Normal Distribution Calculator
The bell curve at your μ and σ: tail probabilities from the erf series, interval probabilities between two bounds, and the inverse direction — a percentile in, an x out — all computed, never table-read.
Normal Distribution Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Assumes the normal MODEL fits your data — the page cannot check that for you
- P(X = x) is 0 for any continuous curve; only intervals are meaningful
In short: Heights at μ = 70, σ = 5: the chance of 85 or more is P(X ≥ 85) = 0.001350 — z = 3, the three-sigma tail. The middle 60-to-80 window holds 0.954500 of the probability, the empirical rule’s “95%” priced exactly (the page derives 68.268949% / 95.449974% / 99.73002% from its own erf series rather than reciting them). The inverse direction: which height cuts the top 5%? Bisect the curve for z* = 1.644854, land on x = 70 + 1.644854 × 5 = 78.224268. σ ≤ 0 is refused — a normal distribution with no width is not a distribution.
Formula
z = (x−μ)/σ · P(X ≤ x) = Φ(z) · x = μ + z*σ (percentile → x by bisection)
Φ comes from the erf power series to 1e-18 and z* from bisection on Φ — the same engine the confidence-interval and margin pages bisect, so every 95% on this site is the same 1.959964.
Worked Example
- Set the curve: μ and a strictly positive σ.
- The interval card prices P(a ≤ X ≤ b) for your bounds; the tail card names both remainders.
- Type a percentile to run the inverse: the page bisects for z* and lands on x = μ + z*σ.
- Compare the empirical-rule card — it derives the 68/95/99.7 percentages from your own σ.
μ = 70, σ = 5: P(60 ≤ X ≤ 80) = 0.954500, P(X ≥ 85) = 0.001350, 95th percentile at x = 78.224268. The empirical card derives 68.2689 / 95.4500 / 99.7300 live.
Strengths & Limits Of This Model
Where this engine is strong
- Both directions: interval probabilities in, percentile-to-x out
- Empirical rule derived from the same erf engine, so rule and answers agree
Where it stops
- No bivariate or mixture normals
- Tails beyond |z| = 6 stated as boundaries, not printed as exact decimals
Practical Use Cases
Quality
specification limits vs process width
Education
what score cuts the top 5%?
Medicine
reference ranges from μ and σ
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Normal Distribution Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What is the difference between this and the z-score page?
The z-score page standardizes ONE point: x, μ, σ in, z and its percentile out. This page owns the whole curve — interval probabilities between two bounds AND the inverse direction, where a percentile goes in and a raw x comes out. Standardize here, invert there; the two pages cross-link.
Where do the 68–95–99.7 numbers come from?
From Φ itself: Φ(1)−Φ(−1) = 0.682689, ±2σ = 0.954500, ±3σ = 0.997300. They are derived properties of the curve, not folklore — the empirical card computes all three from the same erf series that answers your interval, so the rule and your answer cannot disagree.
How does the page invert the curve?
Bisection: it holds a percentile p and squeezes z until Φ(z) = p, then converts x = μ + zσ. The same zstar routine the confidence-interval page uses — which is why the 95th percentile here (z* = 1.644854) and the 95% margin there agree to the last digit.
Why is σ = 0 refused?
A normal distribution with zero width is a spike at μ — every probability mass at one point, every z infinite. The formulas divide by σ, so the honest answer is the refusal, not a decimal that hides a division by zero.
What happens at extreme z?
The tails are stated, not faked: beyond |z| = 6 the page prints “below 1e-9” (or “above 1 − 1e-9”) instead of a decimal that machine arithmetic cannot back. A probability with no honest digits is a sentence, and the page says the sentence.
Is my data normal just because I am here?
No — the curve is a MODEL you are borrowing. Check the shape before quoting: skewed counts and heavy-tailed sums break the tails first, exactly where decisions get expensive. The z-score page’s standardization still works; the probabilities are the part that inherits the assumption.
Can I get P(X = 85) exactly?
For a continuous distribution it is 0 — the curve has no atoms, only intervals. That is why every card answers ≤ or ≥: the meaningful questions are about ranges, and the page refuses to print a fake point mass.
Why does the percentile card not take ‘top ’ phrasing?
The field wants the percentile itself: type 95 for “the value below which 95% sits”. For the top 5%, that is the same x — 78.224268 on the defaults — and the tail card prints the remainder so both phrasings are visible at once.