Shape of Normal Curves Shape of Normal Curves

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Shape of Normal Curves

Shape of Normal Curves

Shape of Normal Curves

Shape of Normal Curves

68%-95%-99. 7% Rule

68%-95%-99. 7% Rule

Areas under Normal Curve

Areas under Normal Curve

Areas under Normal Curve(cont)

Areas under Normal Curve(cont)

Example: Normal Distribution The brain weights of adult Swedish males are approximately normally distributed

Example: Normal Distribution The brain weights of adult Swedish males are approximately normally distributed with mean μ = 1, 400 g and standard deviation = 100 g. (No real life population follows a normal distribution exactly!) a) What is the probability that an adult Swedish male has a brain weight of less then 1, 500 g? b) What is the probability that an adult Swedish male has a brain weight between 1, 475 g and 1, 600 g?

Example: Normal Distribution (cont) μ = 1, 400 g and = 100 g a)

Example: Normal Distribution (cont) μ = 1, 400 g and = 100 g a) What is the probability that an adult Swedish male has a brain weight of less then 1, 500 g?

Example: Normal Distribution (cont) μ = 1, 400 g and = 100 g b)

Example: Normal Distribution (cont) μ = 1, 400 g and = 100 g b) What is the probability that an adult Swedish male has a brain weight between 1, 475 g and 1, 600 g?

Area under the normal curve above

Area under the normal curve above

Example: Normal Distribution The brain weights of adult Swedish males are approximately normally distributed

Example: Normal Distribution The brain weights of adult Swedish males are approximately normally distributed with mean μ = 1, 400 g and standard deviation = 100 g. (No real life population follows a normal distribution exactly!) c) What is the 55 th percentile for the distribution of brain weights?

Example (Ex. Dispersion. sas) 7 21 12 4 16 12 10 13 6 13

Example (Ex. Dispersion. sas) 7 21 12 4 16 12 10 13 6 13 13 13 12 18 15 16 3 6 9 11 Determine the percentage of data points within 1 SD? 2 SD?

Example: Normality (Ex. Normal. sas) 7 21 12 4 16 12 10 13 13

Example: Normality (Ex. Normal. sas) 7 21 12 4 16 12 10 13 13 13 12 18 15 16 3 6 6 9 13 11

Example: QQPlots – Normal (Ex. QQplot. sas)

Example: QQPlots – Normal (Ex. QQplot. sas)

Example: QQPlots – Right Skewed

Example: QQPlots – Right Skewed

Example: QQPlots – Left Skewed

Example: QQPlots – Left Skewed

Example: QQPlots – Long Tail

Example: QQPlots – Long Tail

Example: QQPlots – Tails?

Example: QQPlots – Tails?

Example 4. 4. 5: Nonnormal Data

Example 4. 4. 5: Nonnormal Data

Interpretation of Shapiro-Wilk Test P-Value < 0. 001 < 0. 05 < 0. 10

Interpretation of Shapiro-Wilk Test P-Value < 0. 001 < 0. 05 < 0. 10 Interpretation Very strong evidence for nonnormality Strong evidence for nonnormality Moderate evidence for nonnormality Mild or weak evidence for nonnormality No compelling evidence for nonnormality

Objective Measure: SAS Tests for Normality Test Shapiro-Wilk Statistic p Value W 0. 98762

Objective Measure: SAS Tests for Normality Test Shapiro-Wilk Statistic p Value W 0. 98762 Pr < W 0. 8757 Kolmogorov-Smirnov D 0. 092489 Pr > D >0. 1500 Cramer-von Mises W-Sq 0. 042289 Pr > W-Sq >0. 2500 Anderson-Darling A-Sq 0. 233462 Pr > A-Sq >0. 2500

Objective Measure: SAS Tests for Normality Test Statistic p Value Normal W 0. 98762

Objective Measure: SAS Tests for Normality Test Statistic p Value Normal W 0. 98762 Pr < W 0. 8757 Right Skewed W 0. 949844 Pr > W 0. 4226 Left Skewed W 0. 925624 Pr > W 0. 0479 Long Tailed W 0. 927118 Pr > W 0. 0043 Short Tailed W 0. 949227 Pr > W 0. 0317

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