Normal Distribution Calculator
Calculate normal distribution probabilities or percentile values, visualize the shaded bell curve and export SVG or JSON.
This tool processes its input contents in your browser instead of sending them to a processing service. Page, asset and fixed usage-measurement requests are described on the Privacy page.
Find normal-model probabilities and percentile values, with a shaded bell curve and downloadable results.
How to use it
- 1Enter the mean and a positive standard deviation.
- 2Choose below, above, between or a percentile value.
- 3Enter bounds in the original units or the left cumulative probability.
- 4Calculate, inspect the shaded curve and export SVG or JSON.
Example
- Input
- Mean 100, standard deviation 15, between 85 and 115
- Output
- About 68.26894921% within the interval.
What happens to your data
Values and local SVG/JSON files stay in this tab unless you share the downloads. Anonymous workflow records omit the entered numbers.
Last updated November 2026
Find the chance of a value falling below, above or between your chosen bounds in a normal model. Enter the mean and standard deviation in the original units, such as test points, then see the probability, standardized z-scores and the shaded bell curve. You can also go in the other direction to find the value at a chosen percentile. Keep a chart as an SVG or export the calculation as JSON.
How it works
The normal probability calculator standardizes each bound using z = (x − mean) ÷ standard deviation. Its normal CDF calculator integrates the standard bell curve to find the left-hand area. The upper tail uses symmetry to compute very small right-hand probabilities directly. Between two values, it integrates the selected interval directly. That helps preserve a narrow interval even when subtracting two rounded cumulative probabilities would give zero. The normal percentile calculator searches for the z-score that gives your entered cumulative probability, then converts it back to the original units. The result and JSON also retain the offset from the mean. At very large mean values, the sum can round back to the mean even when the offset is nonzero. The result flags that case.
Integration uses composite eight-point Gauss–Legendre quadrature. Positive-tail integration rescales the integrand and bounds the omitted tail. This is approximate binary64 arithmetic. A tiny probability can fall below the browser’s numeric range, in which case the displayed log probability stays informative. An interval with identical bounds has exactly zero area because the model is continuous. Probabilities near 1 can display as 1 because of rounding.
Use decimal-point values with up to 12 digits on either side of the point. Standard deviation must be positive. Enter bounds within ±38 standard deviations of the mean and a percentile probability strictly between 0 and 1, with up to 12 decimal places. The bell curve calculator draws the z-axis out to at most ±8 for readability; the calculation includes the full selected area. The curve is based on the mean and spread you supply and cannot determine whether a dataset follows this model.
Worked example: One standard deviation on either side
- Input
- Mean 100, standard deviation 15, probability between 85 and 115
- Result
- Approximately 0.682689492137, or 68.26894921%. The bounds are z = −1 and z = 1.
The model describes a continuous distribution; inclusive and exclusive endpoints have the same probability.
Common use cases
- Find the fraction of a normal model between two test scores.
- Calculate a small upper-tail probability without subtracting a near-one CDF.
- Convert a percentile such as the 95th into a value in original units.
- Download a shaded normal curve for a handout or analysis note.
Frequently asked questions
What should I enter for the 95th percentile?＋
Choose Value at a percentile and enter 0.95 as the left cumulative probability. The calculator returns the value with about 95% of the model below it.
Can I calculate standard normal probabilities?＋
Yes. Set the mean to 0 and standard deviation to 1. The entered bounds are then z-scores directly.
Why is the selected area hard to see in the chart?＋
Tiny tails can be too small to see and the drawing stops at most eight standard deviations from the mean. The numerical probability still accounts for the entire selected interval or tail.
Does this test whether my data is normal?＋
No. The tool evaluates the assumed model for the mean and standard deviation you entered. It does not take a dataset or test a distribution fit.
Related tools
Binomial Probability Calculator
Calculate point and cumulative binomial probabilities with a full table, chart and exports.
Standard Deviation Calculator
Analyze a list or CSV column with sample/population spread, quartiles, outlier flags and exports.
Histogram Maker
Make a histogram from numbers or a CSV column. Export charts and audited bin counts.
Break-Even Calculator
Find the minimum whole units to cover costs, calculate target-profit sales and compare three pricing scenarios.