Normal Distribution Calculator

What this calculator does

Find out what share of a bell curve falls below, above or between the values you choose. The normal distribution is the familiar bell curve. Many measurements bunch up around an average, with fewer and fewer values further out on either side. The normal curve describes that pattern with just two numbers. The mean sets the center, and the standard deviation sets the spread. Test scores, measurement errors and many natural quantities are often modeled this way.

This calculator works in both directions. Give it a value, and it finds the share of the distribution below and above that value. Or give it two values, and it finds the share between them. In inverse mode, you give it a cumulative probability, meaning the share that lies below the value you want. It then finds the value that cuts off that share. That's how percentile cutoffs are found.

Where it's used

Why this math matters outside the classroom:

• Test scores. On a test with mean 500 and SD 100, about 84% of students score below 600, one SD above the mean.
• Bolt lengths. A machine making bolts with mean length 50 mm and SD 0.1 mm makes about 95% of them between 49.8 and 50.2 mm.
• Heights. If adult heights average 175 cm with SD 7 cm, about 2.3% of adults are taller than 189 cm.
• Package weights. A packer filling cereal boxes to a mean of 510 g with SD 5 g finds about 2.3% weigh under 500 g.
• Cutoff scores. A program admitting the top 10% on a test with mean 70 and SD 8 sets its cutoff at 70 + 1.2816 × 8, about 80.3.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSS.ID.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

1. Choose what you want: the probability below or above a value, the probability between two values, or the value for a given probability.
2. Enter the mean and standard deviation of the distribution.
3. Depending on the mode, enter the value x, the bounds a and b, or the cumulative probability.
4. Read the probability or value and the z-score. The working shows how your numbers are converted to the standard bell curve.

What the inputs mean

Mean (μ)

The center of the distribution. It can be any number, including negative values.

Standard deviation (σ)

The standard deviation, a measure of how spread out the values are. It must be greater than 0. Larger values give a wider, flatter curve.

Cumulative probability, P(X < x)

The area to the left of the value you want, strictly between 0 and 1. For the 90th percentile, enter 0.9. For the cutoff of the top 5%, enter 0.95.

Formula and method

z = (x − μ) ÷ σ
P(X < x) = Φ(z),  P(X > x) = 1 − Φ(z)
P(a < X < b) = Φ(z_b) − Φ(z_a)
Inverse: x = μ + σ × Φ⁻¹(p)

In these formulas, μ (mu) is the mean and σ (sigma) is the standard deviation. x is your value, and a and b are the two bounds. X stands for a value picked at random from the distribution, so P(X < x) is the chance it falls below x. p is a cumulative probability, and Φ⁻¹ ("inverse Phi") turns it back into a z-score.

Every normal distribution is a stretched and shifted copy of the standard normal, which has mean 0 and standard deviation 1. The z-score measures how many standard deviations a value sits from the mean. Φ(z), read "Phi of z", is the area under the standard curve to the left of z.

There's no simple exact formula for Φ, so the calculator uses a very accurate approximation. Areas far out in the upper tail are found by symmetry, as Φ(−z). That way, tiny probabilities aren't lost to rounding. The inverse mode reverses the process. It finds the z-score with the requested area to its left, then converts it back to your original scale.

Worked example

Scores on a test are normally distributed with a mean of 100 and a standard deviation of 15. What share of people score below 120?

1. Standardize: z = (120 − 100) ÷ 15 = 1.333333.
2. Look up the area to the left: Φ(1.333333) = 0.908789 (rounded to six decimals).
3. The area above is 1 − 0.908789 = 0.091211.

Result: About 90.9% of scores fall below 120 and about 9.1% fall above it. In the between mode, scores from 85 to 115 (one standard deviation either side of the mean) cover about 0.682689 of the distribution.

Understanding the result

A probability here is a share of the whole distribution. 0.908789 means about 90.9% of values lie below x. Read as a percentile, the same number says x is at roughly the 91st percentile.

For a smooth (continuous) distribution, the chance of hitting any exact value is zero. So P(X < x) and P(X ≤ x) are the same. About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.

Assumptions and limitations

• The quantity really follows a normal distribution. Check a histogram (a bar chart) of your data before relying on tail probabilities.
• The mean and standard deviation you enter are the true values for the whole group, or good estimates of them.
• Real data are often lopsided (skewed), or have more extreme values than the normal curve predicts (heavier tails). So the chances this calculator gives for extreme values can understate how often rare events happen.
• For small samples where the standard deviation is estimated, the t distribution is a better fit than the normal for confidence intervals and tests.
• Probabilities smaller than about 10^-300 are reported as 0, because of the limits of how computers store numbers (double precision).

Frequently asked questions

How this calculator is checked

Method: Standard normal CDF and its inverse (Hart/West rational approximation for Φ; Acklam's approximation with a Halley refinement for Φ⁻¹), applied after the z-score transformation. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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