Probability Distribution Calculator
Calculate normal distribution probabilities - P(X < x), P(X > x), or the probability of a range - given a mean and standard deviation.
P(X ≤ 1.96)
97.5002%
z-score
1.9600
Recommended Settings
Pro Tips
- •The normal distribution is defined entirely by its mean (center) and standard deviation (spread) - most real-world measurements like test scores and heights approximate this shape
- •The z-score tells you how many standard deviations a value is from the mean - a z-score of 1.96 marks roughly the top and bottom 2.5% of a standard normal distribution
- •Use the standard normal distribution (mean 0, standard deviation 1) to compute a generic z-score probability, or enter your own mean and standard deviation for a specific dataset
- •This uses a high-precision numerical approximation for the standard normal cumulative distribution function, accurate to about 7 decimal places
Most Popular
Most students start with the standard normal distribution (mean 0, standard deviation 1) before applying their own dataset's parameters
When to Use This Tool
Check normal distribution probability calculations for coursework or exam practice.
Estimate what percentage of test-takers scored above or below a given score.
Estimate the probability that a measurement falls within an acceptable range.
See how a specific value translates into a z-score and corresponding probability.
How It Works
Convert your input value into a z-score based on the given mean and standard deviation
Evaluate the standard normal cumulative distribution function using a high-precision numerical approximation of the error function
For a range, calculate the difference between the cumulative probabilities at each endpoint
100% Private
Files never leave your device. All processing happens locally in your browser.
Lightning Fast
Powered by Client-side normal distribution calculation (Abramowitz-Stegun approximation) for optimal performance on modern browsers.
Open Source
Built with verified, open-source libraries. Fully transparent.
Frequently Asked Questions
What is a z-score?
A z-score measures how many standard deviations a value is above or below the mean. A z-score of 0 means the value equals the mean; a z-score of 2 means it's two standard deviations above the mean.
Does this only work for the normal distribution?
Yes, this tool specifically calculates probabilities for the normal (Gaussian) distribution, the most common distribution used in statistics for continuous data.
How accurate is this calculation?
It uses the well-established Abramowitz-Stegun numerical approximation for the error function, accurate to within about 0.0000015 (1.5×10⁻⁷) of the true value - more than precise enough for coursework, research, or general analysis.
Is my data sent anywhere?
No. All calculations happen locally in your browser.