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Where Are You? Looking at the various data we collected (height, shoe size, etc.), can you determine where you are? Compared to what? Are you average? Below/above average? How far? How can you measure “how far?” It is helpful to have a standard measure of variation to answer such questions. IF the distribution is roughly symmetric and mound-shaped, we can use a mathematical model to help us estimate the answers to such questions. This mathematical model is called the normal model. (See http://en.wikipedia.org/wiki/Normal_distribution for more background.) Notice that the normal model does not PERFECTLY describe the data—it is only an approximation of the shape of the data. After all, data are discrete, and a normal model is continuous. This is another example of the difference between “stats world” and “math world…” A Standard Normal Model is drawn below. If the center is zero, then one standard deviation from zero turns out to be directly below the inflection point of the curve. And by the time you get out to three standard deviations from the center, there is barely any area Fun cti on Pl ot left under the curve. no data One great advantage to using a perfect mathematical model: we can calculate lots of numbers! For instance, we can calculate the area (or percentage or probability or percentile) above or below any point on the xaxis. We just need to know how many standard deviations above/below zero (the z-score). On the TI calculator, we can use normalcdf(left, right), where “left” and “right” are the boundaries in z-scores. Or we can use a z-table found in most statistics textbooks. 0.4 0.3 0.2 0.1 0.0 -4 -3 -2 -1 0 1 x y = normalDensity x , 0, 1 2 3 4 Common exercises using the normal model: 1. Find the z-score for a student who scored 660 on the SAT Verbal, where the mean is 505 and the standard deviation is 110. 2. What percentile did this student score? 3. What SAT Verbal score represents the first (lower) quartile? 4. Approximately what percent of students scored between 450 and 560? 5. What score would a student have to earn to be in the top 5% of all SAT Verbal scorers? 6. Approximately what percent of students scored between 395 and 615? 7. Approximately what percent of student scored within two standard deviations of 505?