Normal Distribution Calculator
This normal distribution calculator estimates probabilities below, between or above two values when a normal model’s mean and standard deviation are supplied.
How to use this calculator
- Enter the mean, positive standard deviation and ordered lower and upper bounds in the same units.
- Check the labels and units before applying the result to a real task.
- Change one input at a time when comparing scenarios.
Formula and method
z=(x−μ)/σ; P(a≤X≤b)=Φ((b−μ)/σ)−Φ((a−μ)/σ)
- z=(x−μ)/σ; P(a≤X≤b)=Φ((b−μ)/σ)−Φ((a−μ)/σ)
- The result assumes normally distributed values. Real data may be skewed, bounded, multimodal or contain outliers.
Worked examples
Worked example
For a normal model with mean 100 and standard deviation 15, find the range from 85 to 115.
The bounds are one standard deviation below and above the mean, enclosing approximately 68.27% of the distribution.
The bounds are one standard deviation below and above the mean, enclosing approximately 68.27% of the distribution.
Understanding the result
This normal distribution calculator estimates probabilities below, between or above two values when a normal model’s mean and standard deviation are supplied.
The result assumes normally distributed values. Real data may be skewed, bounded, multimodal or contain outliers.
Important notes
The result assumes normally distributed values. Real data may be skewed, bounded, multimodal or contain outliers.
Displayed values are rounded for readability; retain appropriate precision when continuing a calculation.
Frequently asked questions
What is a z-score?
A z-score describes how many standard deviations a value lies above or below the mean, using (x−μ)/σ.
Does the calculator test if my data are normal?
No. It calculates probabilities under a normal model that you specify. It does not test whether your real data follow that model.
Why must the standard deviation be positive?
A positive standard deviation defines a distribution with spread. A zero or negative value cannot be used in the normal z-score formula.
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Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
- How to Calculate Probability — Learn basic probability formulas, complementary probability, and how to calculate simple event chances.
- How to Calculate a Z-Score — Learn how a z-score shows how far a value is from the mean in standard-deviation units.
- Probability — Understand probability as a measure of how likely an event is and review simple probability formulas.
This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.