Calculate probabilities and statistics for the normal (Gaussian) distribution. The most important continuous distribution in statistics.
Last updated: March 2026
The Normal Distribution (also called Gaussian distribution) is the most important continuous probability distribution in statistics. It has a characteristic bell-shaped curve that is symmetric around the mean, with most data concentrated near the center and tails extending to infinity in both directions.
This distribution appears everywhere in nature and science: heights of people, measurement errors, test scores, IQ scores, and countless other phenomena. Its prevalence is explained by the Central Limit Theorem, which states that the sum or average of many independent random variables tends toward a normal distribution, regardless of the underlying distribution.
The distribution is completely defined by two parameters: μ (mean) determines the center, and σ (standard deviation) controls the spread. About 68% of data falls within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ (the 68-95-99.7 rule).
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It's ubiquitous in nature due to the Central Limit Theorem. Many phenomena (heights, test scores, errors) follow it. It's mathematically tractable, symmetric, and completely defined by just two parameters (μ, σ). Foundation for most statistical inference.
μ (mean) shifts the curve left/right along the x-axis - it's the peak location. σ (standard deviation) controls spread: small σ = narrow/tall curve, large σ = wide/flat curve. Together they define any normal distribution.
z = (x - μ)/σ standardizes any value to the standard normal (μ=0, σ=1). z=0 is at mean, z>0 is above, z<0 below. |z|>2 indicates unusual values (outside 95% of data). Z-scores let you compare values across different distributions.
PDF (Probability Density Function) is the height of the curve at x - not a probability itself, but shows relative likelihood. CDF (Cumulative Distribution Function) gives P(X ≤ x) - the area under PDF curve from -∞ to x. CDF ranges [0,1].
Theoretically, normal distribution extends from -∞ to +∞. In practice, >99.7% of data is within μ±3σ. Extreme values beyond ±4σ or ±5σ are extremely rare but possible. For real-world constraints, use truncated normal.
P(a < X < b) = CDF(b) - CDF(a) = Φ((b-μ)/σ) - Φ((a-μ)/σ). This is the area under the curve between a and b. Use the 'between' mode in this calculator for convenient computation.
Empirical rule for normal distributions: approximately 68% of data falls within μ±1σ, 95% within μ±2σ, and 99.7% within μ±3σ. Quick mental check for outliers and confidence intervals. Valid only for normal data.
Visual: histogram, Q-Q plot (quantiles vs theoretical normal). Statistical tests: Shapiro-Wilk (best for n<50), Kolmogorov-Smirnov, Anderson-Darling. Check skewness and kurtosis. Shapiro-Wilk p>0.05 suggests normality.
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