Normal Distribution Calculator
Three tools in one: turn a raw score into tail probabilities, turn a probability into a Z score, or find the cut-off value that has a given probability below it. Set the mean and standard deviation, or leave the standard normal defaults.
Related calculators
Created by the Reeliy Editorial Team. Last updated: Oct 19, 2026
How to use the normal distribution calculator
Choose one of three tabs with the Calculate control.
- P from score: enter the mean (μ), the standard deviation (σ) and a raw score x. You get the Z score and four probabilities: above x, at or below x, the two-tailed area, and the central area.
- Z from probability: enter a probability p. You get the Z score that has an upper-tail area of p (one-tailed) and the pair of symmetric Z scores that together leave p in both tails (two-tailed). No mean or standard deviation is needed because the answer is on the standard scale.
- Quantile: enter μ, σ and a probability p to get the value below which a fraction p of the distribution lies, and the central interval that contains a share 1 − p of it.
The Precision box sets how many decimals are printed (2 to 25), but never more than 15 significant digits are shown, because a computer stores no more than that. The defaults, mean 0 and standard deviation 1, describe the standard normal distribution, for which the raw score and the Z score are identical.
The two-tailed rows describe the area at least |z| standard deviations from the mean on either side, measured around your mean. The row labels print your raw score for reference, so with a mean other than 0 read them as "as far from the mean as x is".
What is a normal distribution?
The normal (Gaussian) distribution is the symmetric bell-shaped curve that describes many natural and measured quantities: heights, measurement errors, test scores and the averages of many small independent effects. It is fully described by two numbers, the mean μ, which is the centre and the peak, and the standard deviation σ, which sets how spread out the bell is.
A useful rule of thumb gives the share of values inside a window around the mean:
| Window | Share of values | Outside the window |
|---|---|---|
| μ ± 1σ | 68.27% | 31.73% |
| μ ± 2σ | 95.45% | 4.55% |
| μ ± 3σ | 99.73% | 0.27% |
The calculator extends this table to any score. For standard deviation estimates from data see the standard deviation calculator.
Normal distribution formulas
Three functions are involved: the density (the height of the curve), the cumulative distribution (the area up to a point), and its inverse.
Probability density function (PDF)
f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²))
The density at x is the height of the bell. It is not a probability by itself; probabilities are areas under the curve between two points.
Standard normal distribution
Subtracting the mean and dividing by the standard deviation turns any normal variable into one with mean 0 and standard deviation 1:
z = (x − μ) ÷ σ
Every normal distribution is then handled by the one table, or one calculation, for the standard normal curve φ(z) = e^(−z²/2) ÷ √(2π).
Cumulative distribution function (CDF)
Φ(z) = P(Z ≤ z) = ½ × (1 + erf(z ÷ √2))
Φ(z) is the area to the left of z. The right tail is 1 − Φ(z), and by symmetry the two-tailed area beyond ±|z| is 2 × (1 − Φ(|z|)). The calculator evaluates Φ through the complementary error function, using rational Chebyshev approximations that are good to about 15 significant digits. Each tail area is computed directly rather than as 1 minus a number close to 1, so even a very small tail keeps its leading digits, and a score equal to the mean gives exactly 0.50 on each side. For the underlying function see the error function calculator.
Inverse distribution function (quantile function)
x = μ + σ × Φ⁻¹(p)
The quantile function answers the reverse question: which value has probability p below it? Φ⁻¹ has no closed form. The calculator starts from a rational approximation (Acklam's algorithm) and sharpens it with one Halley correction step against the exact tail area. The result, used by both Z from probability and the Quantile tab, is accurate to about 15 significant digits across the whole range, tails included.
Worked examples
From a score to a probability. Scores on a test are normally distributed with μ = 100 and σ = 15. What share of people score above 130? Z = (130 − 100) ÷ 15 = 2, and the calculator gives P(X > 130) = 0.02275, about 2.3%. So roughly 1 in 44 people exceed 130.
From a probability to a Z score. For a one-sided test at the 5% level the critical value is the Z with 5% in the upper tail. Choose Z from probability, enter p = 0.05 and read 1.6449. The two-tailed row shows ±1.96, the cut-offs that leave 2.5% in each tail.
A quantile. Heights in a group have μ = 170 cm and σ = 8 cm. The 90th percentile is 170 + 8 × 1.2816 = 180.25 cm; enter p = 0.9 on the Quantile tab to confirm. The same tab also reports the 10% mid range, 168.99 to 171.01 cm, which contains the central tenth of heights.
For a ready-made score in standard units see the z-score calculator, and for turning a Z score into a significance level see the p-value calculator.
Why the standard normal distribution is useful
- Comparison. Z scores put values from different scales on one footing, so a 130 on one test can be compared with an 85 on another.
- Central limit theorem. Averages of many independent observations are approximately normal even when the individual observations are not, which is why normal critical values drive confidence intervals and hypothesis tests.
- One table for all. After standardising, a single set of probabilities serves every mean and standard deviation.
- Approximation of other distributions. Binomial, Poisson and t distributions are close to normal under the right conditions.
Remember the assumption: the answers are only as good as the claim that your data are close to normal. Skewed data, heavy tails and outliers will make the probabilities misleading.
Frequently asked questions
How do I find the probability above a score?
Convert the score to a Z score with z = (x − μ) ÷ σ and read the upper-tail area. In the calculator choose P from score, enter μ, σ and x, and use the row P(X > x).
What is the difference between one-tailed and two-tailed Z scores?
A one-tailed Z score leaves the whole probability p in one tail, for example 1.645 for p = 0.05. A two-tailed pair splits p between both tails, so each side holds p/2 and the cut-offs are further out, for example ±1.96 for p = 0.05.
What does the quantile mid range mean?
It is the interval centred on the mean that holds a share 1 − p of the distribution. With p = 0.95 the mid range holds the central 5%.
How precise are the results?
Probabilities and Z scores are accurate to about 15 significant digits, which is the limit of double-precision numbers. The Precision box accepts up to 25 decimals, but the calculator stops printing at 15 significant digits, since anything further would only be the binary expansion of the number and not real information.
References
- Abramowitz M, Stegun IA (eds.). Handbook of Mathematical Functions, chapter 26 (Probability functions). National Bureau of Standards, 1964.
- Cody WJ (1969). Rational Chebyshev approximations for the error function. Mathematics of Computation 23(107), 631-637.
- Acklam PJ. An algorithm for computing the inverse normal cumulative distribution function. 2003.
Cite this calculator and page
If you use this page in your work, you can cite it as:
Reeliy Editorial Team. (2026). Normal Distribution Calculator. Reeliy. https://reeliy.com/calculators/normal-distribution-calculator“Normal Distribution Calculator.” Reeliy, 2026, https://reeliy.com/calculators/normal-distribution-calculator.