Calculate probabilities and percentiles for standard and general normal distributions, z-scores, and interval areas.
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The Normal (Gaussian) Distribution is the quintessential continuous probability distribution in statistics. Symmetrical and bell-shaped, it arises naturally in measurement errors, physical traits (height, blood pressure), standardized test scores (SAT, IQ), and financial market returns.
f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}, \quad z = \frac{x-\mu}{\sigma}, \quad \Phi(z) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{z}{\sqrt{2}}\right)\right]Where μ is the mean, σ is the standard deviation, z is the standardized score, and erf is the Gauss error function.
Scenario: IQ scores follow a normal distribution with mean μ = 100 and standard deviation σ = 15. Find the probability of a person scoring between 100 and 115.
Calculate z1 for x1 = 100: z1 = (100 - 100) / 15 = 0.00 → Φ(0) = 0.5000.
Calculate z2 for x2 = 115: z2 = (115 - 100) / 15 = +1.00 → Φ(1) ≈ 0.8413.
Subtract CDF values: P(100 ≤ X ≤ 115) = 0.8413 - 0.5000 = 0.3413 (34.13%).
A z-score (standard score) indicates how many standard deviations an element is from the mean: z = (x - μ) / σ. A positive z-score is above average; a negative is below.
The standard normal distribution is the special case of the normal distribution with mean μ = 0 and standard deviation σ = 1.
Approximately 4.55% of total data lies outside ±2 standard deviations (2.28% in each tail).
The Central Limit Theorem proves that the sum or average of a large number of independent random variables tends toward a normal distribution, regardless of the shape of the underlying population distribution.
Calculate Student t critical values (t*) and p-values for one-tailed and two-tailed tests from degrees of freedom and significance level alpha.
Compute exact P(X=k) and cumulative binomial probabilities P(X≤k), P(X≥k), distribution mean, variance, and standard deviation.
Calculate the probability of single events, union of events, intersection, conditional probability P(A|B), and complements.