Normal Distribution Calculator
This normal distribution calculator gives the probability that a value is at most, at least or between two limits for a mean μ and standard deviation σ. Example: for IQ scores with μ = 100 and σ = 15, 97.72% fall below 130.
Example: What do you want to find: P(X ≤ x): at most x · Mean μ 100 · Standard deviation σ 15 · Value x (or limit a) 130 → Probability: P(X ≤ 130) = 97.725%. Source: 1.3.6.6.1, NIST/SEMATECH e-Handbook – Normal distribution. Updated: .
How it is calculated
The normal distribution is the bell-shaped distribution of many measurements such as height, measurement error or IQ. It is fixed by two numbers: the mean μ (the peak of the bell) and the standard deviation σ (its width). The calculator finds areas under the bell curve, and those areas are the probabilities.
How to calculate with the normal distribution
- Standardise: z = (x − μ) / σ. The z-score says how many standard deviations x is away from the mean.
- Look up Φ(z): Φ is the cumulative distribution function of the standard normal distribution (once read from a table, here computed). Φ(1.96) = 0.975.
- Combine: at most x: Φ(z); at least x: 1 − Φ(z); between a and b: Φ(z₂) − Φ(z₁).
Example
Suppose adult men's height is normally distributed with μ = 180 cm and σ = 7 cm. Share below 190 cm: z = (190 − 180) / 7 = 1.43, Φ(1.43) ≈ 0.924, so about 92.4%. Between 173 and 187 cm lies exactly μ ± 1σ, which is 68.27%.
The 68-95-99.7 rule
About 68.3% of all values lie within μ ± 1σ, about 95.4% within μ ± 2σ and about 99.7% within μ ± 3σ. The list under the result shows these ranges for your numbers.
Percentile: from probability to value
In "Percentile" mode you enter the probability and get the value x below which that share lies. For 95% with μ = 100, σ = 15 this gives x ≈ 124.67. This is the inverse of Φ (used for confidence intervals and cut-offs).
Frequently asked questions
How do you calculate a probability for a normal distribution?
First standardise: z = (x − μ) / σ. Then take Φ(z) of the standard normal distribution. For "at most x" that is already the probability; for "at least x" use 1 − Φ(z); for an interval use Φ(z₂) − Φ(z₁).
What is a z-score?
The z-score tells you how many standard deviations a value lies above (positive) or below (negative) the mean. It makes values from different normal distributions comparable.
What does the 68-95-99.7 rule mean?
For a normal distribution about 68.3% of the values are within one standard deviation of the mean, 95.4% within two and 99.7% within three.
Why is the probability of exactly one value zero?
The normal distribution is continuous: the area under the curve above a single point is zero. Only intervals make sense ("at most", "at least", "between"), so P(X ≤ x) and P(X < x) are the same.
How do I find the value for a given probability, such as the 95% cut-off?
Choose "Percentile", enter μ, σ and p = 95. The calculator returns the value x below which 95% of the distribution lies. For the standard normal distribution that is 1.645; for 97.5% it is 1.96.
Sources and legal basis
- 1.3.6.6.1: NIST/SEMATECH e-Handbook – Normal distribution
- DLMF §7: NIST Digital Library of Mathematical Functions – error function erfc and Φ
- Wikipedia – 68–95–99.7 rule
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