Calculate probabilities and z-scores for any normal (Gaussian) distribution.
The normal distribution (Gaussian distribution) is a symmetric, bell-shaped probability distribution characterised by two parameters: the mean μ (centre) and the standard deviation σ (width).
It is the most important distribution in statistics because, due to the Central Limit Theorem, the average of a large number of independent measurements will be approximately normally distributed regardless of the original distribution.
In any normal distribution:
This is why values with z > 3 are considered very unusual — only 0.3% of a normal distribution lies beyond ±3σ.
The PDF (probability density function) gives the relative likelihood of a specific value. The area under the PDF between two points gives the probability of falling in that range.
The CDF (cumulative distribution function) gives P(X ≤ x) — the probability that a randomly drawn value is at most x. The CDF ranges from 0 to 1 and is equivalent to the percentile rank.
The standard normal distribution has μ = 0 and σ = 1. Any normal distribution can be converted to standard normal by computing z-scores: z = (x − μ) / σ.
Use the Z-Score Calculator to perform this standardisation and look up tail probabilities.