Normal Distribution Calculations

Statistics · Axiom Academy

LESSON Normal Distribution Calculations Master finding probabilities and percentiles using the normal distribution 1. Finding P(X < a) - Left Tail Probability To find the probability that a value is less than a given threshold, we calculate the area under the normal curve to the left of that value. Identify parameters: μ = 1000, σ = 200, x = 1200 Calculate z-score: z = (1200 - 1000) / 200 = 1.0 Find area: P(X < 1200) = P(Z < 1.0) = 0.8413 Interpret: About 84.13% of students score below 1200 2. Finding P(X > a) - Right Tail Probability To find the probability that a value exceeds a threshold, we calculate the area under the curve to the right. This equals 1 minus the left tail area. Identify parameters: μ = 1000, σ = 200, x = 1300 Calculate z-score: z = (1300 - 1000) / 200 = 1.5 Find left area: P(Z < 1.5) = 0.9332 Find right area: P(X > 1300) = 1 - 0.9332 = 0.0668 Interpret: About 6.68% of students score above 1300 3. Finding P(a < X < b) - Middle Area To find the probability between two values, we subtract the left tail area of the lower value from the left tail area of the upper value. Identify parameters: μ = 1000, σ = 200, a = 900, b = 1200 Calculate z-scores: z₁ = (900 - 1000) / 200 = -0.5 z₂ = (1200 - 1000) / 200 = 1.0 Find areas: P(Z < -0.5) = 0.3085 Subtract: P(900 < X < 1200) = 0.8413 - 0.3085 = 0.5328 Interpret: About 53.28% of students score between 900 and 1200 4. Finding Percentiles - Inverse Problems

This is the written version of the interactive lesson above. See the full Statistics course.