Frequently Asked Questions
Everything you need to know about the Normal Distribution Calculator.
What is a normal distribution?
A normal distribution is a symmetric, bell-shaped probability distribution defined entirely by its mean (center) and standard deviation (spread). Many natural measurements approximate it.
What is a z-score?
A z-score measures how many standard deviations a value is from the mean, calculated as z = (x − μ) / σ. It lets any normal distribution be compared to the standard normal distribution.
How does this calculator find probability?
It converts your value(s) to z-scores, then uses the cumulative distribution function (CDF) of the standard normal distribution to find the area under the curve — which equals the probability.
What does 'P(X < a)' mean?
It's the probability that a randomly drawn value from the distribution is less than a — the area under the curve to the left of a.
What percentage of values fall within 1 standard deviation of the mean?
About 68.27% of values fall within ±1 standard deviation, 95.45% within ±2, and 99.73% within ±3 — the empirical rule.
Can I use this for the standard normal distribution?
Yes — set the mean to 0 and the standard deviation to 1, and the calculator behaves exactly like a standard normal (z-table) lookup.
What does the shaded area on the curve represent?
The shaded region is the probability you're calculating — its area, as a fraction of the total area under the curve, equals the result shown.
Why must standard deviation be greater than zero?
A standard deviation of zero means no spread at all — every value equals the mean, which isn't a normal distribution and makes the z-score calculation undefined (division by zero).
What's the difference between 'between' and 'outside'?
'Between a and b' finds the probability a value falls inside that range; 'outside' finds the probability it falls below a or above b — they always sum to 1.
Is this calculator free to use?
Yes, completely free with no sign-up, and it runs entirely in your browser.