🔔 Normal Distribution Calculator
Normal Distribution Calculator is a free statistics tool that computes probabilities for the bell-shaped normal distribution. Enter the mean μ and standard deviation σ, then choose a mode — below a value P(X ≤ x), above a value P(X ≥ x), between two values P(a ≤ X ≤ b). The tool returns the probability, the equivalent z-score, the probability density f(x), and a plain-language interpretation. All calculations run locally in your browser with no data uploaded and no account required. Students and analysts use it for homework, quality control, risk assessment and any task that needs Gaussian probabilities.
Compute cumulative probabilities, tail probabilities, interval probabilities for any normal distribution. Enter μ and σ — fully offline.
What is this tool?
The normal distribution — also called the Gaussian distribution or bell curve — is the most important continuous probability distribution in statistics. Its symmetric, bell-shaped curve is completely described by two parameters: the mean μ, which locates the centre of the curve, and the standard deviation σ, which controls how wide or narrow the spread is. The famous 68-95-99.7 rule says that roughly 68% of all observations fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
Because so many natural and social phenomena approximate a normal distribution — heights, blood pressure, measurement errors, test scores, manufacturing tolerances — the normal distribution sits at the heart of inferential statistics. Six Sigma quality programmes, IQ scaling, financial risk models and confidence intervals all lean on it.
This calculator gives you those probabilities instantly. Pick a mode, enter μ, σ and your value(s), and it returns the probability density f(x), the cumulative probability, and the equivalent z-score. Everything runs in your browser — nothing is uploaded. For related work, try the z-score calculator, the standard deviation calculator and the confidence interval calculator to extend your analysis.How it works
The calculator converts your input into a z-score, then evaluates the cumulative distribution function (CDF) of the standard normal.
f(x) = (1 / (σ√(2π))) · e^(−(x − μ)² / (2σ²))
P(X ≤ x) = Φ(z) = (1/√(2π)) · ∫−∞z e^(−t²/2) dt, where z = (x − μ) / σ
Z-Score to Percentile Table
Cumulative probability Φ(z) = P(Z ≤ z) for common z-scores. Use this as a quick lookup instead of a printed z-table.
| z-score | P(Z ≤ z) | Percentile | Tail P(Z > z) |
|---|---|---|---|
| −3.00 | 0.00135 | 0.135% | 99.865% |
| −2.58 | 0.00494 | 0.494% | 99.506% |
| −2.00 | 0.02275 | 2.275% | 97.725% |
| −1.96 | 0.02500 | 2.500% | 97.500% |
| −1.645 | 0.05000 | 5.000% | 95.000% |
| −1.00 | 0.15866 | 15.866% | 84.134% |
| −0.50 | 0.30854 | 30.854% | 69.146% |
| 0.00 | 0.50000 | 50.000% | 50.000% |
| 0.50 | 0.69146 | 69.146% | 30.854% |
| 1.00 | 0.84134 | 84.134% | 15.866% |
| 1.645 | 0.95000 | 95.000% | 5.000% |
| 1.96 | 0.97500 | 97.500% | 2.500% |
| 2.00 | 0.97725 | 97.725% | 2.275% |
| 2.58 | 0.99506 | 99.506% | 0.494% |
| 3.00 | 0.99865 | 99.865% | 0.135% |
Empirical Rule (68-95-99.7)
For a normal distribution, the percentage of values within ±1, ±2 and ±3 standard deviations of the mean.
| Range | Interval | Probability inside | Probability outside |
|---|---|---|---|
| ±1σ | μ − σ to μ + σ | ≈ 68.27% | ≈ 31.73% |
| ±2σ | μ − 2σ to μ + 2σ | ≈ 95.45% | ≈ 4.55% |
| ±3σ | μ − 3σ to μ + 3σ | ≈ 99.73% | ≈ 0.27% |
Values beyond ±3σ (roughly 0.27% of cases) are often treated as outliers in quality control and risk analysis. The exact two-sided 95% interval is ±1.96σ, which is where the 95% confidence interval comes from.
How to use
- Enter the mean μ of your distribution (use 0 for a standard normal).
- Enter the standard deviation σ. It must be a positive number.
- Pick a mode — below a value, above a value, or between two values.
- Type the value x (or the bounds a and b for interval mode).
- Press Calculate. Read the probability, percentage, z-score, density f(x) and the plain-language interpretation.
Frequently Asked Questions
What is the difference between PDF and CDF?
The probability density function f(x) gives the height of the bell curve at a single point — it is not itself a probability. The cumulative distribution function Φ(x) = P(X ≤ x) is the actual probability of observing a value up to and including x, which equals the shaded area under the curve to the left of x.
How is the z-score related to the normal distribution?
The z-score converts any value to standard-normal units via z = (x − μ) / σ. A z-score tells you how many standard deviations x sits above or below the mean. For example z = 1.96 means x is 1.96 standard deviations above μ, and the upper-tail probability beyond it is about 2.5%.
What is the empirical 68-95-99.7 rule?
For a normal distribution, about 68% of values lie within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ. It is a quick mental shortcut for estimating probabilities without integrating, and it only holds for the normal distribution.
Why must σ be positive?
The standard deviation measures spread, so it must be a positive real number. σ = 0 would mean every observation equals the mean exactly, collapsing the distribution into a single point; σ < 0 is mathematically meaningless. The calculator rejects zero or negative values.
Can this tool find the inverse — the x value for a given percentile?
{'No — the tool supports three modes only': 'less-than P(X ≤ x), greater-than P(X ≥ x), and between P(a ≤ X ≤ b). It does not compute an inverse (the x for a given probability).'}
Does the calculator work for the standard normal distribution?
Absolutely. Set μ = 0 and σ = 1 and every result is in standard-normal form, identical to what a z-table gives. The tool still reports the z-score, density and cumulative probability.
Tips & Advice
A z-score of ±1.96 corresponds to the middle 95% of the distribution — this is where the familiar 95% confidence interval comes from. Remember that P(X ≤ μ) is always exactly 0.5, because the normal curve is symmetric around the mean. When σ is very large, the curve is flat and wide, so tail probabilities shrink slowly; when σ is small, probability concentrates sharply around μ. Always double-check that σ is positive — a zero or negative standard deviation is invalid and the tool will reject it. For interval mode, make sure the lower bound a does not exceed the upper bound b. All calculations stay in your browser.
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Sources & References
- Abramowitz M, Stegun IA. Handbook of Mathematical Functions, formula 7.1.26 — Error function approximation. dlmf.nist.gov.
- NIST/SEMATECH. e-Handbook of Statistical Methods — Normal Distribution. itl.nist.gov.
- Wikipedia. Normal distribution. en.wikipedia.org.
Last reviewed: August 2026.
Limitations
This tool computes Gaussian probabilities for educational and analytical purposes. It assumes that your data actually follows a normal distribution — real data sets are often skewed or heavy-tailed.
What this tool does not account for:
- Non-normal data: heights, measurement errors and similar quantities are often approximately normal, but income, reaction times and many financial returns are not. Always verify distributional assumptions before relying on the probabilities.
- Numerical precision: the error-function approximation is accurate to roughly 1×10⁻⁷, which is more than enough for everyday work but not for cryptographic or high-precision research.
- Discrete data: the normal distribution is continuous. For counts or proportions, a binomial or Poisson model may be more appropriate.
For mission-critical decisions, verify results with dedicated statistical software and consult a qualified statistician.