Module 3 · Distributions · Topic 7
Standard Normal Distribution. The standard normal distribution has μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using z-scores.
Interpreting Z-Scores. A z-score tells you how many standard deviations a value is from the mean. Positive z: above the mean. Negative z: below the mean. z = 0: at the mean.
z = (x − μ) / σ
P(X = k) = C(n,k)p^k(1-p)^(n-k) — Exact probability of k successes in n independent Bernoulli trials.z = (x - mu) / sigma — Standardizes a value by measuring standard deviations from the mean.SAT scores: μ = 1060, σ = 195. A student scores 1350. z = (1350 − 1060) / 195 = 290 / 195 ≈ 1.49. This score is about 1.49 standard deviations above the mean.
A z-table gives P(Z ≤ z). To find P(Z > z), use 1 − P(Z ≤ z). To find P(a < Z < b), compute P(Z ≤ b) − P(Z ≤ a).
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