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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 ( , ). 2 2 1 2 1 ) ( = π x e x f x x 0 2 4 6 8 1 01 21 41 61 82 02 22 42 62 83 0 A family of density curves 0 Here the means are different ( = 10, 15, and 20) while the standard deviations are the same ( = 3). Here the means are the same ( = 15) while the standard deviations are different ( = 2, 4, and 6). mean μ = 64.5 standard deviation = 2.5 N ( μ , ) = N (64.5, 2.5) All Normal curves N ( , ) 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 ( ) of the mean ( ). ± About 95% of all observations are within 2 of the mean . ± Almost all (99.7%) observations are within 3 of the mean. Inflection point x
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2 Because all Normal distributions share the same properties, we can standardize our data to transform any Normal curve N ( µ , σ ) into the standard Normal curve N (0,1). What about other distance? The standard Normal distribution For each x we calculate a new value, z (called a z -score). N (0,1) => z x N (64.5, 2.5) Standardized (in the unit of σ ) z = ( x ) A z- score measures the number of standard deviations that a data value x is from the mean . Standardizing: calculating z -scores When x is larger than the mean, z is positive. When x is smaller than the mean, z is negative.
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This note was uploaded on 04/26/2008 for the course BIOL 7 taught by Professor Luo during the Fall '07 term at UC Irvine.

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

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