Normal Distribution Calculator
Probability0.908789
As Percentage90.88%
Z-Score1.3333
The Normal Distribution Calculator answers probability questions about a Gaussian bell curve defined by its mean and standard deviation. Choose whether you want the area below a value, above a value, or between two bounds, and the tool converts each cutoff to a standard z-score and evaluates the normal cumulative distribution function. Results are shown as a probability, an equivalent percentage, and the underlying z-scores.
Formula
z = (x − μ) / σ; P(X < x) = Φ(z)
- x
- The value being evaluated
- μ
- Mean (center) of the distribution
- σ
- Standard deviation (spread)
- Φ(z)
- Standard normal cumulative distribution function
How it works
- Enter the mean μ and standard deviation σ of your normal distribution, then pick a tail mode: below x, above x, or between x₁ and x₂.
- Each cutoff is standardized with z = (x − μ) / σ so it can be compared against the standard normal curve.
- The probability comes from the normal CDF Φ(z): "below" returns Φ(z), "above" returns 1 − Φ(z), and "between" returns Φ(z₂) − Φ(z₁).
Worked example
IQ scores are normal with μ = 100 and σ = 15; find P(90 < X < 120).
- Lower z = (90 − 100) / 15 = −0.6667 and upper z = (120 − 100) / 15 = 1.3333.
- Φ(1.3333) = 0.908789 and Φ(−0.6667) = 0.252493.
- P(90 < X < 120) = 0.908789 − 0.252493 = 0.656296.
About 0.6563, or roughly 65.63% of scores fall between 90 and 120.
Frequently asked questions
- How accurate is the probability the calculator returns?
- It uses the Abramowitz and Stegun 26.2.17 rational approximation for the normal CDF, which is accurate to better than 1e-7 across the practical range, so the displayed six-decimal probabilities are reliable.
- What is the difference between this and a z-score calculator?
- A z-score calculator focuses on standardizing a single value and reading off its percentile. This tool adds tail selection (below, above, between) so you can directly answer interval probability questions for any mean and standard deviation.
- What does the empirical rule say about the bell curve?
- About 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. You can confirm these by entering symmetric bounds in the "between" mode.
- Can I use any units for the mean and value?
- Yes, as long as the mean, standard deviation, and value all share the same units. The z-score is dimensionless because the units cancel in (x − μ) / σ.