Standard Normal and Z-Scores

Probability · Axiom Academy

LESSON Standard Normal and Z-Scores Standardizing any normal distribution to mean 0 and variance 1 1. The Standard Normal Distribution When μ = 0 and σ² = 1, we have the standard normal distribution. This is the reference distribution from which all other normal distributions are derived through shifting and scaling. It's symmetric around 0, with most values between -3 and 3. The Z-score tells us how many standard deviations a value is from the mean. Subtracting μ centers the distribution at 0, and dividing by σ scales it to have variance 1. This transformation converts any N(μ, σ²) to N(0, 1). Example: If X ~ N(100, 225) and x = 115, then Z = (115-100)/15 = 1, meaning 115 is exactly 1 standard deviation above the mean. 3. Using Standard Normal Tables Standard normal tables (Z-tables) give P(Z ≤ z) for any value z. Since all normal distributions can be standardized, we only need one table. To find P(X ≤ x) for any normal X, we convert x to a Z-score and look it up in the table.

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