🔔 Normal Distribution Calculator
Find normal-distribution probabilities and z-scores — the area below, above or between values — with a shaded bell curve, from any mean and standard deviation.
P(X < 1.96)
0.975002
= 97.5002%
z-score = 1.96
Uses the exact normal CDF via a high-accuracy error function (erf). 🔒 Computed entirely in your browser.
How the normal distribution calculator works
Enter a mean and standard deviation, then a value (or z-score). The calculator uses the exact normal cumulative distribution function (via the error function) to give the probability below, above, or between your values, and shows a shaded bell curve. It also converts a z-score to a p-value and back, and finds the value at a given percentile (the inverse).
The normal CDF has no elementary closed form, so a chatbot only estimates it — this uses a high-accuracy erf implementation for exact results, and shows the shaded area so you can see what the probability means.
Frequently asked questions
How do I find the probability from a z-score?
The p-value for a z-score is the area under the standard normal curve. P(Z < z) is the cumulative probability; for a two-tailed test use 2 × P(Z > |z|). Enter your z-score and this tool gives the exact area below, above and in both tails.
What is a z-score?
A z-score measures how many standard deviations a value is from the mean: z = (x − μ) / σ. A z of +1.96 is 1.96 SD above the mean and marks the 97.5th percentile.
How do I find P(a < X < b)?
It is the area between two values: P(X < b) − P(X < a). Enter both bounds and the tool computes and shades the region between them.
How do I find the value at a given percentile?
Use the inverse (quantile) mode: enter a probability like 0.95 and the tool returns the value x such that P(X < x) equals it — for the standard normal that is z ≈ 1.645.
Is the calculation exact?
It uses a high-accuracy approximation of the error function (erf), accurate to about 7 decimal places — far beyond what any statistical table or manual estimate provides, and everything runs in your browser.
What is the empirical (68–95–99.7) rule?
About 68% of a normal distribution lies within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. You can verify these exactly by entering the corresponding bounds.
Related statistics tools
From the blog
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How Many People Do You Need to Survey? Sample Size Explained
The sample size for a proportion is n = z²·p(1−p) / E². At 95% confidence with a ±5% margin you need 385 responses — whether your population is 20,000 or 20 million. Why p = 0.5 is the safe choice, why halving the margin quadruples n, and when population size matters.
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Z-Score to P-Value: How to Read the Normal Distribution
A p-value is the tail area of the normal curve beyond your z-score. z = 1.96 gives a two-tailed p of 0.05; z = 1.645 a one-tailed p of 0.05. How the conversion works, one- versus two-tailed, and why a table is coarser than the exact erf calculation.