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# lec04

Course Number: STAT 5021, Fall 2008

College/University: Minnesota

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Lecture 4 Stat 5021 September 10, 2008 Cumulative Proportions The cumulative proportion at x is the area under the curve to the left of x. -4 -2 0 2 4 Figure: The cumulative proportion at x = 1 for a standard normal is about 0.84. Cumulative Distribution Functions The cumulative distribution function (CDF) maps x to the cumulative proportion at x. 1.0 0.6 0.8 q q 0.0 -4 0.2 0.4 -2 0 2 4...

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4 Stat Lecture 5021 September 10, 2008 Cumulative Proportions The cumulative proportion at x is the area under the curve to the left of x. -4 -2 0 2 4 Figure: The cumulative proportion at x = 1 for a standard normal is about 0.84. Cumulative Distribution Functions The cumulative distribution function (CDF) maps x to the cumulative proportion at x. 1.0 0.6 0.8 q q 0.0 -4 0.2 0.4 -2 0 2 4 Figure: CDF of a standard normal Quantile Functions The inverse of the CDF is the quantile function. For a standard normal, the quantile function maps 0 to -, 0.5 to 0, 0.84 to around 1, and 1 to . Characterizing Distributions A distribution may be characterized through its 1. density, 2. cumulative distribution function, or 3. quantile function. Q-Q Plots Since a distribution can be described through its quantiles, one way to compare distributions is to compare their quantiles. A Q-Q plot plots the quantiles of two distributions against one another. If the distributions have similar shapes, the plot will resemble a straight line. We will usually be interested in comparing a sample distribution to a normal distribution. Normal Q-Q Plots Here is a Q-Q plot comparing 50 observations from a N(5, 2) to a standard normal. The standard normal quantiles are theoretical values. Normal Q-Q Plot q q q q q 8 q qqqq q q q q q q q q q q q q qq qq qq q q q q q q q q q q qq q qqq Sample Quantiles 4 6 q q q q q 2 q -2 -1 0 Theoretical Quantiles 1 2 Normal Q-Q Plots Here is a Q-Q plot comparing 50 observations from a right-skew distribution to a standard normal. The standard normal quantiles are theoretical values. Normal Q-Q Plot q 3 4 Sample Quantiles q qq q q q q q q q q q q q q qq q qq q qq qq q q q q q q q qq q qqq qq q q 1 2 0 q q q q q qq -2 -1 0 Theoretical Quantiles 1 2 Normal Q-Q Plots Since normal distributions are indexed by their mean and standard deviation , they are linear in one another. X - X = + Z Z= Quantiles of X = + Quantiles of Z So, we can just compare sample quantiles to a standard normal. Associations between Variables Two variables measured on the same individuals are associated if some values of one variable tend to occur more often with some values of the other variable. Examples: heights and weights blood alcohol content and beer consumed circumferences and heights of haystacks Some Terminology A response variable is an outcome of interest. An explanatory variable is thought to affect the response variable. Examples: Want to predict weights (response) based on heights (explanatory). Want to predict blood alcohol content (response) based on beer consumed (explanatory). Want to estimate beer consumed (response) based on blood alcohol content (explanatory). Scatterplots Let Y be a response and variable X an explanatory variable. q q qq q q q q 5 q q q q q q q q q q q q q q q q q q q q Y 0 q q q q q q q q q q qq q q q q -5 q q q q q q -2 -1 0 X 1 2 Scatterplots are most useful when at least one variable is fairly continous. Talking about Association We usually talk about the 1. form, 2. direction, and 3. strength of an association. Talking about Association: Example Strong positive linear association q q 5 q q q q q q q qq q qq q q q q q q q qq q q 0 q q qq qq q q q Y q q q q qq q q -5 qq q q q -10 q q -3 -2 -1 X 0 1 Talking about Association: Example Strong negative linear association q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q 5 q Y q qq qq q q q 0 q -5 q q q q q q q q -10 q -2 -1 0 X 1 2 Talking about Association: Example Strong sinusoidal association q q q q q q q q qq q qq q q q qq 5 q qq q Y q 0 q q qq q qq qq q q q q q q qq q q q q q q -5 q q q q q -4 -2 0 X 2 4 Talking about Association: Example Strong parabolic association q 20 q q q q q q 15 q Y 10 q q q q q q q q qq qq q q q q q q q q q qq q q q q q q q q q qqq q 0 5 q q q q q q -1 0 X 1 2 Talking about Association: Example Weak positive linear association q q q q 2 q q q q q q q q q q q q q q q q q q q q q q q q 1 q q q q q q q 0 Y q q q q -1 q q q q q q q -2 q q q q -2 -1 0 X 1 2 Outliers Note that an outlying observation in the joint distribution need not be outlying in the marginal distributions. 10 q q q q q q q q q q qq q q q q q y 5 q q q q q q qq q q q q q q q q 0 q q q q q q q qq q q q q -5 q qq q q -1 0 x 1 2...

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Minnesota - STAT - 5021
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