Quick answer: enter the mean, standard deviation and value above — e.g. IQ scores (mean 100, SD 15): the chance of scoring below 115 is about 84.1%. The shaded curve shows the exact area.
Normal distribution calculator
Short answer: this normal distribution calculator finds the probability that a value falls below, above, or between given points on a bell curve. Enter the mean (μ), standard deviation (σ) and your value — it converts to a z-score, computes the probability and shades the area on the curve.
Z-score formula with a worked example
Short answer: z = (x − μ) / σ. For an IQ of 115 with mean 100 and SD 15: z = (115 − 100) / 15 = 1.00, and Φ(1.00) = 0.8413 — so about 84.1% of people score below 115. The calculator shows this working for every input.
Once you have the z-score, the probability is the cumulative standard normal value Φ(z) — the area under the standard bell curve to the left of z. For a range P(a < X < b), it is Φ(z₂) − Φ(z₁). Everything above runs live in the tool.
The 68-95-99.7 rule
Short answer: about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. Hit the "68-95-99.7 demo" preset above to see the middle 68.27% shaded on the curve.
This empirical rule is a quick sanity check: if roughly two-thirds of your data sits within one SD of the mean, a normal model is plausible. Exact probabilities for any other bounds come from the calculator.
When to use a normal distribution
Short answer: use it for measurements that cluster around an average with symmetric spread — test scores, heights, measurement errors, IQ scores. Don't force it on skewed data like incomes or waiting times.
Many natural quantities are approximately normal thanks to the central limit theorem: averages of many small independent effects pile up into a bell shape. If your data is heavily skewed or has hard boundaries (like percentages capped at 100), pick a different model.
Normal vs standard normal distribution
Short answer: the standard normal is the special case with mean 0 and SD 1 (its values are z-scores). Any normal distribution converts to it via z = (x − μ) / σ — which is exactly what the calculator does before looking up the probability.
Use the "From z-score" tab when you already have a z-score and just need Φ(z). Use the other tabs when you're starting from raw values with a known mean and standard deviation.
Frequently asked questions
How do I calculate normal distribution probability?
Convert the value to a z-score with z = (x − μ) / σ, then look up the cumulative probability Φ(z). The calculator does both steps and shows the working.
What is a z-score?
A z-score measures how many standard deviations a value sits from the mean. z = 1 means one SD above the mean, which corresponds to about the 84th percentile.
How do I find the area under the bell curve?
The area under the curve between two points equals the probability of landing between them. This calculator shades that exact area on the bell curve for you.
What is the 68-95-99.7 rule?
About 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. Try the demo preset above to see it on the curve.
Can a probability be greater than 1?
No — a probability is always between 0 and 1 (0% to 100%). The whole area under the bell curve equals exactly 1.
What is the difference between PDF and CDF?
The PDF is the bell curve itself — its height is the density at a point. The CDF is the accumulated area under the curve up to a point, which is the probability. This calculator computes CDF values.
Is this normal distribution calculator free?
Yes — unlimited calculations, free forever. No account, no limits. Everything runs in your browser.
