Normal Distribution Calculator
Calculate left-tail, right-tail, interval, or outside probabilities for a normal distribution, or find a value from a percentile.
Normal distribution formulas
A value is standardized with z = (x−μ)/σ, where μ is the mean and σ is the positive standard deviation. The cumulative distribution function Φ(z) gives the probability to the left of z.
- Left tail: P(X≤x)=Φ(z)
- Right tail: P(X≥x)=1−Φ(z)
- Between two values: P(a≤X≤b)=Φ(zᵦ)−Φ(zₐ)
- Outside two values: 1−[Φ(zᵦ)−Φ(zₐ)]
- Percentile value: x=μ+σΦ⁻¹(p)
Checked examples
| Distribution and request | Result |
|---|---|
| Standard normal, P(X≤0) | 50% |
| Standard normal, P(X≤1) | ≈84.1345% |
| Standard normal, P(X≥1) | ≈15.8655% |
| Standard normal, P(−1≤X≤1) | ≈68.2689% |
| Standard normal, outside −1 to 1 | ≈31.7311% |
| μ=100, σ=15, P(X≤115) | ≈84.1345% |
| μ=100, σ=15, 97.5th percentile | ≈129.3995 |
How to use the calculator
- Select a left-tail, right-tail, between, outside, or inverse-percentile calculation.
- Enter the distribution mean and positive standard deviation.
- Enter one boundary, two ordered boundaries, or a percentile as requested.
- Select Calculate normal distribution and review the probability and z-scores.
Limits and numerical method
- Inputs must be finite, with magnitudes no greater than 1e100.
- Percentiles must be strictly between 0 and 100; endpoints correspond to infinite theoretical values.
- The normal CDF uses a complementary-error-function approximation. Inverse percentiles use a rational approximation.
- Displayed results are rounded; calculations retain full double-precision values.
- This tool assumes a continuous normal model. Real datasets may not be normally distributed.
Frequently asked questions
What is a z-score?
It is the number of standard deviations a value lies above or below the mean. Positive z-scores are above the mean and negative z-scores are below it.
What does cumulative probability mean?
It is the area under the normal curve to the left of a selected value.
Why can the percentile not be exactly 0 or 100?
A normal distribution extends without finite endpoints, so its theoretical 0th and 100th percentiles are negative and positive infinity.
Does this test whether my data are normal?
No. It calculates probabilities under an assumed normal distribution; it does not perform a normality test.
Related calculators
Use the Standard Deviation Calculator to calculate spread from a dataset, the Probability Calculator for event probabilities, the P-Value Calculator for z, t, chi-square and F statistics, or the T-Test Calculator for inference about means.