Statistics / Probability
Normal Distribution Calculator: CDF, Density and Inverse
Calculate a normal cumulative probability, probability density or inverse quantile. Enter the mean and standard deviation for a standard normal or any other normal distribution.
Normal Distribution Calculator: CDF, Density and Inverse: A normal distribution describes a continuous bell-shaped model with centre μ and spread σ. The starting example uses μ = 0, σ = 1 and x = 1.96; its cumulative probability is about 0.9750021. Enter a different mean and positive standard deviation to work in the units of your own measurement. Cumulative distribution, or CD, gives the probability that a modelled value is at most x. For an interval from a to b, subtract CD(a) from CD(b). Probability density, or PD, is the height of the curve at x, not the probability of observing that exact value. Inverse mode reverses the calculation: enter a cumulative probability and find the value below which that fraction of observations falls. This can be useful for percentile thresholds, confidence limits and checking z tables. A normal model is an assumption about the data, so check that it suits your measurement before interpreting the result. The [z-score calculator](/z-score-calculator) also shows how a raw value is standardised, and the [scientific calculator](/scientific-calculator) includes the same probability controls alongside statistics and regression. Runs 100% locally in your browser with zero server file uploads.
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More functions, constants and conversions
Use semicolons between arguments: ncr(52;5), dms(30;15;30), randint(1;6). Rnd rounds to the selected significant digits. E is Euler's number until you store a value in E; C is the speed of light until you store C.
- GCD
- 6
- LCM
- 144
- Integer quotient and remainder
- 2; 12
- 48 = 2 × 18 + 12
- 18 = 1 × 12 + 6
- 12 = 2 × 6 + 0
Constants: c, h, hbar (ħ), qe (elementary charge), me, mp, na, kb, gasr, grav (G), g (standard gravity), eps0, mu0, amu and sigma. Measured constants carry uncertainty; the displayed values are not exact. NIST / CODATA constants
Type or tap. Multiplication can be implied, as in 2π or 3(4+1); ^ is a power, ! a factorial, and % divides by 100. Enter calculates; Escape clears.
Normal PD is a density, not a point probability. CD includes the upper endpoint. Discrete inverse returns the smallest integer with CDF ≥ p; use 0 < p < 1. Tested central probabilities have about 12 decimal places of absolute accuracy; extreme tails may lose precision.
Normal distribution
NIST Engineering Statistics Handbook, Normal Distribution (https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm), documents the density, cumulative distribution and location-scale parameters. The calculator follows those definitions.
Special functions and accuracy
NIST DLMF, Error Functions (https://dlmf.nist.gov/7), relates normal probabilities to the error function. Cumulative calculations use incomplete gamma methods; inverse normal is refined against the CDF. Central-range tests target about 1e-12 absolute accuracy; extreme tails can lose precision.
How to use it
- Choose normal PD, CD or inverse cumulative probability.
- Enter x, or enter a probability for inverse mode, then set the mean and standard deviation.
- Read and copy the result; switch to binomial or Poisson for discrete outcomes.
Privacy & limitations
All parameters and probability calculations stay in your browser.
Related tools
Frequently asked questions
How do I find a right-tail probability?
Calculate CD at your threshold and subtract it from 1. For example, a standard normal value above 1.96 has probability about 0.0249979. Very small tails can lose precision when subtracting nearly equal numbers.
What is inverse normal at 0.975?
With mean 0 and standard deviation 1 it is approximately 1.95996398454. Enter 0.975 in inverse mode, not 97.5. With another mean and spread, the result is rescaled into your original units.
Can a density exceed 1?
Yes. Density has reciprocal measurement units and is not a probability on its own. Probabilities are areas under the curve and remain between 0 and 1. The standard deviation must be strictly positive.
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