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Enter a mean, standard deviation, and values to see the probability on the bell curve.
What Is a Normal Distribution?
A normal distribution is a symmetric, bell-shaped probability distribution that appears everywhere in statistics: heights, weights, test scores, measurement errors, and quality-control measurements. It is defined by two numbers, the mean (μ) and the standard deviation (σ). The mean tells you where the centre of the bell is, and the standard deviation tells you how wide the bell is.
Many statistical tools assume a normal distribution because of the central limit theorem, which says that averages of large samples tend to follow this shape even when the underlying data does not. That makes the normal distribution one of the most important ideas in data analysis.
Normal Distribution Formulas
- μ = mean (centre of the bell)
- σ = standard deviation (spread of the bell)
- erf = Gauss error function, used for the cumulative probability
How to Find P(X < a)
To find the probability that a normally distributed value is less than a given number, evaluate the cumulative distribution function at that number. This gives the area under the left side of the bell curve up to the boundary.
For example, if IQ scores are normally distributed with mean 100 and standard deviation 15, the probability of an IQ below 120 is about 90.9%. The calculator shades the area under the curve and shows the exact steps.
How to Find P(a < X < b)
When you want the probability that a value falls between two boundaries, subtract the cumulative probability at the lower boundary from the cumulative probability at the upper boundary. This captures the area under the curve between the two points.
For a standard normal distribution, the probability between -1.96 and 1.96 is about 95%. That interval is the basis for many 95% confidence intervals used in statistics.
How to Find P(X > a)
The probability that a value is greater than a boundary is one minus the cumulative probability up to that boundary. It is the right-hand tail area under the bell curve.
If you want the chance that a value exceeds the mean by more than two standard deviations, this mode gives you the upper-tail probability directly.
How to Find the Inverse Normal Value
Inverse normal calculations answer questions like "what score puts someone in the top 10%?" or "what value marks the 95th percentile?" Given a probability and a tail, the calculator reverses the cumulative distribution using the inverse error function.
The formula is x = μ + σ × √(2) × erfinv(2p - 1) for a lower-tail probability. For an upper-tail probability, substitute 1 - p before applying the formula.
The Empirical Rule
Normal Distribution FAQ
What is the normal distribution formula? +
The probability density function is PDF(x) = (1 / (σ√(2π))) · e^(-½((x-μ)/σ)²), where μ is the mean and σ is the standard deviation.
How do I find the probability that X is less than a value? +
Use the lower-tail cumulative distribution function: P(X < a) = ½ · (1 + erf((a - μ) / (σ√2))). The calculator returns this probability and shades the left tail on the bell curve.
How do I find the probability between two values? +
Subtract the two cumulative probabilities: P(a < X < b) = CDF(b) - CDF(a). This gives the area under the curve between the two boundaries.
What is the empirical rule? +
For a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
How do I find the value for a given percentile? +
Use the inverse normal formula: x = μ + σ · √(2) · erfinv(2p - 1), where p is the lower-tail probability. Choose the upper tail when looking for a top-percent cutoff.
Why must the standard deviation be greater than zero? +
A standard deviation of zero means every value equals the mean, so there is no spread and the bell curve collapses into a single vertical line. The normal model breaks down in that case.
External Resources
Authoritative references for normal distributions and statistics.
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