Enter the mean and standard deviation of a normal distribution, then choose whether you want the probability below a value, above a value, or between two values. You'll get the probability as a percentage, in plain English.
How the normal distribution calculator works
The normal distribution — the bell curve — is fully described by two numbers: its mean μ (where the peak sits) and its standard deviation σ (how wide the bell is). Probability is the area under the curve, so "what's the chance X is below 130?" means "how much of the bell's area lies to the left of 130?" The calculator converts your bounds to z-scores and uses the standard normal cumulative distribution function Φ to measure those areas: left of a bound, right of it, or the slice between two bounds.
The formula
P(X < b) = Φ( (b − μ) ÷ σ )
P(X > a) = 1 − Φ( (a − μ) ÷ σ )
P(a < X < b) = Φ( (b − μ) ÷ σ ) − Φ( (a − μ) ÷ σ )
Here μ is the mean, σ is the standard deviation (which must be positive), and Φ is the standard normal CDF — the share of a standard bell curve at or below a given z-score.
Worked example
IQ scores follow a normal distribution with mean μ = 100 and standard deviation σ = 15. What share of people score between 85 and 130?
Lower z = (85 − 100) ÷ 15 = −1, upper z = (130 − 100) ÷ 15 = +2.
P = Φ(2) − Φ(−1) = 0.9772 − 0.1587 = 0.8186, i.e. about 81.86% of people.
Why the normal distribution shows up everywhere
The bell curve's ubiquity isn't a coincidence — it's the central limit theorem. Whenever a quantity is the sum of many small, independent influences (genes and nutrition on height, a hundred tiny errors in a measurement), its distribution drifts toward normal no matter what shape the individual influences have. That's also its limit: quantities driven by multiplication or by rare extreme events — incomes, stock crashes, viral posts — are not normal, and applying bell-curve math to them famously understates the odds of extremes. Before trusting these probabilities, ask whether your variable is really the sum of many small nudges.
Frequently asked questions
How do I find the probability in a normal distribution?
Convert your value to a z-score by subtracting the mean and dividing by the standard deviation, then look up that z-score in the standard normal distribution to get the area (probability) below it. This calculator does both steps: probabilities above a value or between two values come from subtracting those areas.
What is the 68–95–99.7 rule?
In any normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It's a quick sanity check — if this calculator gives you P(mean − σ < X < mean + σ), you should see roughly 68.27%.
What is the probability of exactly one value in a normal distribution?
Zero, technically. The normal distribution is continuous, so probability lives in ranges, not points — P(X = 70) is 0, but P(69.5 < X < 70.5) is a perfectly good number. That's why this calculator always asks for a bound or a range.
How do I know if my data is normally distributed?
Plot it. A histogram of normal data looks like a symmetric bell; a normal quantile (Q-Q) plot hugs a straight line. Heights, measurement errors, and test scores tend to be close to normal, while incomes, wait times, and city sizes are usually skewed and need different tools.
What happens if the standard deviation is zero or negative?
The distribution isn't defined. A standard deviation must be positive — zero would mean every value equals the mean (no curve at all), and negative spread is meaningless. This calculator requires σ > 0 and won't return a result otherwise.