Normal Distribution Calculator
Calculate z-scores, left-tail, right-tail, between-values, outside-range probabilities and inverse percentiles for any normal distribution.
Normal probability calculator
Find left-tail, right-tail, between, outside, and z-score probabilities.
Inverse normal calculator
Find the x-value at a selected cumulative probability (percentile).
Normal distribution reference
Common central ranges for a normal distribution.
Step-by-step calculation
- Standardize x: z = (x − μ) / σ = 1.9600.
- Find the left-tail area: P(X ≤ x) = 97.5002%.
- Find the right tail: P(X ≥ x) = 1 − P(X ≤ x) = 2.4998%.
- For a range: subtract the two cumulative probabilities to get 0.0000%.
How to use the normal distribution calculator
- Enter the mean μ and standard deviation σ.
- Enter x for a single-value probability or use x and b for a range.
- Read the z-score and probability results.
- For an inverse calculation, enter a cumulative probability such as 0.95.
Normal distribution formula
The normal probability density is f(x) = 1/(σ√(2π)) × e−(x−μ)²/(2σ²). Probabilities are areas under the normal curve.
Worked example
For a standard normal distribution with μ = 0 and σ = 1, x = 1.96 gives z = 1.96 and P(X ≤ 1.96) ≈ 97.5%.
Normal distribution rules
- About 68.27% lies within μ ± 1σ.
- About 95.45% lies within μ ± 2σ.
- About 99.73% lies within μ ± 3σ.
Frequently asked questions
What is a z-score?
A z-score tells you how many standard deviations a value is above or below the mean.
How do I calculate a z-score?
Use z = (x − μ) / σ, where μ is the mean and σ is the standard deviation.
What is P(X ≤ x)?
It is the probability that a normally distributed value is less than or equal to x.
How do I find the probability between two values?
Convert both endpoints to z-scores and subtract the lower cumulative probability from the upper cumulative probability.
What is an inverse normal calculation?
An inverse normal calculation finds the x-value corresponding to a chosen cumulative probability or percentile.
What are the 68–95–99.7 rules?
Approximately 68.27%, 95.45%, and 99.73% of observations fall within 1, 2, and 3 standard deviations of the mean.