MARE 250 Dr. Jason Turner The Normal Distribution

Define Normal A variable is normally distributed if it is in the shape of a normal curve (Bell-Shaped Curve) Normal Curve Associated with a Normal Distribution is: Bell Shaped Centered at μ Range is between +3 and -3 std dev from the mean

So, am I Normal? Standardized Normal Distribution – Mean 0, Std Dev 1 Associated curve – Standard Normal Curve You can standardize a variable by subtracting its Mean and then dividing by its Std Dev

Properties of Normality 1.Total Area under Standard Normal Curve (SNC) is 1 2. SNC extends indefinitely in both directions, approaching, but not touching the horizontal axis 3.SNC is symmetric about 0; mirror image right/left 4.Most area under SNC lies between -3 and 3 (std dev)

Properties of Normality % of all possible observation lie w/in 1 std. dev. of the mean μ – σ and μ + σ % of all possible observation lie w/in 2 std. dev. of the mean μ – 2σ and μ + 2σ % of all possible observation lie w/in 3 std. dev. of the mean μ – 3σ and μ + 3σ

Assessing Normality Large samples: Histogram can give a rough estimate of Normality Small sample: difficult to tell with histogram need a more sensitive graphical technique

Normal Probability Plot: plot of the observed values of the variable versus the Normal Scores (observations expected for a normally dist. variable) A normal distribution should have highly sample data which is highly correlated (1:1 ratio, linear relationship) with normally distributed values Assessing Normality

Probability Plots - PP

When Using Probability Plots Decision of whether PP plot is linear is subjective Using a of sample observations to assess all

Guidelines for Probability Plots Plot is roughly linear – accept as reasonable that variable is approximately normally distributed Plot shows deviations from linear – conclude variable probably not normally distributed

Testing for Normality How do we test for normality? Use Linear Correlation Coefficient: Compute the linear correlation coefficient between the sample data and normal scores

Normality Tests Many Statistical Tests require normal data You must verify normality with a test Three primarily utilized include: Anderson-Darling Ryan-Joiner (Shapiro-Wilk) Kolmogorov-Smirnov More powerful

Probability Plots - PP H 0 hypothesis: data normally distributed If p value is less than α, then reject H 0 Data does not follow a normal distribution

This is not a Test… Hypothesis testing – used for making decisions or judgments Hypothesis – a statement that something is true Hypothesis test typically involves two hypothesis: Null and Alternative Hypotheses Hypothesis Testing 101