Normal Distribution Calculator
Calculate P(X < x), P(X > x), and P(a < X < b) for any normal distribution instantly with our free normal distribution calculator. See the z-score, PDF value, and a live bell curve with the shaded probability region — all processed locally in your browser. No signup required.
Enter the mean (μ), standard deviation (σ), and x value(s), then click Calculate. The probability is computed using the exact normal CDF with a bell curve visualization. All calculations run instantly in your browser — no data is ever sent to any server.
Population or sample mean
Must be > 0
Quick Examples
Empirical Rule (68-95-99.7)
Why Use Our Normal Distribution Calculator?
Instant Normal Distribution Calculations
Calculate P(X < x), P(X > x), and P(a < X < b) for any normal distribution in milliseconds. Our normal distribution calculator uses the exact error function for maximum precision — no lookup tables, no approximations.
Secure Normal Distribution Calculator Online
All normal distribution calculations run 100% locally in your browser. Your values never leave your device — use our normal distribution calculator online with complete privacy and zero data collection.
Normal Distribution Calculator — No Installation
Use our normal distribution calculator directly in any browser with no downloads, plugins, or app installs required. Calculate normal probabilities from any device, anywhere, with no setup needed.
Interactive Bell Curve Visualization
Every calculation renders a live bell curve with the shaded probability region highlighted in blue. The visualization updates in real time as you change μ, σ, and x — making the normal distribution intuitive and visual.
Common Use Cases for Normal Distribution Calculator
Statistics & Probability Education
Students studying statistics need to calculate normal distribution probabilities for hypothesis testing, confidence intervals, and z-score problems. Our normal distribution calculator shows the bell curve, z-score, and probability simultaneously.
Medical & Clinical Research
Clinical researchers use the normal distribution to model biological measurements like blood pressure, height, and test scores. Our calculator computes P(X < x) and P(a < X < b) for any μ and σ, making reference range calculations straightforward.
Quality Control & Manufacturing
Six Sigma and statistical process control rely on normal distribution probabilities to calculate defect rates and process capability. Use our normal distribution calculator to find the probability of a measurement falling outside specification limits.
Finance & Risk Analysis
Financial models use the normal distribution to model asset returns, value at risk (VaR), and option pricing. Our calculator computes tail probabilities and between-value probabilities for any mean and standard deviation.
Scientific Research & Measurement
Measurement errors, experimental results, and natural phenomena often follow a normal distribution. Our normal distribution calculator helps researchers determine the probability of observing a value within a given range.
Machine Learning & Data Science
Data scientists use the normal distribution for feature normalisation, anomaly detection, and Gaussian naive Bayes classifiers. Our calculator verifies probability computations and z-score transformations instantly.
Understanding the Normal Distribution
What is the Normal Distribution?
The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It is characterised by its mean (μ), which determines the centre of the distribution, and its standard deviation (σ), which determines the spread. The normal distribution is symmetric around the mean, and approximately 68% of values fall within one standard deviation, 95% within two, and 99.7% within three (the empirical rule). Our normal distribution calculator computes exact probabilities using the cumulative distribution function (CDF) for any μ and σ.
How Our Normal Distribution Calculator Works
- 1. Enter μ and σ: Set the mean and standard deviation of your normal distribution. For the standard normal distribution, use μ = 0 and σ = 1.
- 2. Choose probability type: Select P(X < x) for a left-tail probability, P(X > x) for a right-tail probability, or P(a < X < b) for the probability between two values.
- 3. Get instant results: Click Calculate to see the probability, z-score, PDF value, and a live bell curve with the shaded region — all processed locally in your browser.
Key Formulas
- Probability Density Function (PDF): f(x) = (1 / (σ√(2π))) × e^(−(x−μ)² / (2σ²)) The height of the bell curve at x. Note: for a continuous distribution, P(X = x) = 0.
- Cumulative Distribution Function (CDF): P(X ≤ x) = Φ((x − μ) / σ) Φ is the standard normal CDF. Our calculator uses the error function (erf) for exact computation.
- z-Score (Standardisation): z = (x − μ) / σ Converts any normal distribution value to a standard normal (μ=0, σ=1) value. A z-score of 1.96 corresponds to the 97.5th percentile.
The Empirical Rule (68-95-99.7)
- μ ± 1σ: Contains approximately 68.27% of all values. About 1 in 3 values falls outside this range.
- μ ± 2σ: Contains approximately 95.45% of all values. About 1 in 20 values falls outside this range.
- μ ± 3σ: Contains approximately 99.73% of all values. Only about 1 in 370 values falls outside this range.
- Critical z-values: z = 1.645 (90% CI), z = 1.960 (95% CI), z = 2.576 (99% CI) — used in hypothesis testing and confidence intervals.
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Frequently Asked Questions About Normal Distribution Calculator
A normal distribution calculator computes probabilities for a normal (Gaussian) distribution given the mean (μ), standard deviation (σ), and one or two x values. Our calculator supports P(X < x), P(X > x), and P(a < X < b), shows the z-score and PDF value, and renders a live bell curve — all processed instantly in your browser with no signup required.
The standard normal distribution is a special case of the normal distribution with mean μ = 0 and standard deviation σ = 1. It is denoted N(0,1) or Z. Any normal distribution N(μ, σ) can be converted to the standard normal by computing z = (x − μ) / σ. Our calculator handles any μ and σ, not just the standard normal.
A z-score (or standard score) measures how many standard deviations a value x is from the mean: z = (x − μ) / σ. A z-score of 0 means x equals the mean; z = 1 means x is one standard deviation above the mean; z = −2 means x is two standard deviations below the mean. Our normal distribution calculator shows the z-score for every calculation.
P(X < x) is the probability that a randomly selected value from the distribution is less than x. It equals the area under the bell curve to the left of x. For example, P(Z < 1.96) ≈ 0.975 means 97.5% of standard normal values are below 1.96.
The empirical rule states that for a normal distribution: approximately 68.27% of values fall within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ. This rule is used in quality control, statistics education, and data analysis to quickly estimate the spread of normally distributed data.
The normal distribution is a continuous probability distribution, meaning it assigns probability to intervals, not individual points. The probability of any single exact value is zero because there are infinitely many possible values. Probabilities are only meaningful for ranges: P(a < X < b) = area under the curve between a and b.
Yes, completely. All normal distribution calculations run 100% locally in your browser using JavaScript. Your values are never sent to any server, stored, or logged. Use our normal distribution calculator online with full confidence in your privacy.
Yes. Our normal distribution calculator is 100% free with no signup, no account, no premium tier, and no usage limits. Calculate normal distribution probabilities as many times as you need — completely free, forever.
The most commonly used critical z-values are: z = 1.282 (80% CI), z = 1.645 (90% CI), z = 1.960 (95% CI), z = 2.326 (98% CI), and z = 2.576 (99% CI). These correspond to the z-scores where P(−z < Z < z) equals the confidence level. Enter these as x values in our calculator to verify.