Normal Distribution
Compute left-tail, right-tail and between probabilities for any normal distribution, or find the value x that cuts off a given probability (inverse CDF). The shaded area on the graph is the probability.
What is the Normal Distribution?
The normal (Gaussian) distribution is the familiar bell curve. It is fully described by its mean μ (the centre) and standard deviation σ (the spread). Heights, measurement errors and test scores are all approximately normal.
Probabilities are areas under the curve: P(a ≤ X ≤ b) is the area between a and b. The total area is exactly 1. By the central limit theorem, averages of many independent measurements are close to normal — whatever the original distribution.
Live Graph
N(0, 1)Critical z-values
| Central area | Each tail | z |
|---|---|---|
| 68.27% | 0.1587 | 1.000 |
| 80% | 0.1000 | 1.282 |
| 90% | 0.0500 | 1.645 |
| 95% | 0.0250 | 1.960 |
| 95.45% | 0.0228 | 2.000 |
| 98% | 0.0100 | 2.326 |
| 99% | 0.0050 | 2.576 |
| 99.73% | 0.00135 | 3.000 |
| 99.9% | 0.0005 | 3.291 |
Key Techniques
Every normal distribution becomes the standard normal N(0, 1) after this shift and rescale, so a single table of Φ(z) covers them all.
The curve is symmetric about μ, so right-tail and negative-z probabilities follow from left-tail values.
Subtract the left-tail area up to a from the left-tail area up to b.
Sources & Verification
- S. M. Ross, A First Course in Probability, 10th ed., Pearson, 2019 — Sec. 5.4 (Normal Random Variables).
- D. C. Montgomery & G. C. Runger, Applied Statistics and Probability for Engineers, 6th ed., Wiley, 2013 — Ch. 4 (Continuous Random Variables and Probability Distributions).
- M. Abramowitz & I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1964 — Ch. 26 (Probability Functions).