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Normal Distribution Calculator

Compute probabilities, z-scores, and CDF for any bell curve — visualized step by step

The normal distribution (bell curve) is the most important distribution in statistics. It describes heights, test scores, measurement errors, and countless natural phenomena. Any normal distribution is fully defined by just two numbers: the mean μ (where the peak sits) and the standard deviation σ (how wide the bell is).

The PDF (probability density function) gives the shape of the curve. The CDF (cumulative distribution function) gives the probability that a value falls below a threshold — P(X ≤ x). A z-score measures how many standard deviations a value is from the mean: z = (x − μ) / σ.

Tell the AI your mean, standard deviation, and the probability you want to compute — it will draw the bell curve, shade the relevant area, and explain each step.

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FAQ

What is the 68-95-99.7 rule?
In any normal distribution: approximately 68% of values fall within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3. This rule is so useful it is called the "empirical rule." For example, if test scores are normal with μ = 70 and σ = 10, about 95% of scores fall between 50 and 90.
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-score of +2 means the value is 2 standard deviations above average. Converting to z-scores lets you use the standard normal table (μ = 0, σ = 1) regardless of the original units.
What is the difference between PDF and CDF?
The PDF (probability density function) describes the shape of the distribution — the height of the curve at each x value. The CDF gives the cumulative probability P(X ≤ x) — the area under the PDF curve from −∞ to x. To find P(a ≤ X ≤ b), compute CDF(b) − CDF(a).
How do I find the probability that X is between two values?
Use the CDF: P(a ≤ X ≤ b) = CDF(b) − CDF(a). In z-score terms: convert both bounds to z-scores, then look up or compute Φ(z₂) − Φ(z₁), where Φ is the standard normal CDF. The AI will do this calculation and shade the area under the curve for you.
What does "standard normal distribution" mean?
The standard normal distribution is the special case with mean μ = 0 and standard deviation σ = 1. Any normal distribution can be converted to standard normal by computing z-scores. Z-tables and calculators typically work with the standard normal, then you scale back to the original units.
When is the normal distribution a good model?
The normal distribution is a good model when data is approximately symmetric and bell-shaped with a single peak. By the central limit theorem, averages of large samples from any distribution become approximately normal — which is why the normal distribution appears so widely in statistics.

Related Topics

Z-Score CalculatorDistribution FittingConfidence Interval Calculator