lecture_1_3 - Looking at data distributions Density curves...

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    Looking at data: distributions Density curves and Normal distributions IPS chapter 1.3 Copyright Brigitte Baldi 2005 ©
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Objectives (IPS 1.3) Density curves and Normal distributions Density curves Normal distributions The standard Normal distribution Standardizing: calculating “z-scores” Using Table A
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Density curves A density curve is a mathematical model of a distribution. The total area under the curve, by definition, is equal to 1, or 100%. The area under the curve for a range of values is the proportion of all observations for that range.
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Density curves come in any imaginable shape. Some are well known mathematically and others aren’t.
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Median and mean of a density curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and mean are the same for a symmetric density curve. The mean of a skewed curve is pulled in the direction of the long tail.
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Normal distributions 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 ). x x
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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 A family of density curves Here means are different ( μ = 10, 15, and 20) while standard deviations are the same ( σ = 3) Here means are the same ( μ = 15) while standard deviations are different ( σ = 2, 4, and 6).
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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 x ¯ is the mean of a sample.
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