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The Normal Distribution (Gaussian Distribution) Honors Analysis Learning Target: I can analyze data using the normal distribution. German mathematician Influenced statistics, algebra, number theory, geometry, physics. Child prodigy! Constructed heptadecagon Triangular numbers Proved Fundamental Theorem of Algebra Influenced development of statistics, including Normal Distribution (Gaussian Distribution) Carl Friedrich Gauss (1777-1855) Imagine you took a test in two different classes. In the first class, you made a 93%. The class mean was a 96%, and the standard deviation was 3%. In the second class, you made a 78%. The class mean was a 74%, and the standard deviation was 2%. Which test performance was better? 𝜎 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝜇 = 𝑚𝑒𝑎𝑛 Normal Distribution (Gaussian Distribution) (Approximately) 68% within 1 std dev. of mean 95% within 2 std. deviations of mean 99.7% fall within 3 standard deviations of mean 68-95-99.7 Rule Calculate the mean (central value on curve) Each region increases or decreases by one standard deviation from the mean Ex: Test score mean: 74% Std. dev: 2% Labeling a Simple Normal Curve So what happens if you want to calculate a percentage for a value that ISN’T on your normal curve? Ex: PSAT math test with mean of 48 and a std. deviation of 3. What percent of scores are below 50? Normal distribution with a mean of 0 and a standard deviation of 1. Total area under curve = 1 Area to left of a given value on the curve gives the percentile rank – percent of scores LOWER than a given score. Standard Normal Distribution You can convert values to standard normal distribution form by calculating a z-score: 𝑋−𝜇 𝑍= 𝜎 Z-Score percentages can be looked up in a table or on a calculator. Z-Scores Example Solution Example Part II Unrepresentative Sample Undercoverage (Convenience sample, voluntary sample) Non-response Bias Voluntary response Bias Measurement Error Response Bias (Leading questions, social desirability) Types of Bias