Normal Distribution Calculator
Calculate probabilities and areas under the normal (Gaussian) curve.
Area/Probability
Percentile
Result
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Enter parameters and calculate
P(a < X < b)
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Z-Score(s)
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Percentage
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X Value
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Normal Distribution
The normal distribution (Gaussian) is a symmetric, bell-shaped probability distribution that is fundamental to statistics.
f(x) = (1/σ√(2π)) × e^(-(x-μ)²/(2σ²))
Properties
- Symmetric: About the mean
- Bell-shaped: Highest at the mean
- Mean = Median = Mode: All equal
- 68-95-99.7 Rule: Percent within 1, 2, 3 standard deviations
Empirical Rule
| Range | Coverage |
|---|---|
| μ ± 1σ | 68.27% |
| μ ± 2σ | 95.45% |
| μ ± 3σ | 99.73% |
Frequently Asked Questions
What is the standard normal distribution?
The standard normal distribution is a special case with mean = 0 and standard deviation = 1. Any normal distribution can be standardized using z-scores.