The normal distribution calculator computes probabilities and Z-scores for the Gaussian (bell curve) distribution. Enter mean μ, standard deviation σ, and a value x to find cumulative probabilities, percentile rank, and between-bounds area.
Normal Distribution Calculator
Calculation
How to Use the Normal Distribution Calculator
The normal distribution is the most important distribution in statistics, appearing in test scores, heights, measurement errors, and as the limiting distribution of sample means (Central Limit Theorem).
Step 1: Enter Mean and Standard Deviation
Enter μ (mean) and σ (standard deviation). For the standard normal distribution, use μ=0 and σ=1. For SAT scores, μ=1060 and σ=195.
Example: SAT Scores
SAT scores have μ=1060, σ=195. What fraction scores above 1300? Z = (1300-1060)/195 = 1.23. P(X>1300) = 1 - Φ(1.23) ≈ 10.9%. About 11% of test-takers score above 1300.
Between Two Values
Use "Between two values" mode to find P(a < X < b). Example: P(900 < SAT < 1200) = Φ((1200-1060)/195) - Φ((900-1060)/195) = Φ(0.72) - Φ(-0.82) = 0.764 - 0.206 = 55.8% of test-takers.
FAQ
What is the normal distribution?
The normal (Gaussian) distribution is a symmetric bell-shaped probability distribution defined by mean μ and standard deviation σ. About 68% of values fall within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean (the 68-95-99.7 rule).
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 1.96 corresponds to the 97.5th percentile; Z=−1.96 is the 2.5th percentile. Z-scores allow comparison across different normal distributions.
How do I find P(a < X < b)?
Convert both a and b to Z-scores, then P(a < X < b) = P(Za < Z < Zb) = Φ(Zb) - Φ(Za), where Φ is the standard normal CDF. This calculator does this automatically when you enter both bounds.
Is this calculator free?
Yes, completely free with no signup required. All calculations use the Abramowitz & Stegun approximation for the error function, running entirely in your browser.
What is the central limit theorem and why does it matter?
The Central Limit Theorem states that the sample mean of n independent measurements approaches a normal distribution as n increases, regardless of the original distribution's shape. This is why the normal distribution is so widely used in statistics.