CalcaTools

Normal Distribution Calculator

Normal probabilities below, above or between values, plus the inverse from a probability.

Last updated: July 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

RangeProbability
Within ±1σ68.27%
Within ±2σ95.45%
Within ±3σ99.73%
Below μ + 1.645σ95%
Below μ + 1.960σ97.5%

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

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Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

tips_and_updates Pair with the spoke articles

Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

z-scoreReading
|z| < 1Typical — within one SD of the mean
1 ≤ |z| < 2Notable — roughly the outer third
2 ≤ |z| < 3Unusual — outer ~5%
|z| ≥ 3Rare — outer 0.3%

Formula & methodology

Formula: z = (x − μ)/σ; P(X < x) = Φ(z); inverse: x = μ + σ·Φ⁻¹(p)

  1. Enter the mean μ and standard deviation σ.
  2. Choose the question: below, above, between, or inverse.
  3. Values are standardized to z-scores and the standard normal CDF Φ (or its inverse) gives the answer.

Worked example: IQ scores N(100, 15²): P(X < 130) → z = 2 → 0.9772 (97.72%). Inverse: p = 0.975 → z = 1.96 → x = 129.4.

Frequently asked questions

How do I find the probability below a value?
Standardize first: z = (x − μ)/σ, then read the cumulative probability Φ(z). For x = 130 with μ = 100, σ = 15: z = 2, P = 0.9772.
What is the inverse normal calculation?
Given a probability p, it finds the x with that much area to its left: x = μ + σ·Φ⁻¹(p). p = 0.975 gives the famous z = 1.96.
What is the 68-95-99.7 rule?
In any normal model, about 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
How do I find the probability between two values?
Compute the CDF at both bounds and subtract: P(a < X < b) = Φ(z_b) − Φ(z_a). Between 100 and 130 with μ=100, σ=15: 0.9772 − 0.5 = 0.4772.
What's the difference between this and a z-table?
Identical math — the calculator evaluates the standard normal CDF to full precision instead of a 4-decimal printed table, and handles the standardizing step for you.
How do I calculate the cumulative probability (CDF) for a normal distribution?
Enter the mean, standard deviation, and your cutoff value — the CDF is the area to the left. With mean 100 and SD 15, the probability of a value below 120 is 0.909. On TI calculators the same thing is normalcdf(lower, upper, μ, σ); this tool computes it without the button sequence.
How do I read probabilities between two values on the curve?
Subtract CDFs: P(85 < X < 115) with mean 100, SD 15 is CDF(115) − CDF(85) = 0.841 − 0.159 = 0.683 — the familiar 68% within one standard deviation. Any 'between' question on a normal curve reduces to two CDF lookups and one subtraction.

Explore the full statistics toolkit

Every CalcaTools statistics calculator — descriptive measures, probability distributions, hypothesis tests, confidence intervals, and Six Sigma process metrics — with step-by-step workings on each result.

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