Normal Distribution Calculator
Calculate a normal probability below, above or between entered values. See standardized z scores, a probability breakdown and a bell curve with the selected region.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Standardize each bound with z=(x−μ)/σ. Evaluate the standard-normal cumulative probability with a numerical approximation and subtract the appropriate tails for a bounded interval.
z=(x−μ)/σ; P(a≤X≤b)=Φ((b−μ)/σ)−Φ((a−μ)/σ)
Worked example
For mean 100 and standard deviation 15, the interval 85–115 corresponds to z=−1 to +1. Its probability is approximately 0.6826895, or 68.26895%.
Assumptions and input details
Assume a continuous normal model with positive standard deviation; this tool does not establish that data are normally distributed. Endpoints have zero probability, so strict and inclusive bounds give the same answer. The CDF uses a five-coefficient numerical approximation (absolute error around 1e−7); tiny tail probabilities have weaker relative precision and may underflow. The curve uses standardized z on the horizontal axis and displays −4 to +4; entered bounds outside that view are still calculated. No inverse-percentile or hypothesis test is implied.
References: NIST: normal distribution
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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