Normal Distribution & Z-Score Probability Calculator
Calculate probabilities from a normal distribution for any mean and standard deviation. Find cumulative probabilities, tail probabilities, and percentile values.
Normal Distribution Guide
The Normal Distribution
The normal (Gaussian) distribution is bell-shaped, symmetric about the mean. Defined by two parameters: mean μ (centre) and standard deviation σ (spread). Standard normal: μ=0, σ=1, written Z~N(0,1). Standardise any normal to Z: z = (x - μ) / σ. The 68-95-99.7 rule: 68% of values fall within 1σ of mean. 95% within 2σ. 99.7% within 3σ. This means: in a population with mean IQ 100 and σ=15, 95% of people have IQ between 70 and 130. Only 0.3% fall outside 55-145.
Cumulative Probability
Φ(z) = P(Z ≤ z) is the cumulative distribution function (CDF) of the standard normal. This is the area under the bell curve to the left of z. P(X < a): standardise to z = (a-μ)/σ, then look up Φ(z). P(X > a) = 1 - Φ(z) (complement). P(a < X < b) = Φ(z_b) - Φ(z_a). Exact values require numerical integration (the normal CDF has no closed form) — most calculators and statistical tables give Φ(z) values. Common critical values: z = 1.645 → 95th percentile. z = 1.960 → 97.5th percentile (95% two-tail
Normal Distribution in Practice
Natural phenomena that follow a normal distribution: adult heights in a population. Measurement errors. IQ scores (by design). Blood pressure in a healthy population. Many biological measurements. The Central Limit Theorem: even if individual measurements are not normally distributed, the distribution of sample means approaches normal as sample size increases. This is why normal distribution underpins most parametric statistical tests — the test statistics are normally distributed even when the
When Not to Use Normal Distribution
The normal distribution is not appropriate for: skewed data (income, reaction times, survival times). Count data (use Poisson). Binary outcomes (use Binomial). Data with natural boundaries (cannot be negative — use log-normal). Time-to-event data (use Weibull or exponential). Warning signs: mean and median differ substantially (skewed). Histogram is not bell-shaped. Many zeros or floor/ceiling effects. Tests for normality: Shapiro-Wilk test (p > 0.05 suggests normality). Q-Q plot (data points sh
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