Normal Distribution Calculator
Calculate the cumulative probability and density of a value under a normal (bell curve) distribution given its mean and standard deviation.
Your results
Calculation breakdown
- Z-score
- z = (x − μ) / σ
- Cumulative probability
- P(X ≤ x) = Φ(z), approximated via the error function
- Density
- f(x) = (1 / (σ√2π)) × e^(−z²/2)
Worked example
IQ scores follow a normal distribution with mean 100 and standard deviation 15. The probability of scoring 115 or below is about 84.1%, meaning roughly 15.9% of people score higher.
Assumptions
- Assumes the data genuinely follows a normal (bell-curve) distribution; real-world data may deviate, especially in the tails.
- The cumulative probability uses a standard error-function approximation accurate to about 7 decimal places.
How this calculator works
Enter a mean, standard deviation and value (x). The calculator returns the z-score, the probability of a value falling at or below x, and the probability density at that point, using an accurate error-function approximation of the normal cumulative distribution.
Frequently asked questions
What is a normal distribution?
It's a symmetric, bell-shaped probability distribution where most values cluster near the mean and fewer occur further away, common in natural and measured data.
How accurate is the cumulative probability?
It uses a well-known error-function approximation accurate to about seven decimal places, which is more than sufficient for practical use.
Can I use this on skewed data?
No, this calculator assumes the underlying data is approximately normal; skewed or heavy-tailed data will give misleading results.