Math · Statistics
Normal distribution calculator.
Enter a mean and standard deviation, then find the probability below, above, between or outside values, or work backwards from an area to an x value.
For a normal distribution with mean 100 and standard deviation 15, the area between 85 and 115 is 0.6827 (68.27%), because those values are exactly one standard deviation either side of the mean. Going the other way, the value with 97.5% of the distribution below it is 129.40 (z = 1.96).
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How it works
The normal distribution has density f(x) = e−(x − μ)² / (2σ²) / (σ√(2π)), with location μ and scale σ. Every normal distribution can be converted to the standard normal (μ = 0, σ = 1) with z = (x − μ) / σ. The area to the left of z, Φ(z), has no closed form and is computed numerically, as is its inverse. Area between a and b is Φ(zb) − Φ(za); area outside is the sum of the two tails.
Worked example
With μ = 100 and σ = 15, the values 85 and 115 have z = −1 and z = 1. The area between them is Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827, and the two tails outside hold 0.3173. For the inverse, an area of 0.975 to the left corresponds to z = 1.960, so x = 100 + 1.960 × 15 = 129.40. NIST’s table of commonly used values pairs z = 1.960 with a cumulative probability of 0.975, which this calculator reproduces.
Limits
The answers are only as good as the assumption that your variable follows a normal distribution. For skewed or heavy-tailed data the areas can be far off. Use the z-score calculator when you only need a z-score and percentile.
Sources, opened 8 October 2026: NIST/SEMATECH e-Handbook, 1.3.6.6.1 Normal Distribution (density formula, standard normal with μ = 0 and σ = 1, cumulative distribution and percent point function computed numerically); NIST/SEMATECH e-Handbook, 1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (z = 1.960 pairs with cumulative probability 0.975).
Frequently asked questions
How do you find the area between two values on a normal curve?
Convert both values to z-scores with z = (x - mean) / sd, look up or compute the area to the left of each, and subtract the smaller from the larger. For 85 and 115 with mean 100 and sd 15, that is 0.8413 - 0.1587 = 0.6827.
What does the inverse normal give me?
It works backwards from an area to a value. Enter the area to the left of x and it returns the z-score and the x value. For an area of 0.975 with mean 100 and sd 15, x = 129.40.
What is the difference between area outside and area between?
They add up to 1. Area outside both values is the two tails combined. In the example, 0.6827 lies between 85 and 115 and 0.3173 lies outside.
Do I need a z table?
No. The calculator computes the cumulative normal probability numerically, which is what a z table tabulates. It reproduces NIST's value of 0.975 for z = 1.960.
Can I use it for data that is not normal?
Only as a rough guide. The areas assume a normal distribution. For skewed data the real proportions below or above a value can differ a lot.
How is this different from the z-score calculator?
The z-score calculator gives the z-score and percentile for one value. This one handles ranges, both tails, and the inverse problem.
Learning aid, not for decisions
Educational calculator. Results assume the conditions listed on this page, which real data often only roughly meet. Do not use it for clinical, regulatory, legal or financial decisions.