Normal Distribution Calculator
Calculate normal distribution probabilities, z-scores, percentiles and critical values with a high-accuracy error function, for any mean and standard deviation.
Last reviewed by the Radiatus Cloud team
Need this done properly for your business?
Radiatus delivers secure cloud, DevOps & compliance engineering.
The normal distribution is common because of the central limit theorem
Sums and averages of many independent effects tend towards a normal distribution regardless of the shape of the individual effects, which is why measurement errors, heights and aggregate quantities so often look bell shaped. It is not a law of nature and does not apply everywhere: incomes, city sizes, file sizes and losses from rare events are famously not normal, and treating them as if they were badly underestimates how often extreme values occur.
What a z-score means
A z-score expresses a value as a number of standard deviations from the mean, which makes measurements on different scales directly comparable. The familiar landmarks follow from it: about 68 percent of the distribution lies within one standard deviation, 95 percent within about 1.96, and 99.7 percent within three. Those percentages are properties of the normal distribution specifically and do not hold for other shapes.
Tails are where the accuracy matters
Beyond about four standard deviations, probabilities become extremely small and the simple approximations used in textbooks lose all their significant digits. This calculator uses a high accuracy error function so tail probabilities remain meaningful far out, which matters in quality control, risk work and significance testing where the interesting numbers are precisely the small ones.
Related tools
- Percentage Calculator — Calculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
- Statistical Calculator — Calculate mean, median, mode, standard deviation, and more.
- Probability Calculator — Calculate probabilities for single and combined events, including conditional probability and Bayes' theorem.
- Scientific Calculator — Perform advanced mathematical operations including trig, log, and exponents.
Frequently Asked Questions
What is a z-score?
The number of standard deviations a value lies from the mean. It puts measurements on different scales onto a common footing, so a z of 2 means the same relative position whether the underlying quantity is height, temperature or test score.
What are the 68, 95 and 99.7 figures?
The proportions of a normal distribution lying within one, two and three standard deviations of the mean. The 95 percent figure is really 1.96 standard deviations rather than 2. These proportions are specific to the normal distribution and do not transfer to other shapes.
When is the normal distribution the wrong model?
Whenever the data is bounded, strongly skewed or heavy tailed. Incomes, city sizes, insurance losses and file sizes are all badly modelled by it, and the error is not neutral: it dramatically understates how often extreme values occur.
What is the difference between the PDF and the CDF?
The density is the height of the curve at a point and is not a probability by itself, since a continuous variable has zero probability of taking any exact value. The cumulative function gives the probability of being at or below a value, which is what you usually want.
Why do tail probabilities need special care?
Because beyond about four standard deviations the numbers become so small that ordinary approximations lose all their significant digits. A high accuracy error function keeps them meaningful, which matters in quality control and significance testing where the small numbers are the point.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter a mean, standard deviation and value to get the probability, z-score and percentile.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
Related Tools
Percentage Calculator
MathCalculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
Statistical Calculator
MathCalculate mean, median, mode, standard deviation, and more.
Probability Calculator
MathCalculate probabilities for single and combined events, including conditional probability and Bayes' theorem.