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Normal Distribution Probability Calculator

Find the probability between two bounds for a normal distribution, plus the probabilities in both outside tails.

Data stays on this deviceConvention explained below
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Data explorer

Check the dataset and sample convention before interpreting results.

See the method

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Your result

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Use the Calculate button to see your result.

Review the method below for assumptions and conventions.

How to use this tool

  1. Enter lower bound, upper bound, population mean, population standard deviation.
  2. Select Calculate to view the result.
  3. Check the method and assumptions below before using the result.

Input reference and sample values

Use the units and choices shown for each field. Sample values are provided for practice, not as recommendations.

InputFormat and unitsSample
Lower boundNumeric value.-1
Upper boundNumeric value.1
Population meanNumeric value.0
Population standard deviationNumeric value. Must be greater than zero.1

Result for these sample inputs

0.682689492 probability
Below lower bound
0.158655254
Above upper bound
0.158655254
Lower z-score
-1
Upper z-score
1

For a continuous distribution, including or excluding the endpoints does not change the probability.

Select Try the example in the form to load these values and calculate them again. This replaces the current form inputs. Compare the result with the method and worked explanation below.

The method, explained

Standardize each bound using z = (x−mean)/SD. Subtract normal cumulative probabilities, using survival tails to avoid cancellation in the right tail.

A WORKED EXAMPLE

Using lower bound = -1, upper bound = 1, population mean = 0, population standard deviation = 1, the result is 0.682689492 probability. Change these example inputs to match your task; use the method above to check each step.

What to keep in mind

Assumes the stated normal population model, not merely a dataset with a known mean and standard deviation. A descriptive or probability calculation under the stated assumptions. Check the sample design and model before interpreting results.

Reference: NIST: statistical methods handbook

Common questions

Can I use a normal curve for any dataset?

No. The shape assumption matters. A mean and standard deviation alone do not establish that your population is normally distributed.

Why is a single exact value assigned zero probability?

The normal distribution is continuous. Probability corresponds to an interval of values; the density at one point is a height, not a point probability.

Methodology maintained by ClarityKit. How these tools are built and checked.