Standard Normal Distribution

Statistics · Axiom Academy

LESSON Standard Normal Distribution Understanding the N(0,1) distribution, z-scores, and how to use z-tables to find probabilities 1. The Standard Normal Distribution N(0,1) The standard normal distribution is a special normal distribution with mean = 0 and standard deviation = 1 . We denote a standard normal random variable as Z . This bell-shaped curve is symmetric around zero, with most values falling within 3 standard deviations of the mean. Total area under the curve = 1 Approximately 68% of values fall within [-1, 1] Approximately 95% of values fall within [-2, 2] 2. Standardization: Converting to Z-Scores Any normal distribution can be standardized by converting values to z-scores . A z-score tells us how many standard deviations a value is from the mean. The transformation formula subtracts the mean and divides by the standard deviation, shifting and rescaling the distribution to N(0,1). 3. Using Z-Tables to Find Probabilities A z-table (or standard normal table) gives us the probability that Z is less than or equal to a given value. This is written as P(Z ≤ z). The table shows the area under the standard normal curve to the left of any z-score. We can use this to find probabilities for any normal distribution after standardizing. Convert your value x to a z-score: z = (x - μ) / σ Look up the z-score in the z-table The table gives P(Z ≤ z), the area to the left For P(Z > z), use: 1 - P(Z ≤ z) For P(a < Z < b), use: P(Z ≤ b) - P(Z ≤ a)

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