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# l5 - Box Plots: A graphical way to show the...

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Unformatted text preview: Box Plots: A graphical way to show the shape of a distribution Example: weights of 57 children Example: Weights of 57 children Boxplot of Childrens’ weights 80 Possible outlier Upper adjacent value ! Weight (lbs) 60 Upper Quartile 40 Median Lower Quartile 20 Lower adjacent value 0 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit The Q Q plot: Comparison of the shapes of the distribution of two samples or a sample and the theoretical distribution Comparing quartiles of two populations Result: if the Q ­Q plot is linear, then what does it imply? e Page QQ-plot: chest size data Page (a) Histogram tents (b) QQ plot ! ! !! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! !! !! ! ! ! ! !! ! ! ! ! !! 0.15 Sample Quantiles 35 40 Chest (in) 45 ￿￿ Density 0.10 ￿ 0.05 0.00 50 of 61 35 40 45 Back −4 −2 0 2 4 Theoretical Quantiles creen lose uit Review Keywords: Box-plots, QQ-plots. • • • • First Prev Next Last •Go Back •Full Screen •Close •Quit Bivariate Data • Suppose two variables (X,Y) are measured on a sample of n units to give data (x1,y1), (x2,y2),…….(xn,yn) • The variables X and Y might be associated • Example: yi = response variate for unit i while xi is the corresponding explanatory variable • The problem: Is there an associative relationship between the two variates? If there is, is the relationship a causal one? Graphical Solutions: Scatter Plot: More than two variables Scatter Plot matrix me Page Example: (from the notes) Scatterplots Scatterplot of Votes/Seats 80 tle Page Plot of Seats by Time ! ontents ! ! 70 ￿￿ Seats % ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! !! ! ! ! !! ! ! 60 ￿ 50 Seats % 40 30 o Back ! 40 ll Screen 45 50 Votes % 55 30 1900 40 e 52 of 61 50 60 70 80 1920 1940 Year 1960 Close Quit •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Scatter Plot matrix matrix Scatter-plot 40 45 50 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 55 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Year ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! !! ! ! !! ! ! ! 55 50 !! ! ! ! ! ! ! ! ! !! ! ! ! ! Votes % ! !! !! 40 45 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! ! ! ! ! ! ! 1900 1920 1940 1960 30 40 50 60 70 80 30 40 50 Seats % 60 70 80 1900 ! ! 1920 1940 1960 ! •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Algebraic Measures of Association: Categorical Variates: The ideas from STAT 230 (the idea of independence) Continuous Variates: The correlation coefficient. Example: Investigating the relationship between being a smoker and risk of heart attack Smoke/Risk Yes No Total High 42 12 54 Low 7 39 46 Total 49 51 100 Dependence & Relative Risk • Two events A and B are said to be independent if P(A|B) = P(A|not B) = P(A) • We can measure dependence by calculating the ratio P(A|B) P(A|not B) And see how far it is from 1 • What is true for probabilities also holds for proportions. So use Proportion of smokers who are high risk Proportion of non ­smokers who are high risk • The above measure is called Relative Risk Relative Risk = 3.64 > 1 How high is too high? Correlation • A commonly used measure of strength of a linear relationship between two continuous variables (xi, yi) i=1,2,..n • This is usually denoted by r and is defined as r = SXY/(SXXSYY)1/2 where SXX = Σ (xi ­x)2 SYY = Σ(yi ­y)2 SXY = Σ {(xi ­x)(yi ­y)} Properties of the correlation coefficient: •  ­1≤ r ≤ 1 • When the points lie exactly on a straight line with positive (negative) slope, r=1. (r = ­1) when there is a reasonably strong positive (negative) relationship, r will be appreciably positive (negative), when there is no relationship, r will be near zero • Is the converse true? Time Series Data • The investigation concerns a process • We have data points of the form (t, y(t)) t an be discrete (more likely), or continuous (weekly sales figures, # of visits to the doctor’s office every month, etc) • Scatter plot • Yt = Tt +St + εt Tt = Trend Term St= seasonal term εt = Error Term Examples: Home Page Example: Dow street crash Title Page Contents Dow Jones Index (1920−1941) 400 400 Dow Jones Index 1925 1930 1935 1940 Dow Jones Index (1929−1931) ￿￿ ￿ ￿￿ 300 ￿ Dow Jones Index 200 Page 60 of 61 100 Go Back 0 0 1929.5 100 200 300 Full Screen 1930.5 Years(1929−1931) 1931.5 Years(1920−1941) Close Quit •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit ...
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