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Normal Distribution Calculator
Calculate probabilities for normal distributions instantly. Enter your population mean and standard deviation, then choose whether you want to find the probability above, below, or between specific values.
How to use this calculator
- Enter your parameters: Input the mean ($\mu$) and standard deviation ($\sigma$) of your dataset.
- Select your probability type: Choose whether you want to find the area below a value, above a value, or between two values.
- Get instant results: The calculator will immediately output the corresponding Z-score and the probability (as both a decimal and a percentage).
Normal Distribution
Calculate probabilities and Z-scores instantly.
Probability
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*Technical note: Area under the curve is calculated using the Abramowitz and Stegun approximation for the cumulative distribution function, ensuring precision without external dependencies.
Understanding Your Results
The normal distribution (often called the “bell curve”) is the most important probability distribution in statistics because it accurately describes many natural phenomena.
- Z-Score: This tells you exactly how many standard deviations your value ($x$) is away from the mean. A Z-score of 0 means your value is exactly average. A Z-score of +2.0 means your value is two standard deviations above the average.
- Probability (Area under the curve): The total area under a normal distribution curve always equals 1 (or 100%). The calculated probability tells you the percentage of the data that falls into the specific region you selected. For example, if you calculate the probability “Below $x$” and get 0.84, it means 84% of the population falls below your chosen value.