calculator
The normal (Gaussian) distribution is the most important probability distribution in statistics. It is symmetric, bell-shaped, and fully described by its mean and standard deviation. About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three (the empirical rule).
Frequently Asked Questions
What is the standard normal distribution?
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be converted to the standard form using z-scores: z = (x - mean) / standard deviation.
What does P(X < x) mean?
P(X < x) is the probability that a randomly chosen value from the distribution will be less than x. It is the area under the curve to the left of x.
How is the normal distribution used in practice?
It appears in confidence intervals, hypothesis testing, quality control, finance, natural measurements, and anywhere the Central Limit Theorem applies.