1Key concepts from week 2Idealized/Smoothed density curvesA smoothed density curveis 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 distributionse = 2.71828… The base of the natural logarithmπ= 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). 22121)(⎟⎠⎞⎜⎝⎛−−=σµπxexfxx024681012141618202224262830A family of density curves024681012141618202224262830Here 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= 2.5N(μ, σ) = N(64.5, 2.5)All Normal curves N (µ, σ) share the same propertiesReminder: 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 σof the mean µ.Almost all (99.7%) observations are within 3 σof the mean.Inflection pointx
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