Math 4030 – 7b Normality Issues (Sec. 5.12) Properties of Normal? Is the sample data from a normal population (normality)? Transformation to make it Normal? 1

Why Normality Check? Many statistical methods require that the numeric variables we are working with have an approximate normal distribution. For example: t-tests; F-tests; Regression analyses; ……

3 Standardized normal distribution with empirical rule percentages. Without normality, we may use Chebyshev's Theorem. 75% 89% Table 3 1 – 1/k 2

Tools for Assessing Normality Histogram Boxplot Normal Scores Plot Goodness of Fit Tests

Graphic observation: 5

6 x Sample values: Normal Scores Plot: To test whether a sample (of size n) is from a normal population. Also called: Rankit Plot Normal Probability Plot Normal Q-Q Plot

To make a normal scores plot, 7 Normal scores Ordered data values 1. Order sample values from the smallest to the largest 2. Based on sample size n, use Normal table to find all the normal scores: 3. Make Normal Scores Plot 4. Sample is from a normally distributed population if the Normal Scores Plot looks like a straight line.

Normal Scores Plots: 8

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Transformations (Sec. 5.13) 10 If we have a skewed histogram from data set, the distribution cannot be normal. However, by carefully choose the transformation, we might be able to obtain a normal (or near normal) distribution. -Skewed to the right: common transformations are -1/x, ln x, x 1/4, x 1/2, etc. -Skewed to the left: common transformations are x 2, x 3, etc.

Normal Scores Plots for LN(AGE): 11