Normal Distribution Calculator
Calculate probabilities and z-scores for a normal distribution.
Z-Score
Probability
Percentage
Details
Probability Statement
Probability Density f(x)
Z-Score
The Normal Distribution & Z-Scores
The normal distribution (or Gaussian distribution) is a symmetric, bell-shaped curve defined by its mean (μ) and standard deviation (σ). A z-score measures how many standard deviations a value x is from the mean: z = (x − μ) / σ. Converting to a z-score lets you use the standard normal distribution (μ = 0, σ = 1) to find probabilities for any normal distribution.
The 68-95-99.7 Rule
- About 68% of values fall within 1 standard deviation of the mean.
- About 95% of values fall within 2 standard deviations of the mean.
- About 99.7% of values fall within 3 standard deviations of the mean.
How to Read the Probability
The cumulative probability Φ(z) gives the area under the curve to the left of x, which equals P(X < x). For P(X > x), subtract from 1. For the probability between two values, subtract the two cumulative probabilities: P(x1 < X < x2) = Φ(z2) − Φ(z1). The result is reported as a decimal (0 to 1) and as a percentage.