Normal Distribution Calculator
Calculate Z-scores, probabilities, and percentiles for normal (Gaussian) distribution. Enter mean, standard deviation, and either an X value or Z-score.
What is a normal distribution?
Normal distribution (Gaussian/bell curve) is a symmetric probability distribution where data clusters around the mean. Properties: 1) Bell-shaped, symmetric. 2) Mean = median = mode. 3) 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD (68-95-99.7 rule). Examples: Heights, test scores, measurement errors, IQ (mean 100, SD 15). Formula: f(x) = (1/rho�sqrt2PI)e^(-(x-μ)^2/2rho�^2). Used in: Statistics, quality control, finance, social sciences.
What is a Z-score and how do I calculate it?
Z-score measures how many standard deviations a value is from the mean. Formula: Z = (X - μ) / rho�. Where X = value, μ = mean, rho� = standard deviation. Example: Test scores mean 75, SD 10. Score of 85: Z = (85-75)/10 = 1.0 (1 SD above mean). Score of 60: Z = (60-75)/10 = -1.5 (1.5 SD below mean). Z = 0 means value equals mean. Positive Z = above mean. Negative Z = below mean. Used to compare different scales.
How do I find probability from Z-score?
Use Z-table or calculator to find area under curve. P(Z < z) = probability below Z-score. Example: Z = 1.0 → P(Z < 1) = 0.8413 (84.13% below). Conversions: P(Z > z) = 1 - P(Z < z). P(a < Z < b) = P(Z < b) - P(Z < a). Real example: IQ test (mean 100, SD 15), score 115: Z = 1.0, you scored better than 84% of people. Between Z = -1 and 1: 68% of data.
What is the 68-95-99.7 rule?
Empirical rule for normal distributions: 68% of data within μ +/- 1rho�. 95% within μ +/- 2rho�. 99.7% within μ +/- 3rho�. Example: SAT scores mean 1050, SD 200. 68% score 850-1250 (+/-1 SD). 95% score 650-1450 (+/-2 SD). 99.7% score 450-1650 (+/-3 SD). Used for: Quality control (6 sigma = 99.99966% within spec), outlier detection (>3 SD is unusual), confidence intervals, process capability.
How do I convert between raw scores and percentiles?
Steps: 1) Calculate Z-score: Z = (X - μ) / rho�. 2) Find probability P(Z < z) from table/calculator. 3) Multiply by 100 for percentile. Example: Height mean 170cm, SD 10cm. Your height 180cm: Z = (180-170)/10 = 1.0. P(Z < 1) = 0.8413 → 84th percentile. Reverse: 90th percentile → P = 0.90 → Z ≈ 1.28 → X = μ + Zrho� = 170 + 1.28(10) = 182.8cm. Used in: Test scores, growth charts, performance rankings.