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Lecture5-note - Idealized/Smoothed density curves A...

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1 Key concepts from week 2 Idealized/Smoothed density curves A smoothed density curve is a mathematical model of a distribution. It plots percent density distribution. The total area under the curve is 100%. The area under the curve for a range of values is the percentage of all counts for that range. Density curves come in any imaginable shape. Some are well-known mathematically and others aren’t. Normal distributions e = 2.71828… The base of the natural logarithm π = pi = 3.14159… Normal—or Gaussian—distributions are a family of symmetrical, bell- shaped density curves defined by a mean µ ( mu ) and a standard deviation σ ( sigma ): N ( µ , σ ). 2 2 1 2 1 ) ( = σ µ π x e x f x x 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 A family of density curves 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Here the means are different ( µ = 10, 15, and 20) while the standard deviations are the same ( σ = 3). Here the means are the same ( µ = 15) while the standard deviations are different ( σ = 2, 4, and 6). mean μ = 64.5 standard deviation σ = 2.5 N ( μ , σ ) = N (64.5, 2.5) All Normal curves N ( µ , σ ) share the same properties Reminder : μ (mu) is the mean of the idealized curve, while is the mean of a sample. σ (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample. About 68% of all observations or counts are within 1 standard deviation ( σ ) of the mean ( µ ). About 95% of all observations are within 2 σ of the mean µ . Almost all (99.7%) observations are within 3 σ of the mean. Inflection point x
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